Acoustics & Music Theory
why it sounds like that
Theme

The instrument is not the model

Every argument here starts from an idealised object: a perfectly flexible string, a spectrum of exact harmonics, an ear with unlimited resolution. Real strings are stiff, real spectra are not harmonic, and the ear has a bandwidth. The differences are measurable and they matter.
How far each system sits from equal temperament. Deviation from equal temperament, in cents, for each degree of the chromatic scale. Zero is equal temperament by definition; the just major third is about fourteen cents flat of it and the Pythagorean major third about eight cents sharp. Pitch and tuning

Where to hide the comma, which is the only real question

Four tuning systems, four different places to put an error that cannot be removed. Each one is coherent, each was standard somewhere, and the choice between them decided what music could be written.

 The chain of fifths in quarter-comma meantone. The fifths laid end to end as the chain they are. The bar under each shows how far that fifth departs from a pure three-to-two, and one of them — G♯ to D♯, the 12th link, where the chain is forced to close — is the wolf, at 35.7 cents. Pitch and tuning

The wolf at the end of the chain

Put all the tuning error into one interval and eleven others become perfect. The twelfth becomes a howl, and for two centuries keyboards were built that way on purpose.

 Roughness across an octave. Sensory dissonance between two complex tones as the upper one is swept through an octave, computed by summing the roughness between every pair of their partials. Nothing here is placed by hand. The wells this spectrum produces sit on 4/3, on 3/2, on 5/3, found by scanning the curve rather than by marking them. And the tempered minor third sits 198 cents below the nearest well, and the tempered major third sits 98 cents below the nearest well, which is a real feature of this model and not a defect of the drawing. Intervals and chords

Roughness can be computed, and the answer looks like a scale

Feed two spectra into a model of how nearby frequencies interfere, sweep one past the other, and the curve that comes out has dips exactly where the consonant intervals are. Nobody put them there.

 A note with its first partial removed. The spectrum of a 220 Hz tone with the lowest partial deleted, and the wave that remains. The wave still repeats 220 times a second, because the repeat rate of a sum of harmonics is fixed by the spacing between them and the spacing has not changed. The pitch heard is the one that is no longer in the sound. Intervals and chords

The note that is not there

A telephone reproduces nothing below about three hundred hertz, and a bass line down at eighty comes through it perfectly. The pitch that is heard is not a frequency present in the sound, and one nineteenth-century experiment settles what it is instead.

 Four spectra of the same note. The amplitude of each partial for 4 timbres at the same pitch — pure, string, clarinet, bell. These are the exact lists the sound buttons here synthesise from, so the picture and the sound are the same data. Timbre and acoustics

The ear hears the list, not the shape

Two sounds with the same partials and different phases have completely different waveforms and sound identical. What the ear extracts is a list of frequencies and strengths, and everything else is discarded.

 A small room's lowest modes. The first few axial standing waves of a room, drawn in plan, with the frequency of every mode below 160 hertz listed underneath. The low modes are far apart in frequency, so some bass notes are loud in one corner and absent in another. The sound buttons play these two octaves above their real pitch, because a room's lowest modes are below what most speakers reproduce. Timbre and acoustics

The room is part of the instrument

A room has frequencies it supports and frequencies it will not. In a small one those frequencies are far apart, so some bass notes are loud in one corner and absent in another — and no equipment fixes it.

 The vowel in "hod", sung at 110 Hz. The partials of a 110 Hz note, each drawn at the amplitude the vocal tract's resonances give it. The peaks of the curve are the formants — 730 Hz and 1090 Hz — and they stay where they are when the pitch changes, because they are a property of the shape of the mouth and not of the note being sung. Timbre and acoustics

A vowel is two resonances

The vowel in "heed" is the same vowel sung high or low, and nothing about it is a property of the note. It is two peaks in the response of the mouth, sitting at fixed frequencies while the partials of the voice slide underneath them.

 Where a tuned piano actually sits. The departure from equal temperament of every key of a small upright, computed from the stiffness of its strings and the fact that a tuner sets octaves without beats rather than at a ratio of two to one. The treble ends up 52 cents sharp and the bass 18 cents flat, and neither is an error. Timbre and acoustics

The piano is tuned wrong on purpose

Every well-tuned piano has a sharp treble and a flat bass, by up to a third of a semitone at the extremes. It is not an error, it is not a compromise about keys, and it follows from one property of a steel wire that can be computed from its diameter and its length.

 Beat rates for laying equal temperament. Each link in the chain of fifths, inside one bearing octave, with the rate at which its coincident partials beat — the third partial of the lower note against the second of the upper for a fifth, the fourth against the third for a fourth. Rates run from 0.59 to 1.12 beats per second, a spread of 0.52. This temperament narrows every fifth equally, and the rates still differ, because a beat rate is a difference in hertz and scales with the register. Pitch and tuning

A tuner counts beats, and that is the whole method

Laying a temperament sounds like the most subjective job in music, and it is arithmetic. Every interval in the bearing octave has a target rate in beats per second, the rate follows from the ratio and the register, and a tuner who hits the numbers has produced equal temperament without ever thinking about a cent.

 Where A has been. Documented pitch standards and surviving instruments, plotted as cents from A440. The extremes are 392 Hz and 465 Hz, which is 296 cents apart — 3.0 semitones, close enough to a minor third that a piece written at one and played at the other is in a different key. Nothing here is a preference; each is a decision somebody recorded. Pitch and tuning

The pitch nobody agreed on, for four hundred years

A440 is a committee decision from 1955. Before it, the same written note was played anywhere between about 392 and 465 hertz depending on the town, the building and the decade — a spread of very nearly a minor third, and every piece of theory here is untouched by it.

 Roughness across an octave. Sensory dissonance between two complex tones as the upper one is swept through 19.02 semitones, computed by summing the roughness between every pair of their partials. Nothing here is placed by hand. The wells this spectrum produces sit on 7/5, on 11/7, on 5/3, on 9/5, on 11/5, on 7/3, on 13/5, found by scanning the curve rather than by marking them. The wells are where this spectrum's partials coincide, and for a spectrum with no even partials they are not where an octave-based scale would put them. Scales and modes

A scale without an octave, and the spectrum that asks for it

Every scale so far repeats at the 2:1. That looks like a law and it is a consequence — the octave fuses because every partial of the upper note lands on one of the lower. Take a spectrum with no even partials and the 2:1 stops doing that, while the 3:1 starts.

 Which partials two notes have in common. The first 12 partials of a note and of the note an interval above it, on a logarithmic frequency axis, with every partial of the upper tone that lands on one of the lower marked. octave: 12 of 12, fifth: 6 of 12, fourth: 4 of 12, major third: 3 of 12, minor third: 2 of 12, minor second: 0 of 12. Sharing partials is what fusion is: two voices whose spectra mostly coincide stop being heard as two voices, which is a measurable property of the interval rather than a matter of taste. Harmony and voice leading

Two voices that stop being two

The ban on parallel fifths is the most famous rule in Western music and it is usually taught as taste. It is not taste. At an octave the upper voice contributes no frequency the lower one did not already have, at a fifth it contributes half of them, and the number can be counted — which turns a prohibition into a measurement.

 Reverberation time, by two formulas. Sixty-decibel decay time against average absorption, divided by the room's volume-to-surface ratio so that every room sits on the same pair of curves. Sabine's equation, which is the one every textbook gives, and Eyring's correction to it. They agree in the reflective rooms Sabine measured and separate above ᾱ ≈ 0.18: at 0.6 Sabine reads 53% high, and at ᾱ = 1 — a room whose walls absorb everything, which is the outdoors — it still returns a positive time for a space with no reverberation at all. Marked: a concert hall 1.59 s, a stone church 3.37 s, a studio live room 0.45 s, a carpeted bedroom 0.13 s, an anechoic chamber 0.03 s. Timbre and acoustics

How long a room rings, and where the formula stops

Sabine's reverberation time is one line of arithmetic — volume over absorption — and it built the modern concert hall. It also predicts that a room whose walls absorb everything still rings, which is a room with no reverberation at all, and the error is largest in exactly the rooms most music is now made in.

 Partial 3, mistuned by 3%. A 10-partial tone on 220 Hz with one partial treated differently from the rest. Mistuning it moves it off the harmonic grid by 3.0 per cent, which is 19.8 Hz — slow enough to be heard as a beat rather than as a separate pitch, and enough for the partial to be heard out of the note as a whistle of its own. An onset difference does the same to a partial that is exactly in tune. Timbre and acoustics

What makes two partials one note

A note is a stack of ten or twenty simultaneous tones and is heard as one thing. The obvious explanation is that they are whole-number multiples of a fundamental — and the obvious explanation is not sufficient. Mistune one partial by three per cent and it leaves the note; give a perfectly harmonic partial a thirty-millisecond head start and it leaves too. Shared behaviour beats arithmetic.

 Three octaves, and none of them is 2:1. How far above an exact doubling the upper note of an octave is set, against frequency. The listener's octave is measured with pure tones, which have no partials to beat against each other, so nothing about a stiff string can account for it. The piano's stretch is a different quantity with a different cause, and the two are drawn together only so that the difference is visible. Perception and the listener

The octave that is not two to one

The octave is the one interval nobody argues about: two to one, exact, in every tradition that has one. Asked to set an octave by ear, listeners set it wide — and they do it with pure tones, which have no partials to beat against each other. Whatever is stretching the octave, it is not the stiffness of a piano string.

 The end correction, for a bore of radius 7.5 mm. How flat a tube sounds against what its physical length alone would predict, because the wave carries on past the opening before it turns round. The correction is 4.6 mm at every note — Levine and Schwinger's 0.6133 times the radius for an unflanged end — and the error it causes is 13 cents on a 60 cm sounding length and 52 cents on 15 cm. It is the same millimetres in both cases. Instruments and their design

The tube ends after it ends

A wave does not turn round at the opening. It carries on into the room for about six-tenths of the bore radius and reflects there, so every tube is acoustically longer than it is. The correction is a fixed number of millimetres against a wavelength that halves every octave — a rounding error at the bottom of an instrument's range and most of a semitone at the top.

 The same note, hit at a middling dynamic. The spectrum of a struck string with the hammer's own contact time applied as a low-pass. Contact lasts 1.60 ms at this force, against 2.26 ms at the softest and 0.95 ms at the loudest drawn — felt is a nonlinear spring, so a harder blow is a shorter contact and a brighter note. The spectral centroid moves from partial 1.5 to partial 2.2, which is a change of timbre and not of loudness. Instruments and their design

A hammer is not an impulse

Contact lasts a couple of milliseconds, which low-passes the note — any partial whose half-period is shorter than the contact is barely excited. Piano felt is a spring that stiffens as it compresses, so a harder blow makes the contact shorter, the corner higher and the note brighter. A loud note is not a scaled-up quiet one, and no linear model gives that.

 A bowed string on 196 Hz, through a violin body. The source is a sawtooth at one over n; the filter is the body's measured response, with A0 at 275 Hz, B1− at 460 Hz, B1+ at 540 Hz, bridge hill at 2500 Hz. What is radiated is their product, drawn as the bars. The resonances stay where they are when the note changes, exactly as a vowel's formants do — which is why an instrument has a voice rather than a tone, and why the same argument that identifies a vowel identifies a violin. The body frequencies are measured means over full-size instruments rather than computed from a plate. Timbre and acoustics

The body is the filter

A violin string radiates almost nothing. What reaches a room is the string's sawtooth multiplied by the body's response, and that response is a comb of measured resonances that stays put while the note moves. It is the same arithmetic that identifies a vowel, on wood instead of a mouth — which is why an instrument has a voice rather than a tone.

 A string mode swept through a body resonance at 460 Hz. What the string plays against what comes out. Away from the resonance the two are the same and the line is the diagonal. Near it the mode splits into a pair, and the note warbles at the difference between them — 12.9 Hz at the centre, which is slow enough to be counted and far too fast to be a tremolo. The splitting is a coupled oscillator and has nothing to do with the wolf fifth of a tuning system, which is a twenty-three cent arithmetic residue and shares only the word. Timbre and acoustics

The other wolf

A cellist's wolf note is a string mode landing on a body resonance, at which point the two stop being separable and start exchanging energy — the mode splits in two and the note warbles at the difference. It is a coupled oscillator. The tuning system's wolf is twelve fifths failing to close by 23.5 cents. They share a word and nothing else.

 Where each frequency goes, from a source 18 cm across. Polar response of a circular radiator of radius 9 cm at 200 Hz (ka = 0.3), 800 Hz (ka = 1.3), 2000 Hz (ka = 3.3), 5000 Hz (ka = 8.2). Zero degrees is straight ahead. The low frequency is a circle — it goes everywhere — and the high one is a narrow lobe with nulls either side of it, so a listener off to the side hears the same note with its top missing. Timbre and acoustics

An instrument points

A source radiates evenly while it is small compared with the wavelength and beams once it is not, and the crossover is one number. So the same instrument is omnidirectional in its bottom octave and a searchlight in its top one — which means its spectrum depends on where the listener is standing, and a microphone position is a choice about what the instrument sounds like.

 6 chords in a gothic cathedral. Each chord's reverberant decay in a room with a 8 second reverberation time, at 1 chord a second. Decay is linear in decibels, so each line is straight with a slope of -7.5 dB a second. When a chord arrives, 2 earlier ones are still above 20 dB down. Timbre and acoustics

The room chooses the harmonic rhythm

A chord in a cathedral is still sounding, seven decibels down, when the next one arrives — and the one after that, and the one after that. Reverberation is linear in decibels, so the number of chords audible at once is one number divided by another, and it puts a hard ceiling on how fast a composer writing for that building can change harmony. The ceiling is computable, and the music written for those rooms sits under it.

 The most a 6 cm cone can make of a low note. The maximum sound pressure level at one metre from a circular radiator of effective radius 3.2 cm, moving 1.5 mm at its limit, in a system resonating at 250 Hz. Below resonance the cone is already at that limit and the pressure a piston makes goes as the square of frequency, so the curve falls at twelve decibels an octave: 66 dB at 40 Hz, 78 dB at 80 Hz, 94 dB at 200 Hz. The 40 Hz figure is 28 decibels below the 200 Hz one, and that gap is arithmetic about a radius and a displacement rather than a property of any particular loudspeaker. The dots are the harmonics of a 41.2 Hz note with a one-over-n source spectrum; the loudest of them is the 6th. Instruments and their design

The bass a small loudspeaker does not make

A three-inch cone at its excursion limit produces sixty-six decibels at forty hertz, which the ear converts to seventeen phons — barely above nothing. The note is heard anyway, because its harmonics are radiated and its fundamental is supplied by the listener. Computing what arrives turns the residue from a curiosity into a design decision, and finds that the fundamental of a low note is not the loudest part of it on any system a listener is likely to own.

 Where the two accounts part company. The spread of the error between the hands, in milliseconds, against bar number, averaged over 120 seeded runs of each model at 100 bars a minute. With one timekeeper it is flat at about 9 ms after 24 bars; with two it reaches 106 ms and is still climbing, because a random walk has nothing to return to. Measured players hold 3 against 2 inside about 25 ms indefinitely. Rhythm and metre

One player is not two clocks

Two accounts of a pianist playing three against two, simulated from the same noise. With a timekeeper in each hand the hands drift apart by a hundred milliseconds inside two dozen bars. With one timekeeper they never drift at all. The measurement everybody cites as evidence for a timekeeper turns out to be the same number under both accounts.

 Two players, and the correction that keeps them together. The spread of the asynchrony between two players, in milliseconds, against beat number, for 3 correction gains, averaged over 120 seeded runs each. It reaches 120 ms after 64 beats at a gain of 0, 27 ms after 64 beats at a gain of 0.1, 20 ms after 64 beats at a gain of 0.3. With no correction at all the asynchrony is a random walk and grows without bound; with any correction it settles at a fixed spread within a few beats and stays there. Two people cannot share a timekeeper, so the fact that ensembles do not come apart is itself the evidence that they are correcting. Instruments and their design

Two players and no clock

Two people cannot share a timekeeper, and two independent ones drift a hundred and twenty milliseconds apart inside a minute. Ensembles do not, so something is correcting — and the measurement everybody reaches for recovers the pair's total responsiveness exactly and cannot tell which of the two is doing it. Four tenths from one player and two tenths each give the identical number.

 The same hall, empty and full. A hall of 18700 cubic metres with 900 square metres of audience, designed to 1.9 seconds occupied, with three kinds of seat under the audience. It is 2.76 s empty and 1.90 s full with hard wooden seats, a change of 31 per cent; 2.29 s empty and 1.90 s full with lightly padded, a change of 17 per cent; 1.96 s empty and 1.90 s full with heavily upholstered, a change of 3 per cent. The audience is 45 per cent of the total absorption when the hall is full, which is the largest single term in the equation — and how much the hall changes is decided entirely by what the seats were doing before anybody sat on them. Instruments and their design

The model has nobody in it

Sabine's room is an empty box. The audience is 45 per cent of a full hall's absorption, a hall with hard seats goes from 2.76 seconds empty to 1.90 full, and because an audience absorbs far more treble than bass it does not shorten the decay so much as tilt it. And the players are inside the loop the model has no term for at all.

 The roughness curve for ideal bar. The partials are at 1 : 2.756 : 5.404 : 8.933 : 13.340 times the fundamental, which is not a harmonic series, so nothing about where the wells fall can be read off the small whole numbers. Sensory dissonance between two complex tones as the upper one is swept through 13.00 semitones, computed by summing the roughness between every pair of their partials. Nothing here is placed by hand. The wells this spectrum produces sit 7 cents below 3/2, 14 cents below 5/3, 13 cents above 7/4, found by scanning the curve rather than by marking them. The wells are where this spectrum's partials coincide, and for a spectrum with no even partials they are not where an octave-based scale would put them. Timbre and acoustics

The spectrum that was supposed to explain the gamelan

Run the model that designed the Bohlen–Pierce scale on a bar's partials and it asks for a compressed pseudo-octave at 1166 cents and wells at 694 and 870. A measured slendro's degrees are at 231, 474, 717 and 955, and only one of the four is near anything the model wants. Sweep the partials and a spectrum wanting a 240-cent step can be found — sitting sixty per cent of the way up its own roughness curve, which is what a scale-level test says about the whole comparison.

 Two ways to detune a pair, and they disagree by an octave. A pair of instruments tuned deliberately apart, drawn across 4 octaves. Holding the detuning at 10 cents gives a beat rate that rises from 0.64 to 10.20 beats a second — a factor of 16, one doubling per octave, because a fixed ratio is a growing number of hertz. Holding it at 3 beats a second instead gives a flat rate and an interval that shrinks from 46.6 cents at the bottom to 2.9 at the top. A tuner has to choose, the choice is audible across the range, and a table of cents can only write down the first of the two. Pitch and tuning

A tuning is not a table of cents

Two instruments of a pair are tuned deliberately apart, and the tuner has to choose between a fixed number of cents and a fixed number of beats — which differ by a factor of sixteen across four octaves and cannot both be held. A list of pitches records the first, cannot record the second, and cannot record at all that there are two instruments.

 The flow through the larynx, over two periods of a 110 Hz note. Volume flow against time, in Rosenberg's two-half-cosine model of the glottal pulse — a slow opening, a faster closing, and a closed phase during which no air passes at all. M1 — chest is open for 50 per cent of each period and opens 2.4 times as slowly as it closes. Nothing here is a displacement: the folds are a valve on a steady stream of air, and the flat stretches are the moments they are shut. At 110 Hz each period lasts 9.1 milliseconds, of which 4.5 is silence. Instruments and their design

The other instrument with a reed

The folds do not vibrate the way a string does. They open and shut across a steady stream of air, once per period, and what leaves the larynx is a train of flow pulses with a closed phase in it. Everything said about the voice's tone is a statement about the shape of that pulse — and the shape has two numbers in it.

 Two mechanisms, the notes both of them make, and the seam. The frequency range of each laryngeal mechanism for an adult male voice, on a logarithmic axis, with the band both can produce shaded. M1 — chest runs 82–349 Hz and M2 — falsetto runs 220–698 Hz, so 799 cents of the range — 8.0 semitones — can be sung either way. The two dots inside that band are the measured signature that this is a bifurcation rather than a threshold: the change upward happens at 330 Hz and the change downward at 294 Hz, 200 cents lower. A threshold is crossed at the same place in both directions and this is not. Instruments and their design

Two mechanisms, and the seam between them

Every singer has a place in the range where the voice changes character, and eight semitones of it can be produced either way. The measurement that settles what kind of a place it is takes ten seconds: the change upward happens two hundred cents higher than the change downward, and a threshold cannot do that.

 Long-term average spectra: an orchestra, playing forte against a trained operatic soloist. Each source's mean spectrum over a long passage, in decibels below its own strongest region, on a logarithmic frequency axis. An orchestra, playing forte peaks at 250 Hz and is 30 dB down by 3,150 Hz; a trained operatic soloist peaks at 250 Hz and is 11 dB down by 3,150 Hz. The shapes are the same until about 1 kHz and separate above it: at 3153 Hz the difference is 19.0 decibels, which is the largest anywhere in the range. Nothing here is about level. Both curves are drawn against their own peaks, so what is being compared is shape. Timbre and acoustics

One voice over ninety players

A soloist heard over a full orchestra is not louder than it and could not be. What the trained voice does instead is put a peak of energy at three kilohertz, which is where the orchestra's spectrum has already fallen away and where the ear's own threshold happens to be lowest. Nineteen decibels of advantage, in a place nobody is competing for.

 Every voicing of a second-inversion major triad, least rough first. All 27 arrangements of the same three pitch classes within 3 octaves from 131 Hz, scored for roughness. The best is spaced 19 then 9 semitones — wide below, close above — and the worst is the chord in close position at the bottom of the range, 7.6 times rougher with exactly the same notes in it. Intervals and chords

The inversion that cannot end a phrase

Three positions of one triad, three measurements, and the practice contradicted at every turn. The second inversion is the smoothest of the three by roughness, the best explained of the three by a virtual-pitch model, and the one that three centuries of practice will not let a phrase end on. What forbids it is a rule about a two-voice skeleton.

 A reed that shuts at 5000 pascals, and the air it lets through. Volume flow through the reed channel against the pressure across it, in the quasi-static model: Bernoulli flow through an opening that closes linearly with pressure. The flow peaks at 1667 pascals — exactly a third of the closing pressure, for any reed, because that is where the two effects balance — and it is 0.18 litres a second there. Everything to the right of that peak is the argument: the flow falls as the player blows harder, from 0.18 to 0.05 litres a second by 4500 pascals, and a resistance that behaves that way supplies energy instead of taking it. There is no reed inertia in this model, so it cannot squeak. Instruments and their design

The reed is a valve, not a vibrator

Three essays here have said that a clarinet's reed does not choose the note, and none of them said what it does instead. It chops a steady stream of air, and past a third of the pressure that closes it the flow falls as the player blows harder — a resistance with the wrong sign, which is the only thing in the instrument capable of putting energy into an oscillation that is otherwise losing it.

 Where the series stops being a chord and becomes a scale. The interval between each partial and the next, in semitones, against partial number. It is an octave, then a fifth, then a fourth, a major third, a minor third — and it crosses 2 semitones between partials 9 and 10. Each shaded band is one octave of pitch and holds twice as many partials as the band below it: 2 between 1 and 2, 3 between 2 and 4, 5 between 4 and 8, 9 between 8 and 16. Nothing about the series changes with height; what changes is the density, and a set of notes becomes usable as a melody when its neighbours are a step apart. Instruments and their design

The register where the series becomes a scale

A melody needs steps, and a natural trumpet has only the harmonic series. The gap between one partial and the next falls as the series rises — an octave, a fifth, a fourth, a third — and does not reach a whole tone until the eighth. That single arithmetic fact decides what two centuries of brass writing could be: fanfares below, and melody only in a band one octave wide with nine notes in it, three of them badly out of tune and every one of them lipped into place by a player who had no valve to press.

 How far each system sits from equal temperament. Deviation from equal temperament, in cents, for each degree of the chromatic scale. Zero is equal temperament by definition; the just major third is about fourteen cents flat of it and the Pythagorean major third about eight cents sharp. Pitch and tuning

The note that is sharp because of where it goes

Eleven essays here have tuned notes against other notes sounding at the same time. A melodic line makes the opposite demand of the same note. In C major the leading note wants to be 1088 cents if the argument is vertical — a pure third above the dominant — and 1110 if it is horizontal, because a leading note is a note that is about to arrive somewhere and performers narrow the step. The two answers differ by 21.5 cents, which is a syntonic comma exactly, and no keyboard has ever been able to hold both.

 The vowel in "hod", sung at 110 Hz. The partials of a 110 Hz note, each drawn at the amplitude the vocal tract's resonances give it. The peaks of the curve are the formants — 730 Hz and 1090 Hz — and they stay where they are when the pitch changes, because they are a property of the shape of the mouth and not of the note being sung. Timbre and acoustics

The sound a listener knows best

A voice is recognisable across every vowel it says, across two octaves of pitch, down a bad telephone line and in a whisper where there is no pitch at all. Nothing that survives all of that can be a frequency. What survives is a ratio: the resonances of a vocal tract are set by its length, so a shorter tract multiplies every formant by the same factor, and identity is a scale on the spectral envelope rather than a position within it. Between an adult man and a child the whole pattern moves by a fifth, and the vowel does not change at all.

 The fluctuation stays; the rate goes. A unison of n voices with a spread of 15 cents, averaged over 5 draws. The depth of the amplitude fluctuation does not fall as voices are added — a choir is no steadier than a duet — but the fraction of that fluctuation in any single modulation component falls from 77 per cent at two voices to 24 at 32. Two voices make one beat and it can be counted; 16 make 120 and none of them is a rate. That is why a choir cannot be tuned by nulling anything. Timbre and acoustics

What a choir does that a soloist cannot

Two singers on one note produce one beat and it can be counted. Sixteen produce a hundred and twenty at once, and the amplitude still fluctuates by as much as it did — a choir is no steadier than a duet. What has gone is not the fluctuation but its rate: the modulation energy that sat in a single line at two voices is spread across a band at sixteen, with no line in it. That is the choral sound, and it is also why the just-intonation drift measured for tuned singing describes only ensembles that hold their pitch still.

 Two modes, one set, and every measure of a set that cannot tell them apart. Raga Bhupali and Raga Deshkar drawn as the moves each allows: an arrow from one degree to another means the tradition's ascent or descent goes that way. Raga Deshkar's ascent omits Re, so the two graphs differ by an edge. Below, every standard measure of a scale, evaluated on both — and they are identical in every row, which the drawing checks before it is made. An ascent and a descent that between them use all 5 degrees can be chosen in 63 ways, 62 of them asymmetric. That is how many modes collapse onto one pitch set under the simplest order model there is, and a census over subsets counts the set once. Scales and modes

A degree is where it goes next

The first essay on scales beyond twelve said a mode is not a set and listed four things a set cannot record. This one makes the sharpest of them arithmetic. Specify a mode as an ascent and a descent — which is how a living tradition specifies one — and it becomes a directed graph on its degrees; sixty-three such graphs collapse onto a single pentatonic set, sixty-two of them with an ascent that is not the descent reversed, and every standard measure of a scale returns the same value for all of them.

 A constant bow force cannot start a note. Schelleng's minimum and maximum bow force, evaluated at the speed the bow has after one period rather than at the speed the note will be sustained at. Both bounds are proportional to bow speed and the speed after one period is the acceleration divided by the frequency, so both are proportional to acceleration and the region is a wedge through the origin. Its width as a ratio is 9.0 to 1 at every acceleration — the attack is not a narrower window than the sustain, it is the same window somewhere else. A fixed force, drawn here at 0.02, is inside it at one acceleration, about 0.18, which is why the force has to rise with the speed and why beginning a note is a trajectory through this plane rather than a point in it. Instruments and their design

The note has to start somewhere

Schelleng's diagram says how hard a bow may press at a given distance from the bridge for a steady tone to be possible. It is a map of a note that is already sounding, and it contains no information whatever about how to begin one. Read the same inequality at the speed the bow has after a single period rather than at the speed it will settle to and the admissible region turns out to be a wedge through the origin — same width as a ratio, a hundred and ninety-six times lower in force. A constant bow pressure is inside it for one instant of the attack and outside it for the rest.

 Where to stop, and what stopping there gives. Each prefix of the harmonic series sounded as a chord of pure tones, with its roughness per pair and the pitch classes it contains. The smoothest prefix is the first 4, which is a bare fifth and an octave; roughness rises monotonically from there, so no roughness argument selects six. The prefixes that contain a major triad and nothing else are 5 and 6. One partial further and it is a dominant seventh; four further and there are five pitch classes. Six is the last stopping point that gives the answer the derivation wants, and nothing else in the arithmetic picks it out. Intervals and chords

The series is not a chord

The major triad is derived from the first six partials of the harmonic series in every textbook that derives it from anything, and the derivation works: the first six contain a major triad and nothing else. Take the first four instead and there is no third at all; take seven and there is a dominant seventh with a flat seventh thirty-one cents below any keyboard note; take sixteen and there are eight pitch classes. Six is the last stopping point that gives the answer, roughness prefers three, four or ten depending on the register, and no criterion in the arithmetic selects six over any of them.

 Where A has been. Documented pitch standards and surviving instruments, plotted as cents from A415. The extremes are 392 Hz and 465 Hz, which is 296 cents apart — 3.0 semitones, close enough to a minor third that a piece written at one and played at the other is in a different key. Nothing here is a preference; each is a decision somebody recorded. Pitch and tuning

The note that moved by a minor third

A has been anywhere between 392 and 465 hertz in the surviving record, which is 296 cents — a minor third short of six. The usual conclusion is that absolute pitch level is a convention and nothing musical depends on it. That is true of everything written on paper and false of everything that happens in a throat or a body: a singer's register break sits at a fixed frequency, so across the historical range it lands three semitones further down the written page. Transposing a piece is not a uniform operation, because the performer does not transpose.

 A dynamic mark is an instruction about the spectrum. Six dynamic markings, given a hammer velocity each in a stated sequence of factors of two, with what the string then does. The level rises 35.1 decibels from pp to ff, which is the part everybody means. The contact time falls from 2.26 to 0.95 milliseconds, so the first null of the hammer's own pulse moves from partial 2.5 to partial 6.0 and the spectral centroid rises by 56 per cent. The partials between those two nulls are not quieter at pp; they are not there. Timbre and acoustics

The mark that is not a level

There are six of them, they carry no units, and a performer has to turn one into a number before it means anything. What they instruct is not loudness. On a struck string a harder blow shortens the hammer's contact from 2.26 milliseconds to 0.95, which moves the first null of its own pulse from the third partial to the sixth: the partials between those are not quieter at pianissimo, they are gone. A fortissimo is a different sound, and the page has one word for both things it changes.

 C4, in every place it can be played. A guitar neck with the 4 places C4 can be stopped, drawn at the fret spacing a 64.8-centimetre scale actually has. The stave writes one note and the tablature writes one of these; each notation says exactly what the other leaves out. The speaking lengths run from 61.2 down to 27.2 centimetres, so a hand plucking 12 centimetres from the bridge meets between 20 and 44 per cent of the string. Instruments and their design

What a tablature keeps

Middle C can be stopped in four places on a guitar. The speaking lengths run from 61 to 27 centimetres, so a hand plucking twelve centimetres from the bridge meets between a fifth and nearly a half of the string, and the comb of missing partials is different at every one: the second partial is thirteen decibels stronger in the best position than in the worst. A stave writes one note for all four. A tablature writes four different things and cannot say which note any of them is.

 Where a twelfth comes from. The range of a walk with no walls, against how many notes it runs for, at three settings of the one parameter it has. The parameter is fitted to the post-skip reversal rate and to nothing else; the range is then read off. With no central tendency at all the walk passes two octaves by 60 notes and keeps going. At the setting that reproduces 70 per cent reversal — κ = 0.78 — the range is 11.9 semitones at thirty notes and 17.9 at a hundred and twenty. It grows logarithmically, so over the whole plausible length of a tune it sits between an octave and a fifteenth, and a twelfth is the middle of that. The three tunes carried here are marked and all three fall below the curve. Form and structure

The twelfth, and where it comes from

Melodies occupy about an octave and a fifth, and an earlier essay set out to explain that by the singer's register break and found that it does not: the chest mechanism alone spans two octaves and a semitone. The answer is in a parameter the essay on leaps fitted and then put down. A walk with no walls whose central tendency reproduces the post-skip reversal rate has a range that grows logarithmically — six semitones at eight notes, twelve at thirty, eighteen at a hundred and twenty — so across every length a tune plausibly has, the span is between an octave and a fifteenth.

 How near the breaking point each string already is. Frequency times length, as a fraction of what the material allows. The ceiling is half the square root of specific strength — sheep gut 240, music wire 276, nylon 114, brass 127 hertz metres — and it depends on nothing a maker can change: not the gauge, not the tension, not the workmanship. The guitar top E runs at 187 per cent of its own ceiling, which is why it is the string that breaks and why every complaint about rising pitch in the historical record is about that one string. Pitch and tuning

A standard is a specification

Choosing where to put A looks like a convention and is a mechanical decision. Tension goes as the square of frequency, so a piano built at 440 and tuned to 466 carries twelve per cent more load — a tonne and a half in this model's arithmetic. And there is a hard ceiling nobody can engineer round: frequency times length is capped by half the square root of a material's specific strength, which for gut is 240 hertz-metres. A violin E at A440 runs at 89 per cent of that. At A493 it is at a hundred, and every complaint in the historical record about rising pitch is about that one string.

 One key, nineteen notes, one right answer. A register hole disturbs a mode in proportion to the square of that mode's pressure at the hole, so the place that spoils the fundamental and leaves the third harmonic alone is the third harmonic's own pressure node — a third of the way down whatever length is sounding. The length changes with every fingering and the key does not move, so the two curves cross at one note. At A3 the key sits almost exactly on the node and the twelfth speaks cleanly; at A♭4 it is at 62 per cent of the tube and disturbs the third harmonic by 96 per cent of what an antinode would, which is the throat of the instrument and is exactly where players say the notes are worst. Instruments and their design

The hole that spoils a note

A tone hole shortens the tube. A register hole does the opposite job: it is small enough to shorten nothing and is placed where it will wreck the fundamental's resonance and leave the third harmonic's alone, so the note jumps a twelfth instead of retuning. The place that does both is a pressure node of the harmonic being kept — a third of the way along whatever length is sounding — and the length changes with every fingering while the key does not. One key is at the right place for exactly one note, and the note it is worst for is in the throat of the instrument, which is where players say the instrument is worst.

 What a spread costs a chord. a major triad of 3 notes lasting 600 ms each, with the onsets spread by up to 320 ms. The upper line is the share of each note's length during which every note is sounding; the lower is the chord's roughness weighted by that share, since roughness is a property of two partials sounding at the same time. At a spread of 30 ms — the asynchrony at which a mistimed partial stops belonging to its note — the chord is still 89 per cent simultaneous. It stops being simultaneous at all at 300 ms, which is where the last note arrives after the first has finished. Intervals and chords

The chord that is not played at once

Every chord until now starts its notes at the same instant, and no figure ever set the asynchrony to anything else. A spread chord is not a defective simultaneity: at forty milliseconds a triad of half-second notes is still eighty-seven per cent simultaneous, so it carries almost all its roughness, and it stops being a chord at all only when the last note arrives after the first has finished. What none of this can explain is the one thing every keyboard player knows — that a chord is rolled upward. The masking asymmetry that ought to explain it is 2.4 decibels at a close voicing, which is not enough.

 A perfectly tuned unison is the shortest note on the piano. How long a struck unison takes to fall forty decibels, against how far apart its two strings are tuned, for a note whose single string would ring for 20 seconds and whose bridge takes the in-phase mode down in 1.5. At a perfect unison the hammer excites only the mode that drives the bridge, so the note dies in 1.00 seconds. The longest note is at 2.30 cents, at 3.05 seconds — three times as long. Past 4.48 cents the two modes stop sharing a frequency and start sharing a decay, the note beats instead of ringing, and the sustain collapses. Instruments and their design

Three strings, and the note that comes back

The usual account of a beat adds two independent sources together. A piano's strings are joined at the bridge and are not independent: the pair has normal modes, one of which drives the bridge and dies in a second and a half while the other cancels at the bridge and rings for twenty. A hammer striking a perfect unison excites only the first — so a perfectly tuned unison is the shortest note on the instrument, dying in one second, and the longest is at a deliberate detuning of about two and a half cents, at three. Past four and a half cents the two modes swap what they share and the sustain collapses.

 How near the node a hammer has to land. The level of partial 7, relative to the loudest partial of the same note, against how far the hammer is from that partial's node — for a contact 2.6 per cent of the speaking length wide, which is 16 mm on a 620 mm string. At the node exactly the partial is absent whatever the width, because a mode shape is odd about its own node and integrating an odd function symmetrically gives zero. Twenty decibels of suppression needs the hammer within 2.82 mm, and thirty needs 0.89 mm. The width itself barely matters in the musical range: its sinc reaches its first zero only at partial 77. Instruments and their design

The hammer is not a point either

The comb of missing partials is computed throughout for a contact of no width, and a piano hammer is sixteen millimetres of felt on a string of six hundred. Integrating the mode shape across the felt turns out not to fill the null in: a mode is odd about its own node, so a symmetric contact centred on it gives an exact zero however wide it is. What destroys the null is being in the wrong place, and the tolerance is 2.8 millimetres for twenty decibels of suppression and 0.89 for thirty. The design decision found at the outset is a manufacturing tolerance.

 A note gets duller as it dies. Each partial of a string note against time, with the loss rising as the partial number to the power 1 — so the fundamental takes 6 seconds to fall sixty decibels and the 8th takes 0.75. The heavy line is the power-weighted centroid, falling from partial 1.77 toward the fundamental; it is halfway there after 0.15 seconds. A single-rate envelope would draw all of these as parallel lines and the centroid as a horizontal one, and a struck string does neither: what is left at the end of a long note is very nearly a sine. Timbre and acoustics

The note that gets duller as it dies

Every envelope drawn so far is one curve applied to a whole sound, and no struck string behaves that way. A string loses energy to air, to internal friction and to the bridge, and all three losses rise with frequency — so a note with a six-second fundamental has a sixteenth partial that is gone in under half a second, and the sound moving toward the listener is a spectrum collapsing toward its own fundamental. Which means an instrument is identified twice: once by the fifty milliseconds of its attack, which the earlier essays measured, and again by how fast its colour drains, which they did not.

 One instrument, five spectra. The first 12 partials of a bowed string at 5 pitches, each passed through the same fixed body response and normalised to its own loudest partial. The resonances stay where they are and the partials slide under them, so the pattern is different at every note: the power-weighted centroid runs from 1.02 to 1.71 across the compass and is not monotonic in pitch. A source with no body at all would give 2.35 at every pitch, which is the single number every roughness figure here uses for a string. Timbre and acoustics

An instrument is not one timbre

Every roughness verdict so far uses one partial list per instrument, and a violin does not have one. Its body's resonances stay where they are while the fundamental moves, so the radiated spectrum is different at every pitch: the power-weighted centroid runs from 1.02 to 1.64 across the compass, and it is not monotonic. Which means the result that a spectrum chooses its own scale gives a different scale at every note on the same instrument — three minima in the dissonance curve at the bottom of a violin's range, two in the middle and four at the top.

 Blowing harder is playing sharper. How far the played note is pulled from the bore's own resonance, against blowing pressure, for an air jet crossing a 4 mm embouchure. The jet's preferred frequency goes as the square root of the pressure, so it rises by a factor of 3.16 across the tenfold pressure range drawn, and the bore holds it to 272 cents of that — from -174 at the quietest to +98 at the loudest, about the player's own nominal. The rows below give the same pull for a drive one semitone sharp of the bore, for four valves: a clarinet reed is damped by the lip and pulls 4.9 cents, brass lips are not and pull 23.6. Pitch and tuning

Blowing harder is playing sharper

Every frequency computed so far for an air column is a resonance of the tube, and no wind instrument plays at its bore's resonance. It plays between the bore and whatever is driving it, weighted by how sharply each is tuned — and a flute's driver is an air jet whose own preferred frequency goes as the square root of the blowing pressure. Across a tenfold pressure range the jet's preference rises by a factor of 3.16 and the bore holds it to 112 cents of that. A wind player's dynamics and their intonation are one control, and the size of the coupling between them is the valve's Q.

 How many intervals a duet is smooth at. Every ordered pairing of 5 radiators, with the number of wells in its dissonance curve over an octave from 262 hertz. Rows are the instrument underneath and columns the one above, so the grid is not symmetric about its diagonal and that asymmetry is the result. The count runs from 1 to 7 across the grid, and a cell and its mirror need not agree: a violin under a clarinet has 1 and a clarinet under a violin has 2. The fifth is a well in every one of the 25 pairings and no other interval is. Timbre and acoustics

Which instrument is underneath

Every roughness curve so far compares two tones of the same timbre, which is a duet nobody plays. Give the two notes different instruments and the sum stops being symmetric: the same written interval, on the same two players, is up to five times rougher depending on which of them takes the lower note. From G3 upward the fifth is a well in all twenty-five pairings and no other interval is; below it, three pairings lose even that, and all three have a clarinet on top.

 What a chorus costs on one bridge. The sixty-decibel time of the longest-lived mode of two coupled strings at 262 hertz, against how far apart they are tuned. Below the bifurcation at 4.5 cents the pair splits its decays and one mode rings on; above it the pair splits its frequencies instead, both modes carry the bridge's loss, and the sustain sits at 2.8 seconds however much further the tuning is opened. The flat line is what the same two strings would do if they did not share a bridge. A chorused sound and a long one cannot be had from one bridge, and the first four and a half cents cost all of it. Pitch and tuning

The pair tuned apart on purpose

A piano's unison buys its sustain with a detuning of two and a half cents and loses it past four and a half. Read from the other side that is a design constraint on every instrument that wants the beating instead: past the bifurcation the sustain is gone and no further detuning costs anything more, which is why every chorused voice in the world — the celeste rank, the musette reeds, the paired gamelan — is built out of sources that do not share a bridge.

 Which partials of a natural horn can be lipped into tune. Every partial of the natural series against the nearest note of twelve equal, in cents, with the band the player's lips can actually move it drawn around each one. The band is ±23.1 cents, computed from the Q-weighted mean of a bore at Q 40 and lips at Q 12 rather than chosen. 4 of the first 16 partials fall outside it: 7, 11, 13, 14. The worst is the 11th at -48.7 cents, which would need lips of Q 38 to reach — comparable with the bore's own, at which point the bore has stopped deciding the pitch at all. Pitch and tuning

The partial the lips cannot reach

A wind instrument plays between its valve's preferred frequency and its bore's, which suggested that the natural trumpet's notorious eleventh partial would therefore turn out to be a statement about the lips. It is not. Run the same Q-weighted mean over the whole series and the lips can move that partial twenty-three cents, which is a quarter of what it needs — and the Q that would fix it is the Q at which the bore stops choosing the note.

 Two clocks on one contact, and which of them runs out first. Two times that could end a hammer's contact with a string, across the compass. The flat line is the felt's own recoil, which this site models as a fixed 1.6 milliseconds at a stated force and which does not know what note is being played. The falling line is the string's: a mass m driving a string of impedance Z loses its momentum in m/2Z, which is 11.9 milliseconds at the bottom of the keyboard and 0.76 at the top. They cross at G3. Below the crossing the felt lets go first and above it the string gets there first, and over six octaves the two are within a factor of two of each other — so the contact time is a coupled quantity and not a property of the felt. Instruments and their design

The hammer that is heavier than its string

Three earlier essays varied where the hammer lands, how long it stays and how wide it is, and the third named the fourth variable: the exciter's own mass. It inverts across one instrument — the hammer is lighter than the wire it hits at the bottom of a piano and thirteen times heavier at the top — and it turns the contact time into a quantity the string decides rather than the felt.

 The bow's window across a compass, with the bridge in it. Schelleng's window — the ratio of the largest usable bow force to the smallest — at a bow position of 0.09 of the length, across 196 to 1568 hertz, with the minimum scaled by the body's own admittance at each note. The window is widest between resonances and narrowest on them: it falls from 25 to 1.7 at B♭4. The notes a player finds hard to start are the local minima, and they sit on the body's resonances by construction — C♯4, B♭4, C♯5 for this body. At this bow position it never closes; move the bow toward the bridge and it does. Instruments and their design

The note the body will not let start

Every figure until now treats the string as though it ended at a rigid point, and an earlier essay admitted it: the body feeds back on the string hard enough to make some notes difficult on one instrument and easy on another. Put the body's own admittance into Schelleng's minimum bow force and the window narrows by fifteen to one at the corpus resonances — and near the bridge it closes.

 Which ensembles have this problem and which do not. The width of the heard-moment spread built into 7 standard instrumentations, before any player does anything. An ensemble drawn from one attack family has a spread of zero — every note is heard the same distance after it is started, so a common onset is a common heard moment, and this is true of a string quartet and of a gamelan for the same reason and in the same amount. A mixed ensemble carries between 11 and 33 milliseconds of it. Instruments and their design

The players who have to be early

Ensembles have been measured for fifty years and found to be about forty milliseconds out of alignment, which has always been reported as the limit of human precision. Part of it is not: an ensemble that mixes attack families carries a heard-moment spread of ten to thirty-three milliseconds before anybody plays a note, and an ensemble drawn from one family carries none at all — which is true of a string quartet and of a gamelan for the same reason.

 Six ways to put three players on three notes. The same chord — G3, B♭3, D4 — played by clarinet, oboe, voice in all 6 possible assignments, scored by the roughness each produces. Every bar is the same pitches and the same instruments; only who is on which note changes. The worst is 1.42 times the best, which is a factor a score can control and a chord symbol cannot express at all. Each row is labelled from the bottom note upward. Timbre and acoustics

Which player on which note

An interval's roughness depends on which instrument is underneath, so the pair does not commute. Three players over three notes is the smallest thing that asymmetry has anywhere to go: six assignments, all of them the same chord, and across 450 of them the roughest averages half again the smoothest and reaches six times it. It is orchestration in the only form that can be computed here — not which chord, and not which voicing, but who is on which note.

 What a hand in the bell buys, and what it costs. How far the instrument flattens and how much radiation it loses, against the share of the bell's mouth the hand blocks, for an F horn's 15 centimetre mouth on a 3.7 metre acoustic length. The two curves are the same aperture radius read twice: a narrower mouth adds inertance, which lengthens the tube, and is acoustically smaller, which stops radiating. The mark is the 25.6 cents left owed earlier on the eleventh partial — it needs 62 per cent of the mouth blocked and costs 3.8 decibels of radiated power. Fully stopping is a semitone and nine decibels. Pitch and tuning

The hand that changes the bore

A conjecture refuted here left a question with a number on it: the natural trumpet's eleventh partial is 48.7 cents flat, the lips can move it 23.1, and 25.6 cents are owed by something that is not an embouchure. There is exactly one thing a player can change about the bore while playing, and putting the hand in the bell buys those 25.6 cents at a cost of 3.8 decibels — because the aperture that tunes the instrument is the aperture that radiates it.

 Every modern bell puts the boundary at the same partial. For each brass instrument, the partial at which its bell stops turning the wave round — the frequency at which half the energy escapes at the mouth, divided by the tube's own fundamental. The five modern instruments land between partial 8.4 and partial 10.2 despite tube lengths differing by a factor of four, because a bell is scaled with its instrument. The natural trumpet of the baroque, whose bell is small and whose tube is long, puts it at partial 17.2. Intervals and chords

The fourth top is the maker's

The series has three tops and all three are the ear's: where consecutive partials stop being resolved, where their spacing falls below a semitone, and where it falls below what the ear can hear at all. The fourth is how far up a player can go, and it is the only one somebody chose. Every modern brass bell puts the boundary at about the ninth partial across a family whose tubes differ by a factor of four — and the baroque natural trumpet, whose bell is small, puts it at the seventeenth, which is exactly the top of the clarino register.

 The window is a different width on each string. The width of Schelleng's bow-force window across each string's own playable range, with both terms in it: the body's admittance, which was added earlier, and the string's own characteristic impedance, which every figure had held at one value. The four curves are the same shape displaced vertically, because the impedance is a constant per string — the G string's is 1.81 times the E string's, and the window is one over that. Where the ranges overlap, the same pitch sits on two or three curves at once. Instruments and their design

The same note is a different width

Schelleng's two bounds both carry the string's characteristic impedance and they carry it to different powers — the maximum force as Zc and the minimum as Zc squared — so the window a player has to stay inside goes as one over Zc. A violin has four strings whose impedances differ by a factor of 1.81, the same written pitch is available on two or three of them, and the tolerance is nearly twice as wide on one as on another. The two lightest strings turn out to have almost identical impedance, which nobody chose by accident.

 C4: the pulse computed and the pulse assumed. Above, the force the hammer delivers to the string at C4, integrated forward against the felt's nonlinear force and the string's returning corner, drawn against the half-sine of 1.60 milliseconds that every earlier figure assumed. The computed contact lasts 2.21 milliseconds and the corner comes home 4.6 times inside it. Below, the excitation each pulse gives to each partial. They agree at the bottom and part company higher up — worst at partial 6, by 34 decibels — because the assumed pulse has nulls the computed one does not. Timbre and acoustics

The pulse that was assumed

Every figure until now low-passes the string's excitation with the spectrum of a half-sine, which is what a hammer would deliver against a rigid wall. An earlier essay said so and declined to do better. Doing better takes forty lines and refuses the prediction that came with it: the corner's round trips govern the spectrum as expected, and the contact time is governed by something else entirely — the mass ratio discovered one essay earlier.

 What the climb costs, in tonnes and in inharmonicity. The total string tension a piano frame carries at each historical standard, computed from the string scaling used here. Tension goes as the square of the pitch, so the climb from 392 to 465 hertz is a rise from 11.5 to 16.2 tonnes on one instrument. The same ratio runs the other way through inharmonicity, which is inversely proportional to tension: the same wire at 465 hertz has 29 per cent less of it than at 392, so its partials are that much closer to a true series. Pitch and tuning

What the climb was a search for

The four-hundred-year rise in the pitch standard is usually explained as a search for brilliance, and an earlier essay ended by asking whether that was even coherent: does raising a string's tension change its spectrum as well as its pitch? Mostly it does not. What changes the timbre is something else entirely, and it is a mechanism that applies to instruments with no strings at all — every filter in the chain is fixed in absolute frequency while the notes move against it, so raising the standard is not a brightening but a reshuffling, and some notes lose.

 4 bores of one length, and the series each supports. cylinder, cone, exponential horn, Bessel horn — every one of them 148 cm long, 5.5 mm at the throat and 62 mm at the mouth, so the only thing that differs is the shape between the two. Beside each is the series of resonances Webster's equation gives it, and the number is how far that series is from evenly spaced, in cents. cylinder: 0.0 cents, with each mode sitting at minus 0.50 of a spacing off a whole multiple; cone: 9.6 cents, with each mode sitting at minus 0.12 of a spacing off a whole multiple; exponential horn: 22.8 cents, with each mode sitting at minus 0.16 of a spacing off a whole multiple; Bessel horn: 3.7 cents, with each mode sitting at minus 0.11 of a spacing off a whole multiple. Instruments and their design

Only two shapes make a series

A tube's length sets where its modes are and its shape sets whether they are a series at all — and of every shape a maker could choose, exactly two give evenly spaced modes. Neither of them is the shape of a trumpet. Webster's equation says how far the others miss by, and the answer is a quarter of a semitone for the obvious ones and four cents for the family brass bells actually belong to.

 What a trumpet mouthpiece does to the series. Each bore's modes before a mouthpiece is fitted and after, on the axis that says which harmonic each mode is: zero means the m-th mode sits at m times the spacing and is the m-th harmonic. Every shape starts near -0.14 and ends near 0.05. The cup holds 3.0 millilitres against a throat 3.7 mm across and pops at 642 hertz. What does not improve is the regularity: the spacing stays as uneven as the flare left it. Timbre and acoustics

What the mouthpiece is actually for

A cup and a throat look like a comfort fitting and are a component. Put a real one on the front of a real bore in the horn equation and the whole mode series slides sideways by about a seventh of its own spacing — which is exactly the distance between a series whose second resonance is 1.85 times the spacing and one whose second resonance is the second harmonic. The flare decides whether the modes are evenly spaced; the mouthpiece decides which harmonic each one is.

 cylinder and Bessel flare: the length each mode behaves as though it has. Each mode's own acoustic length, m·c over twice its frequency, for a bore 148 cm long. A cylinder would give one number repeated. This gives 164 cm at the second mode and 154 at the 8th — a spread of 10.1 centimetres, or 110 cents, because a flare's end correction is a length that shrinks as the note rises. The first mode is off the top of this axis and is not a mode a player uses. Pitch and tuning

A horn has one length per partial

Every tube until now has had an acoustic length: its physical length plus a correction for the wave carrying on past the opening. A flaring bore does not have one. Its second mode behaves as though the tube were 164 centimetres long and its eighth as though it were 154, and the ten centimetres between them are the same physical fact — a fixed correction against a shrinking wavelength — arriving as a hundred cents.

 The hand closing, and the note it is holding. The fourth resonance of a 370 cm horn against how much of the bore the hand occludes, at 97 per cent of the way along where the radius is 36 mm. Nothing much happens for the first ninety per cent. The note then falls to -401 cents — most of a fourth — over the next few, and between 99.5 and 99.90 per cent it jumps to 96 cents SHARP, because the lowest resonance has left the bottom of the range and every mode has taken the place of the one below it. The series' ratios are unchanged across the jump. Pitch and tuning

The hand goes in, and the note jumps

A horn player's hand closes the bell and the pitch falls — 19 cents, then 55, then 132, then four hundred, accelerating the whole way. Then, in the last half per cent of closure, it stops falling and lands a semitone above where it started. An earlier essay guessed the mechanism was the boundary condition changing kind and the series going odd-only. It is not. The series never changes at all.

 The bow's window along each string, with the bow held still. Schelleng's window — the ratio of the greatest bow force a note tolerates to the least — for every written pitch on every string that can play it, with the bow 35 mm from the bridge and staying there. The window widens up each string because β is the bow's distance over the SOUNDING length and the sounding length is shortening, and it steps down at each new string because a heavier string carries a narrower one. The dashed lines are what was drawn earlier, which held β at 0.09 for every note on every string. The widest disagreement between two strings at one pitch is 1.48 to one, at A♭5. Instruments and their design

Two dials the player turns together

Schelleng's window is a ratio of two bow forces, and an earlier essay found that the string's own impedance is in it twice. What it held still was the bowing point — as though a player kept β constant while moving up the fingerboard, which is the opposite of what a bow does. Let the bow stay where a bow stays and the window widens up every string, the spread between two strings at one pitch goes from 1.4 to 1.5 the other way round, and the string that is easiest changes.

 The partial above which the returning corner has lost its phase. Up a piano, the lowest partial whose extra travel after one round trip to the near end exceeds a quarter of a cycle — above it, the component comes home somewhere other than where a corner needs it. The curve follows the inharmonicity coefficient, which is smallest in the tenor and rises both ways, so the corner survives best at A2 — to the 24th partial — and worst at A7, where only 5 partials are still in step. The earlier coupled model assumes a clean corner at every partial it uses. Timbre and acoustics

The corner does not come back a corner

Stepping a hammer forward against the string's own returning wave showed that the coupling matters most in the bass. It assumed the wave that came back was the shape that left. On a stiff string it is not: partials travel at different speeds, and by the top octave of a piano the sixth partial is the first to arrive out of place. But a periodic wave can only see phase modulo a cycle, and counted that way twenty-one of forty-eight partials are still in step — which is why the returning wave is degraded rather than destroyed.

 The beat rate between two sections, second by second. The 5th partial of the lower section against the 4th of the upper, over 256 pairs of voices each sweeping 100 cents 6 times a second with its own phase. The band is the tenth to the ninetieth percentile of the instantaneous rate and the line is the median. With no vibrato the whole thing would be one flat line at 8.7 hertz, which is what was computed earlier. With it, the pair is inside the beating band 20 per cent of the time and above it for the rest — so what a listener gets is neither a beat nor a roughness but an alternation between them at the vibrato rate. Intervals and chords

Sixteen sweeps against sixteen

Every intonation figure about the voice treats a singer as a frequency. A singer is a frequency being swept a hundred cents wide six times a second, and two sections singing an interval are two hundred and fifty-six pairs of sweeps. The beat rate between the partials the interval brings together stops being a number and becomes a function of time — and the pair spends four fifths of its time above the rate at which beating is beating at all.

 How much of each spectrum a listener can assemble into one note. Each partial of each spectrum at the harmonic number it is nearest, against the whole-number series that fuses the most of them, with anything more than 1 per cent out marked as heard separately. an ideal string keeps 10 of 10; a piano string keeps 9 of 10; a bell keeps 7 of 8; a bar keeps 2 of 6; a kettledrum keeps 3 of 5. The fundamental is capped at a tenth of the top partial, and the cap is load-bearing rather than tidy: a bell's ratios are all whole multiples of a tenth, so an unconstrained search finds a fundamental twenty-five harmonics down, calls every partial exact, and reports that a bell fuses perfectly. Nothing that high is resolved and the low harmonics of it are not there. Perception and the listener

The spectrum that will not fuse

A partial about one per cent off its harmonic is heard as a sound of its own rather than as part of a note. Apply that criterion to a whole spectrum instead of to one mistuned component and it becomes a count: a piano string keeps nine of its ten partials, a bell keeps seven of eight, a bar keeps two of six. The physics of inharmonicity has had an essay here for a long time. This is what it sounds like.

 The heard moment against the pitch, on an instrument whose own attack is 8 ms. A note cannot establish an amplitude in less than 4 of its own cycles, so the attack has a floor of 4 periods — 145 milliseconds at A0 and 1.9 at C7. Below A4 the floor is longer than the instrument's own attack and the pitch decides the heard moment; above it the instrument does. The lag runs from 46.0 milliseconds at the bottom to 2.5 at the top, a spread of 44 milliseconds that no player can play their way out of. Rhythm and metre

A low note cannot start on time

Three earlier essays have held the pitch at one value. A note cannot establish an amplitude in less than a few of its own cycles, so the attack has a floor that rises as the pitch falls — 146 milliseconds at the bottom of a piano and three at the top. On an instrument whose action takes eight milliseconds everywhere, that is a forty-three millisecond spread across the keyboard from the period alone, and no player can do anything about it.

 Re-gauging at a fixed tension: how close sheep gut comes to breaking. Holding the tension at 700 newtons and re-gauging every string to suit the standard, the diameter each note needs goes as one over its frequency — so the stress goes as the frequency squared, and the margin against breaking falls the same way. At A = 392 the worst note has 1.79 times the stress sheep gut will take; at A = 466 it has 1.27. The margin reaches one at A = 525 hertz, which is far above anything the four hundred years of climb reached. Pitch and tuning

It was never the strings that stopped the climb

Every figure so far holds the instrument still and moves the standard. History did the opposite — instruments were rebuilt to suit the pitch, string by string. Hold the tension instead and each string's gauge is forced: the diameter goes as one over the frequency and the stress as its square. Gut breaks at A = 525 hertz and steel at 604, and the climb stopped at 466. The ceiling was somewhere else entirely.

 Every assignment, at equal levels and at its own best balance. The 6 ways of putting 3 players on a chord, each drawn twice: hollow at equal levels, which is what an assignment ranking sees, and filled at the levels that balance the parts and then minimise roughness. Every scoring here is at the same total loudness, 23.3 sones, so two points are comparable. Solving the discrete problem first picks violin · clarinet · oboe; solving both at once picks violin · oboe · clarinet, and the two-stage answer costs 9.0 per cent more roughness. The orderings do not keep their places between the two columns, which is the whole of the argument: a ranking taken at equal levels is not a ranking. Form and structure

Who plays what and how loud is one question

Two lines of argument, one about spectrum and one about loudness, each stopped at the same wall and each said so. One of them can choose who plays which note and has every player at the same level; the other can choose how loud each part is and has nobody assigned to anything. Put together they are a single problem with two kinds of variable, and solving it in stages picks a different answer from solving it at once — nine per cent rougher, at the same loudness, on an ordinary triad.

 Trumpet at three dynamics, as a spectrum rather than a level. The radiated partials of a trumpet at 45, 70, 95 decibels, each normalised to its own strongest partial so that only the SHAPE is compared. A linear source would give three identical pictures. This one does not: the spectral centroid moves from partial 2.19 to 6.41, a factor of 2.92, because the excitation is nonlinear and blowing harder steepens the pressure front rather than scaling it. The tilt used is 3 decibels per octave of partial number per ten decibels of level, referred to 70 dB — a stipulated, ordinal number, not a measurement of any instrument. Timbre and acoustics

A dynamic mark changes what a note is

Every spectrum until now is a shape with a level in front of it, so that playing ten decibels louder raises every partial by ten. That is true of exactly one instrument in an orchestra. Everybody else steepens their own spectrum as they lean on it, and a trumpet's centre of gravity moves from the second partial to the sixth across a dynamic range while an organ flue pipe's does not move at all.

 The first five peaks, followed as the hand closes. Each line is one member of the series, tracked by its rank rather than by its frequency, and each dot's size is that peak's height. The lowest peak falls from 38 hertz to 34 as the hand closes and then jumps to 45, which is the renumbering computed earlier: past the wall the series is one member shorter at the bottom and every peak has taken the place of the one below it. The dots shrink through the middle of the travel and grow again at the far end, so the transition costs the player support as well as pitch — and the cost is temporary, which is why a fully stopped horn is a usable instrument and a nearly stopped one is not. Pitch and tuning

A resonance has a strength as well as a frequency

What eleven earlier essays drew is a row of frequencies, because the solver behind it has no losses and a lossless resonance has no width. Put the losses in and every one of them acquires a height and a Q — and the hand closing a horn's bell turns out to take away nine and a half per cent of the instrument's total support before giving all of it back, in a window a few per cent wide where the horn is genuinely hard to play.

 The cutoff computed from the shape, and the cutoff read off the measurement. Two numbers for the same boundary. The hollow point is Webster's local cutoff — the largest value of (c/2π)√Γ along the bore, which is geometry alone. The filled point is where the impedance peaks stop, which is what a maker measures with a loudspeaker and a microphone. cone: no geometric cutoff at all, and peaks stopping at 4731; exponential horn: 89 hertz from the geometry, and peaks stopping at 2614, a factor of 29.26; catenoidal horn: 115 hertz from the geometry, and peaks stopping at 2499, a factor of 21.76; Bessel horn: 1154 hertz from the geometry, and peaks stopping at 1593, a factor of 1.38; cylinder and Bessel flare: 2945 hertz from the geometry, and peaks stopping at 4417, a factor of 1.50. The published figure for a trumpet is about 1500 hertz, marked. The two agree within about half for a bore whose flare is concentrated at the end and disagree by more than an order of magnitude for one whose flare is spread along it, which says what the local formula is and is not a measurement of. Instruments and their design

The cutoff a maker can actually measure

A trumpet's bell cutoff computed from the geometry alone comes to 1,150 hertz against a published 1,500, and the calibration was the first thing that left owing. Measuring the model the way a maker measures the instrument — sweep a loudspeaker in, find where the impedance peaks stop — gives 1,593. The discrepancy was almost entirely in the formula, and on a bore whose flare is spread rather than concentrated the same formula is out by a factor of twenty-nine.

 The fingerboard, with the hardest place on it. Every written pitch from G3 to G6 on every string that can reach it, shaded by the width of Schelleng's bow-force window there — dark is narrow, which is a note that is hard to start. Two effects are multiplied and neither earlier figure could show the other: the body's admittance, which depends on the frequency, and the bowing fraction and string impedance, which depend on where the hand is. The worst place is C♯4 on the G3 string, 6 semitones up it, at a window of 12.2 against 471 at the easiest — a factor of 39 across the instrument. It sits where the body's A0 resonance crosses the heaviest string played high, which is the compounding this figure was drawn to find: neither variable alone puts a minimum there. Instruments and their design

The hardest place on the fingerboard

Two things narrow a bow's window and each has been drawn alone. The bridge's admittance is a function of frequency; the bowing fraction is a function of where the left hand is. They meet on a real fingerboard, and multiplying the two curves gives a map with a worst place on it — C♯ on the G string, sixth position, where the body's air resonance crosses the heaviest string played short. The window there is twelve, against three hundred and eighty at the easiest.

 Which of the four terms is doing the removing, note by note. Each of three terms has a corner — a partial number above which it is the one taking the energy away — and the lowest corner at each pitch is the term that binds. The comb is flat at 8, because where the hammer lands is a fraction of the length and does not change with the note. The contact time's corner falls from 34 at A0 to 0.5 at A7. They cross at about E3, 165 hertz: below it the strike point decides the spectrum and above it the contact time does. The dispersion's corner is above both at every pitch on the instrument — 5.8 at the top against a contact corner of 0.5 — so that earlier term is real and is nowhere the binding one. That is a statement no one of them could make on its own. Timbre and acoustics

Four terms, and only one of them binds

Six earlier essays each computed one term of a piano's excitation spectrum and handed it to the next, and nobody had multiplied them. Multiplied, three of the four have a corner — a partial above which they are the term doing the removing — and putting the three corners on one axis says which is in charge at each pitch. Below E3 the strike point decides; above it the contact time does; and the dispersion, which is the newest and most laborious term, is nowhere the binding one.

 Which interval gives a tuner the deepest null. A tuner listening to an interval p:q is listening to the lower note's p-th partial against the upper note's q-th, and how deep the beat's trough goes is decided by those two amplitudes rather than by the interval. For a string spectrum, whose partials fall as one over n, the minor third pairs partial 6 against partial 5 at a ratio of 1.18 for a dip of 21.8 decibels; the major third pairs partial 5 against partial 4 at a ratio of 1.25 for a dip of 19.1 decibels; the fourth pairs partial 4 against partial 3 at a ratio of 1.32 for a dip of 17.2 decibels; the fifth pairs partial 3 against partial 2 at a ratio of 1.52 for a dip of 13.8 decibels; the major sixth pairs partial 5 against partial 3 at a ratio of 1.65 for a dip of 12.2 decibels; the minor sixth pairs partial 8 against partial 5 at a ratio of 1.67 for a dip of 12.0 decibels; the octave pairs partial 2 against partial 1 at a ratio of 2.00 for a dip of 9.5 decibels. The best is the minor third at 21.8 and the worst is the octave at 9.5, which is the reverse of the order a tuner is usually taught to trust: the deepest null in the list is on the interval whose coincidence sits highest in the spectrum, where adjacent partials are nearly equal in strength. Intervals and chords

A beat has a depth, and six essays held it at one

Every beat figure so far adds two tones of equal amplitude, which is the single ratio at which the trough of a beat is a true null — and a null is what a tuner is actually listening for. Vary the ratio and the picture changes: at two to one the dip is nine and a half decibels, at ten to one it is under two, and the interval that gives the shallowest null of all is the octave.

 Which notes of a scored chord have to be played early. Four parts of one chord, each with its own instrument, its own pitch and its own dynamic, and the perceptual centre that comes out of all three. piano, sforzando on E1: an attack family of 8 milliseconds against a pitch floor of 97, so the pitch is what limits it, shortened by the dynamic to 65, heard 20.5 after it starts and needing to be played 12.0 early; flute, quiet on A5: an attack family of 60 milliseconds against a pitch floor of 5, so the instrument is, shortened by the dynamic to 69, heard 21.8 after it starts and needing to be played 13.2 early; violin, mezzo forte on E4: an attack family of 90 milliseconds against a pitch floor of 12, so the instrument is, shortened by the dynamic to 90, heard 28.5 after it starts and needing to be played 19.9 early; trumpet, forte on A3: an attack family of 30 milliseconds against a pitch floor of 18, so the instrument is, shortened by the dynamic to 27, heard 8.6 after it starts and needing to be played 0.0 early. The spread is 19.9 milliseconds, which is well above the two or three a listener resolves, so a conductor asking for these four to sound together is asking for four different physical onsets. Rhythm and metre

Which notes have to be played early

There are three separate contributions to one quantity — the instrument's attack family, the dynamic it is played at, and the note's own period — and every figure so far varies one and holds the others. Added together for a real scoring they do not add: a sforzando low piano note is pitch-limited to a hundred-millisecond attack and the sforzando shortens it back to sixty-five, so flattening the dynamics makes the ensemble's spread larger rather than smaller.

 Three ways a note starts, on one pair of axes. Every instrument in this collection, with how long its note takes to speak measured twice: across, in periods of the note itself; up, in milliseconds. The three clusters are three different pieces of physics. A wind instrument accumulates energy in a resonance, so its wait is the resonance's Q — 27, 26, 26, 19, 37 periods here. A bowed string is at full amplitude the moment Helmholtz motion begins and what takes time is the bow reaching a force inside Schelleng's window, which is 3.0, 3.8, 5.1, 7.6 periods. A plucked or struck string is at full amplitude at once and the only duration in it is the exciter's own contact, under a tenth of a period. The two axes do not agree: the slowest instrument in milliseconds is an alto saxophone at 159, and in periods it is an alto saxophone at 37. Instruments and their design

A note takes a number of periods to speak

Two separate accounts stopped at the same object and said so. A wind instrument's note takes as long to arrive as its resonance takes to build, and that time is the resonance's Q divided by its frequency — so the number of periods is the Q and has no pitch in it, while the number of milliseconds does. The two readings order the instruments differently, and both orderings are wanted.

 Up a trumpet, the two clocks disagree. Every impedance peak of a trumpet, with the settling time each implies. The Q rises up the ladder and so does the frequency, and the settling time is their ratio — so in milliseconds the wait falls from 153 at the C♯2 to 24 at the B5, while in periods it rises from 11 to 24. Both curves are monotone and they point opposite ways, which means a player going up the instrument gets notes that arrive sooner and take longer in their own terms. Nothing here is a measurement of an instrument: it is the transmission-line solve of a trumpet-shaped bore, whose peak Qs are sensitive to how finely the sweep is sampled at about five per cent. Pitch and tuning

The higher note speaks sooner and takes longer

Up a brass instrument the settling time in milliseconds falls by a factor of seven and the settling time in periods rises by a factor of three. Both curves are read off the same impedance sweep, both are monotone over most of the compass, and they point in opposite directions — so the slowest note of the instrument depends entirely on which clock is used to time it.

 The violin's attack times, string by string. Every playable cell of the earlier map, with its bow-force window turned into a time. A player aiming at the geometric centre of the window has to wait until the accelerating bow's maximum force rises to that value, which is a fraction one over the square root of the window width of the way through the bow's ramp — so a narrow window is a late note and the exponent is a half. The latest cell is C♯4 on the G3 string at 21.9 milliseconds, against 4.1 at C6 on the A4: a factor of 5.3 in time out of a factor of 29 in window width. The darkest cell is the same hardest place, unchanged — the map is the same map under a monotone change of units, and what is new is that the units are milliseconds, which a player and a listener both have access to. Rhythm and metre

The hardest place is also the latest

A bow-force window is a ratio of forces, which nobody can hear. An attack time is milliseconds, which a player and a listener both have. The map of the violin's windows becomes a map of its attack times under a change of units, and the exponent turns out to be a half — so a window twenty-nine times narrower is only five times later.

 A family resemblance, in the heights rather than in the frequencies. The peak heights of trumpet, F horn, tenor trombone, plotted against peak number rather than against frequency. The three differ in length by a factor of 2.4 and their frequency series cannot be made to overlap; their heights agree to 3.2 decibels on average and their Qs to a factor of 1.30. The agreement improves up the series — 6.9 decibels at the first peak and 1.6 at the 8th — which is an earlier claim arriving as a measurement: a family has one voice because it has one filter, and the filter is visible in what the bore pushes back with and not in where its resonances are. Instruments and their design

A family resemblance in the heights

Trumpet, horn and trombone differ in length by a factor of two and a half, so their frequency series cannot be laid over one another. Their impedance peaks agree to three decibels in height and to thirty per cent in Q, peak for peak, and the agreement improves with peak number. An earlier essay inferred that a family has one voice because it has one filter; the solver can now be asked directly.

 One vent on a cone, over the notes it has to serve. A cone's modes are a full harmonic series, so the mode a register vent has to keep is the octave rather than the twelfth, and its pressure node sits at half the sounding length measured from the virtual apex. The apex is a fixed point of the instrument and the bell end is not, so every fingering moves the ideal position and there is one hole. The upper line is how much the vent spoils the fundamental, which is what it is for; the lower is how much it damages the octave, which is the cost. Their ratio runs from 451.2 at F♯4 down to 1.9 at B♭3 — a factor of 235 across a single register. The vent is placed at 37 per cent of the longest sounding length, which is where the worst fingering is least badly served. Instruments and their design

The vent a cone cannot place

A clarinet's register key has to spoil a fundamental and leave a twelfth. A saxophone's has to spoil a fundamental and leave an octave, whose pressure node sits at half the sounding length from the virtual apex — and the apex is a fixed point while the bell end is not. Over one register the ideal position moves by a factor of two, and one hole is right for one note.

 The flare that makes a series harmonic, and how narrow it is. Every combination of a flare exponent and a station at which the flare begins, shaded by how far the bore's resonance series is from a harmonic series over partials 2 to 8, in cents. The best is 4.6 cents at an exponent of 1.00 beginning 43 per cent of the way along, against 127 cents for a plain cylinder, 21 for a plain cone and 26 for a Bessel horn of the exponent used everywhere else. Only 2.1 per cent of the surface is within five cents of the minimum, so the shape is forced rather than chosen — which is what three centuries of empirical brass design were finding. Timbre and acoustics

The flare that makes a series harmonic

Every figure until now treats a partial as n times a fundamental, and the whole of brass instrument design is the business of making that true. A plain cylinder is 127 cents from a harmonic series and a plain cone is 21; the best flare found by sweeping is 4.6, and only two per cent of the swept surface comes within five cents of it. The shape is forced rather than chosen.

 Where each exciter's contact corner sits against its comb. The merger's four corners, drawn for three exciters. The comb is the strike point and is a horizontal line, because it is a fraction of the string and has no pitch in it. The contact corner is the exciter's, and it falls up the compass because the contact time is a fixed number of milliseconds against a shrinking period. Where they cross, the binding term changes. piano hammer crosses at E3; harpsichord plectrum crosses nowhere in the compass; dulcimer beater crosses at A5. So the crossing found earlier is not a fact about pianos. It is a fact about compliance: a felt hammer stays on the string for about 1.6 milliseconds and a plectrum for 0.05, and a difference of that size is a corner in a different place. Timbre and acoustics

The crossing belongs to the felt

On a piano the strike point decides the top of the spectrum below E3 and the hammer's contact decides it above. The merger's own bookkeeping asked whether that crossing is a fact about pianos or about hammers. A plectrum never crosses at all and a hard beater crosses two octaves higher, and the boundary is a contact time of about an eighth of a millisecond — ten times shorter than felt.

 A loud chord is a smaller chord. The share of a voicing's partials that stand above what the rest of it masks, and the share of its computed roughness that is between partials a listener actually has, from 30 decibels to 100. Both fall: 83 per cent of the partials survive at 30 decibels and 38 at 100, and the roughness share goes from 88 per cent to 67. The direction is the upward spread of masking, which grows faster than linearly with level: a loud partial masks a band above itself much wider than a quiet one does, so the chord's own top disappears into its own bottom. Two earlier essays are drawn at one level, and this is what they were holding. Harmony and voice leading

A loud chord is a smaller chord

Two earlier essays hold the level fixed, and the level decides how much of a chord a listener is given. At thirty decibels twenty of a triad's twenty-four partials stand above what the rest of it masks; at a hundred, nine do. Every roughness figure until now counts partials that are in the score, and a partial the chord masks is not a partial the listener has.

 A written dynamic is an instruction to the listener's impression. Every earlier scoring holds one chord still. A passage is a succession, and the running impression of loudness carries a chord into the one after it, so what a marking asks for and what playing the marking produces are different things. Here is a five-chord passage with a written shape. Playing each chord at its own written loudness gives the running impression 2.4, 3.0, 4.2, 5.6, 4.0 sones against the 2.4, 3.0, 4.2, 5.6, 2.0 that were asked for — right until the last chord, where it misses by 2.0. Solving for levels that make the impression arrive at the marking does not fix it: the last chord's target is I, two parts, and it is unreachable — the correction runs to silence and the impression still sits 1.1 sones above. A subito piano after a full chord is not a level a player can produce. It is a rate of change, and the smoother's two-second release is what refuses it. Form and structure

A subito piano is a rate, not a level

All three earlier essays score one chord held still. An orchestration is a succession, and the running impression carries a chord into the one after it — so a written dynamic is an instruction to the listener's impression rather than to the instantaneous sound, and there are markings that cannot be produced at all. The correction runs to silence and the impression still sits above the target.

 trumpet: what the cup does to every peak. Each impedance peak of a trumpet drawn twice: as the bare bore, and with its own catalogue mouthpiece in front of the throat. The bare heights fall monotonically, from 19.2 at the pedal to 1.96 at the top of the series, which is what a tube does. The fitted heights do not: they fall to 5.4 at E♭6 and rise again to 30.1 at E♭5, giving the instrument a maximum of support in the middle of its compass that the tube alone has nowhere. The dashed line is the mouthpiece's popping frequency, 642 hertz — the note the cup sounds when it is slapped on the palm — and the maximum sits beside it. Instruments and their design

What a cup does to the support

The mouthpiece's job was settled four essays ago and settled in cents: it decides which harmonic each mode is. Measured instead in the currency a player buys one in — how hard the note pushes back, and how narrowly it holds its pitch — the cup does something else entirely. It multiplies the support in the written register by about five, and it puts a ceiling on the instrument that the bell had not put there.

 The bell and the cup, on one surface. Every combination of a flare exponent and a cup depth, shaded by how far the resonance series is from a harmonic series over partials 2 to 8, in cents. The pale line is the best flare for each cup — the ridge — and it runs diagonally: the exponent that suits the shallowest cup here is 0.93 and the one that suits the deepest is 0.70, a spread of 0.22. The ring marks the bell found earlier by sweeping the flare alone, at an exponent of 1.00 — which reaches 4.6 cents with nothing in front of it, and sits where the ring is, at 24.9 cents, once a catalogue mouthpiece is on it. It is on none of the ridge, and that is the whole finding: the two choices are not separable, and a bell optimised alone is not the bell an instrument wants. Timbre and acoustics

The bell is tuned for the cup

Sweeping a brass bell's two flare parameters found a shape that makes the resonance series harmonic to four and a half cents. Bolt the mouthpiece that instrument is actually played with onto it and the series is twenty-five cents out — worse than a plain cone. The flare and the cup are not two independent choices, and the best bell for a maker to build is one that is deliberately wrong.

 Seven holes that all sound 196 hertz, and none of them agrees about the twelfth. Each dot is a hole radius, placed at the station that makes the first resonance 196 hertz. The stations run from 411 millimetres for a 7.5-millimetre hole to 307 for a 1.4-millimetre one, which is a fifth of the tube. Up the axis is what the second resonance does: a cylinder's should be three times the first, and it is -2 cents from it for the widest hole and -453 for the narrowest. The hole's inertance rises with frequency, so a narrow hole lengthens the tube more for the twelfth than for the fundamental — and two holes that are interchangeable in the first register are a fourth apart in the second. Instruments and their design

A hole is a short tube

Four earlier essays have treated an open tone hole as a point where the pressure is released. It is not: the air in a hole has mass, and a hole with mass does not end the bore, it loads it. Seven holes drilled at seven stations all sound the same G — and their twelfths are spread over a fourth. Cross-fingering falls out of the same arithmetic, and it is not made of what everybody says it is.

 Three instruments, three wires, and an order of magnitude between them. The inharmonicity coefficient of each instrument's own string design, across its own compass. At middle-of-the-keyboard C a piano's wire is 315 millimetres long and 1.02 thick and gives B = 1.5e-3; a harpsichord's at the same pitch is 355 millimetres and 0.30, and gives 8.1e-5 — a factor of 19. The harpsichord's line is nearly flat across four octaves because its scaling is Pythagorean: the length doubles for every octave down, exactly, until the case runs out. The piano's is a V, because its case runs out two octaves earlier and everything below the break is a compromise. Every column before now used the top line for all three instruments. Instruments and their design

Three exciters and three wires

Every column drawn so far has run a plectrum and a beater on a piano's string, because a piano's was the only scaling to hand. A harpsichord's wire is an order of magnitude less stiff, for a reason that turns out to be the tensile strength of iron — and with all nine combinations computed, the crossing found before stays exactly where it was, on every wire.

 The violin is the one where the bow's width catches up. Three separate limits on how near the bridge a bow can go, for the four bowed instruments. The ribbon's near edge reaches the bridge at 5, 6, 6, 8 millimetres; the ribbon covers half the corner's bridge-side excursion at 10, 11, 12, 15; and Schelleng's force window narrows to a factor of 3 at 10, 12, 21, 32. On the viola, cello and double bass the force window binds first, by a margin that widens to a factor of two on the bass. On the violin the ribbon binds first, at 10 millimetres against 9.8. A bow's hair is as wide as a hand can control and a string is as long as its pitch requires, so the ratio between them is a fact about the violin rather than about bowing. Instruments and their design

The bow is not a point either

Seven earlier essays set the bowing point to a number between 0.02 and 0.3 and drew it as a point. A violin bow's hair is ten millimetres wide on a string of three hundred and twenty-five, so near the bridge the ribbon is wider than its own distance from it. In the spectrum that turns out to be worth nothing. In the geometry it is the reason sul ponticello has a floor, and the violin is the one instrument of the four where it arrives before the force does.

 Settling and being heard are not the same quantity. Across, how long the instrument takes to reach its steady amplitude, computed from its own physics — a resonance's Q, a bow's capture, an exciter's contact. Up, how long after its physical onset a listener places the note, computed from the measured shape of its envelope. Five instruments both accounts hold. The diagonal is where they would agree and nothing is on it. The ratio between them runs from 0.10 to 6.6, a factor of 69, and it sorts perfectly by mechanism: about 6.6 for a struck or plucked string, 2.2 for a bowed one, and about 0.15 for a wind. Ordering the five by each measure changes the place of 3 of them, and the one that moves furthest is the violin — third slowest to settle and the last to be heard. Rhythm and metre

A note starts twice

One account computes how long an instrument takes to settle, from its own physics. Another computes how long after its onset a listener places a note, from the shape of its envelope. Both come out in milliseconds and neither has ever been shown the other. Paired on the five instruments they share, the ratio between them spans a factor of sixty-nine and sorts perfectly by mechanism — and the violin is third slowest to settle and the last to be heard.

 Four of the five are a whole number of semitones, and one is exactly half of one. Each mismatch in cents, against the ticks at whole semitones — which are the only places a transposing keyboard can put a player. 4 of the 5 land within six cents of a tick: the Chorton–Kammerton gap is 197 cents against a whole tone's 200, and Chorton against French pitch is 296 against a minor third's 300. The exception is an English organ against Handel's fork, at 50 cents — 50 cents from the nearest tick, which is as far as it is possible to be. So the small mismatches are the unsolvable ones, and the large ones were solved by shifting the keys. Pitch and tuning

The instrument that cannot be moved

A string is regauged and a woodwind is scaled. An organ's pitch is the length of its pipes, and metal can be cut off and cannot be put back — so an organ is a ratchet that only goes sharp. The mechanical answer was to shift the keyboard against the pipes, and its cost is not the transposition. It is that the temperament's key colours rotate out from under the notation, by an amount measured in fifths rather than in semitones.

 A mistuned 2:1 makes 8 beats, not one. Every pair of partials that coincides in a 2:1 mistuned by 6 cents, with the rate each one beats at. On a perfectly flexible string the rates are 1.5, 3.1, 4.6, 6.1 and so on — an exact harmonic series of the slowest, because partial 2k is exactly twice partial k. On a stiff one they are 1.3, 0.9, 2.7, 11.1, and the 8th member is at 122 hertz. The family grows as the cube of the member number rather than linearly, so 4 of the 8 are slow enough to be beats at all and the rest are roughness. Intervals and chords

A beat is never one beat

Every beat counted until now has come from one pair of partials. A real spectrum has many, so a mistuned octave produces eight beats at once — and on a perfectly flexible string those eight are an exact harmonic series of the slowest. On a real piano string they are a cubic, half of them are not beats at all, and no single width of octave silences more than one. That is where the tuner's five octaves come from.

 The cue that settles it. Every spectrum to hand, arbitrated by the earlier competition and then again with the onset cue added at equal weight. 3 of the 5 change their verdict, and all 3 change the same way — from splitting into two streams to staying as one: a piano string, a bell, a bar. Nothing changes the other way, because the onset cue on a struck source votes for fusion on every partial and can only ever push toward one stream. The bell is the case worth naming: its partials are wildly inharmonic and it is heard as one sound, which is a fact the harmonicity cue alone cannot produce. Perception and the listener

The cue that settles it

Arbitrating between two grouping cues meant sweeping an exchange rate nobody could supply. The cue it had no term for at all is the one every account calls strongest, and its strength is computable: a struck string's partials start together to within a tenth of a millisecond against a threshold of twenty. Put that into the competition and three of five verdicts change, all the same way — and a bell becomes one sound.

 The passage that separates them, and a listener cannot hear it. A scoring changes at the halfway bar, and the two maps of required leads differ by 18.1 milliseconds at their widest. An ensemble that has internalised the map applies the new one on the first note of it and its spread never leaves zero. An ensemble that is listening to each other has to re-converge: its spread jumps to 11.8 milliseconds and takes 3 beats to get back under 5. The dashed line is twenty milliseconds, which is what a listener notices — and the disagreement never reaches it. So the two accounts are separable on a recording and very nearly not separable by ear, which is why nobody has noticed the distinction and why the measurement is worth making. Rhythm and metre

The passage that separates two players

An ensemble that has learnt where the asynchronies are applies them; one that is listening discovers them. In steady state the two are identical, which is why nobody has separated them. Change the scoring mid-phrase and they are not: one ensemble is wrong by twelve milliseconds for three beats and the other is not wrong at all — and twelve milliseconds is under what a listener notices and far above what a microphone resolves.

 A straight mute where its corks put it, and the annulus left over. The last 220 millimetres of a trumpet's bell, drawn to scale in radius and in length, with a straight mute in it. The mute is a cone 150 mm long running from 35 mm across at the tip to 105 at the base, and it is pushed in until its clearance is the cork's 3.0 mm — which stops it 63 mm inside the rim. The sound goes past it through the annulus between cone and wall, and the narrowest place in that annulus is 393 square millimetres, 59 mm in, which is 27 per cent of the bell's own section there. The lower curve is that annulus turned into the only thing a transmission line cares about: the radius of a round tube of the same area, which falls to 11.2 mm at the throat and recovers to 53 at the rim. Instruments and their design

The resonator at the far end

A cup at the throat owns the top of a brass instrument and puts a ceiling on it. A straight mute is a second closure at the other end of the same air column, and the prediction written down before it was computed was that it would push that ceiling further down. It does the opposite: the ceiling goes from C6 to E♭6, and it goes there by reviving exactly the resonances the cup had killed.

 Four bells on one tube, and the series each of them supports. The same 1.48-metre trumpet bore, drawn to scale, with its mouth opened from 32 millimetres across to 280 — a factor of 8.8 in radius and everything else held. Beside each is the frequency past which its flare stops reflecting, computed from the geometry: 403 Hz, 1212 Hz, 2945 Hz, 7303 Hz. That boundary moves by a factor of 18. The resonances underneath it barely move at all — the fourth peak sits at 409, 421, 428, 432 hertz across all four. Instruments and their design

The mouth that decides nothing

Twelve figures across six earlier essays draw a trumpet with a 124-millimetre bell, and not one of them draws any other. Opened from 32 millimetres across to 280, the bell's own boundary moves by a factor of eighteen and almost nothing else moves at all: the instrument's registration, the sharpness of its notes in the written register, and where it runs out are all within a semitone of themselves. The reason is that the resonances are held by the walls, and a bell cannot reach them.

 A woodwind with holes graduated 12 mm to 6 mm, drilled so that every fingering is in tune. A cylindrical bore 567 millimetres of acoustic length and 15 across, with twelve tone holes through a 4-millimetre wall. Opening them one at a time from the far end takes it up a chromatic scale from D3 to D4. The stations are not copied from a maker's drawing: each was solved so that its own fingering sounds its equal-tempered note in this model, one hole at a time down the tube with every hole below it already open, which is what a reamer and a tuning fork do. The worst fingering is 11.9 cents out. The diameters run 12.0 millimetres at the bell end to 6.0 at the top, and the spacings close from 30 millimetres to 19. Instruments and their design

The cutoff that is a list

Five earlier essays have quoted one number for a woodwind's cutoff — 1,824 hertz for a clarinet — from a formula written for an infinite lattice of identical holes. Solve a whole twelve-hole chart instead and the number is eleven different numbers, running from 2,193 hertz down to 1,574, which is 574 cents. The lowest fingering has no cutoff at all, and which way the list runs turns out to be a design decision rather than a fact about woodwinds.

 How much of each instrument is narrow enough to steepen a wave. For each bore, the quantity that decides how nonlinear it is: the narrowest radius divided by the radius at each station, integrated along the tube. The shading is that integrand, so a bar that stays dark is a tube still doing damage to the wave and a bar that fades is a flare that has thinned it out. Divided by the instrument's own length the integral is a pure number: a plain cylinder 1.00, a tenor trombone 0.88, a trumpet 0.86, an F horn 0.59, a cone of a trumpet's length 0.24. A plain cylinder is 1 by construction, a cone of the same length and mouth is 0.24, and the ordering across the brass family is the one players give when asked which of them can be made to blare. Instruments and their design

The partials the tube makes itself

Eleven earlier essays compute a passive linear resonator, and none of them ever says so. At a real fortissimo the air in a brass instrument is not linear: a compression outruns a rarefaction, the wave leans forward as it travels, and the fourth partial of a loud trumpet note is seventy decibels louder than a scaled-up quiet one — generated in the tube rather than at the lips. How much of it happens is an integral over the bore, and it is why a flugelhorn cannot be blown into being a trumpet.

 A clarinet's partials, each on its own resonance. The 8 partials of the clarinet's chalumeau D that ride an impedance peak, each building toward its steady amplitude as 1 − exp(−t/τ) with τ = Q/πf from that peak's own Q. The time constants run from 11.9 milliseconds to 64.5, so the partials do not arrive at different times — they all begin the instant the reed does — and what differs is how fast each approaches its final level. The horizontal bars are how far apart the first and last are at three criteria: 5.5 ms at 10 per cent, 36.4 ms at 50 per cent, 121.0 ms at 90 per cent. A twenty-millisecond asynchrony is the threshold for hearing a partial out of a note, and this note crosses it at 32 per cent of steady amplitude — so whether a blown note's onset cue is unanimous or divided is decided entirely by how far along a partial has to be before it counts as having started. Perception and the listener

A blown note does not start late, it starts slowly

Computing the onset cue removed a free parameter and turned out to be unanimous, and it predicted that a wind instrument would put it back, because a blown note's partials arrive over tens of milliseconds. They do — 121 on a clarinet — and it is not an asynchrony: every partial begins the instant the reed does and they differ in rate, not in time. Read at a tenth of the steady amplitude the spread is 5.5 milliseconds against a threshold of twenty, so the cue is still unanimous, and the missing number is no longer the exchange rate but the criterion.

 Three impedances meet at the bowing point and only one is in the way. Every impedance at the contact between a violin bow and a violin's strings, on one logarithmic axis in kilograms per second. The four strings run from 0.19 on the E to 0.34 on the G. The hair ribbon's TRANSVERSE impedance, across its own length, is 0.59 — 1.7 to 3.2 times the strings', which is comparable and is the number expected earlier to matter. It lies perpendicular to the string's own motion, because the bow is drawn ACROSS the string and the hair therefore runs along the direction the string vibrates in; what it is in the path of is the bow force. The impedance that does lie along the string's motion is the ribbon's LONGITUDINAL one, 11.4, which is 19 times the transverse and 34 to 61 times the strings'. A point load of that size against the 2Zc a string offers reflects 94.4 per cent of an arriving corner, so the fixed bowing point every earlier figure has assumed is right to within 5.6 per cent on the G string. Instruments and their design

The hair runs the wrong way

Eight earlier essays treat the bow as a contact and the string as the object, and the last of them asked what the bow's own impedance does at the point of contact. It has three, and the one that is comparable to the string's — 0.59 kilograms per second against 0.19 to 0.34 — lies at right angles to the direction the string moves in. The one that lies along it is thirty-four times the string's, which is why a fixed bowing point has been the right assumption all along; and it stops being right at one partial in the middle of the instrument's range.

 Schelleng's window at five states of the rosin. The ratio of the largest usable bow force to the smallest, against distance from the bridge on a violin, drawn at friction contrasts of 0.5, 0.75, 1, 1.5, 2 times the normally-rosined bow every other figure assumes. The window is a hundred times the friction contrast times beta, so each curve is the next one shifted bodily: the position at which it narrows to a factor of 3 moves from 19.5 millimetres at 0.5 to 4.9 at 2. The two vertical marks are the geometric limits found earlier and neither of them moves with the rosin: the ribbon covers half the corner's bridge-side excursion at 10.0 millimetres and its near edge reaches the bridge at 5.0. The friction contrast at which the force limit retreats behind the first of those is 0.98, and behind the second 1.95 — so a violin at ordinary rosin is sitting within a few per cent of the exchange, and at twice it the force window has no say at any playable bow position. Instruments and their design

What the rosin is worth

Schelleng's window has three parameters and it has been drawn thirty-nine times with two of them held at one. The bow speed turns out to cancel exactly — it multiplies both bounds and leaves the ratio alone — and the friction contrast does not: the window is proportional to it, so every limit priced in millimetres moves as one over it and none of the geometric ones move at all. A violin at ordinary rosin is sitting 2.5 per cent from the point where the two exchange places, and at twice it the force window has no say at any playable bow position.

 A plucking point 50 millimetres from the nut, across a harpsichord's compass. The jacks stand in a rail and the rail is one object, so a register plucks at a fixed distance from the nut while the strings shorten by a factor of ten from the bass to the treble. At 50 millimetres that is one 36th of the string at C2 and one 3.5th at C6 — a plucking fraction that changes by a factor of 10.1 without the maker moving anything. The comb corner is one over that fraction and falls with it, from 36 to 3.5; the dispersion corner has the fraction under a cube root and falls only from 106 to 21. They never meet. Setting one over p equal to the cube root of two over three times the inharmonicity and the fraction gives a plucking point at p = √(1.5B), which on this instrument's iron wire is between 3.4 and 9.9 millimetres from the nut — nearer to it than any register a harpsichord has ever carried, the lute stop included. So the maker's one free choice is the binding term at every pitch and every register position, which is exactly what the piano's is not: on a piano the contact time takes the decision away above E3. Instruments and their design

What the second register is for

Nine earlier essays take the plucking fraction as given — an eighth, a sixth, a seventh — and a harpsichord's jacks stand in a rail, so what is given is a distance and the fraction is a consequence: 50 millimetres is one thirty-sixth of the string in the bass and one three-and-a-halfth in the treble. Engage two registers on one string and the amplitudes add exactly, the near one fills the far one's missing partials, the pair digs holes of its own that neither has, and the combination comes out louder and rounder than either — including the bright one.

 What a slow start costs a trumpet, in its own settling time. The amplitude of a trumpet's A4 resonance — the 4th impedance peak, of Q 37, time constant 27.1 milliseconds — driven from rest by a pressure that rises over 10, 40, 80, 140 milliseconds, against the step every earlier figure has assumed. The step reaches 90 per cent of its final amplitude in 62.5 milliseconds. A ramp of 140 takes 157.8, which is 95.3 more — and that excess is 68 per cent of the ramp's own length. Across the whole range a player works in the excess is a little over half the ramp: 0.52, 0.56, 0.61, 0.68 at 10, 40, 80, 140 milliseconds. So the tongued attack is not something added to the note. It is the step, which is what has been computed all along, and what has a price is its absence. Rhythm and metre

What the tongue actually removes

The question this essay was written against asked for an impulse: a tongued attack, a martelé stroke and a struck key all deliver one before the steady drive begins. The arithmetic refuses the framing. A tongue release does carry energy at the note's own frequency, and it is worth 0.17 milliseconds on a trumpet against a settling time of 62 — capped at about 1.13 over the resonance's Q. What articulation is worth is the ramp it removes, and that is about half the ramp's own length: 44 milliseconds, and very nearly the same 44 on every wind instrument in the collection.

 The census with the criterion moved under it. Every instrument's speaking time in milliseconds, against the fraction of the steady amplitude counted as speaking. The criterion is in the wind instruments alone: a resonance takes −ln(1−p)·Q/(πf) to reach a fraction p, so those lines rise across the whole picture, while a bow's capture and an exciter's contact contain no criterion at all and are flat. Every earlier figure sits at 0.9, where the wind instruments are the slowest things in the collection by a factor of 20.5. At 0.05 they are the fastest: a violin's G3 string is the slowest at 15.5 milliseconds and a trumpet takes 1.4. The three clusters cross at a criterion between 0.18 and 0.39, which is inside the range the perceptual measurements work in — their three named criteria are 15 decibels below peak, 6 decibels below peak, and ninety per cent — and the settling figures have only ever used the third of them. Perception and the listener

Read at two different heights

Ten placements of these figures, one value: the settling criterion is nine tenths in every one of them, and nothing is measured behind it. It is a multiplicative constant only inside the mechanism that has it — a bow's capture and an exciter's contact contain no criterion at all — so moving it rescales one of three clusters against two that stand still. The most-quoted number here, a factor of sixty-nine between the instrument's account and the listener's, is 5.9 at the criterion the listener's own measurements use, and the ordering an earlier essay was written about does not exist below a fifth.

 Which pairs blend is a question about the note. The level at which a doubled pair's composite changes owner, drawn for all 15 pairs of 6 radiators over 2.6 octaves from 131 to 784 hertz. A pair blends when that level is inside the shaded band, which is the twenty-four decibels either way two players can manage; a curve outside it, or absent, is a pair one instrument owns at every balance. 6 of 15 pairs blend at the bottom of the range and 12 at the top. Every filter in this collection is fixed in frequency and the fundamental is not, so a radiator's shape is a function of pitch and so is everything computed from two of them — the blend ranking at the bottom and at the top disagree on 70 of 105 comparisons, which is more than half, so the order has turned over rather than merely shuffled. Timbre and acoustics

The blend table has a row for every note

Eight earlier essays sound their instruments at one note, and one of them says why that cannot be innocent: every filter here is fixed in frequency and the fundamental is not. Swept over four octaves, the number of pairs that blend doubles from six to twelve, the ranking turns over rather than shuffles — seventy of a hundred and five comparisons swap — and a clarinet with an oboe goes from the best pair in the collection to the eleventh.

 Four players on three notes, every arrangement. The 36 ways of putting 4 players on a 3-note chord so that every note is covered, ranked by roughness, all at one total loudness of 26.9 sones. Each row is shaded by which note carries the pair. The best is flue | clarinet+violin | oboe and the worst is clarinet | oboe+flue | violin, a factor of 2.21. Every earlier essay puts exactly one player on each note, which is a permutation; a doubling makes the arrangement a surjection instead, and the doubled note sounds neither of its two players but the composite they make. Which note gets the pair explains 7 per cent of the spread here and which players sit on the lowest note explains 89: the fourth player is a much smaller decision than the three that were already there. Instruments and their design

The fourth player is a spectrum, not a decision

Six earlier essays put exactly one instrument on each note, which makes an arrangement a permutation — and the commonest operation in orchestration is a doubling, which does not. Four players on three notes give thirty-six arrangements instead of six, and the extra choice turns out to be the smallest thing on the page: which note carries the pair explains three per cent of the spread and which players sit on the bass explains eighty-nine. A doubled note can be priced as one player, and which one is not the one a spectral account would have named.

 What a stopped tube tuned to the fundamental does to a marimba bar's partials. Power radiated from the mouth of a 311 millimetre stopped tube of 56 millimetre bore, tuned so that its first resonance is the bar's fundamental at 262 hertz, plotted against multiples of that fundamental and referred to what it does at the fundamental itself. Its quality factor there is 80. A stopped tube resonates at the odd multiples and presents an infinite input impedance — a rigid lid — at the even ones, so a partial's fate is decided by the parity of the ratio the arch put it on. The bar's own partials are marked: 1 at 0.0 decibels, 4.00 at -38.5 decibels, 9.03 at -13.1 decibels. Instruments and their design

The tube shuts on the partial the arch placed

A stopped tube tuned to a bar's fundamental resonates at every odd multiple of it and presents a rigid lid — an infinite input impedance — at every even one. A xylophone's arch puts its second partial on 3, which is a resonance, and the tube passes it within five decibels. A marimba's puts it on 4, which is an antiresonance, and the tube takes it thirty-eight decibels down. Same tube, opposite answers, and the difference is parity.

 one measured slendro is the scale least committed to a spectrum. Each scale drawn, placed by how smooth it is against 2000 random scales of the same size, under 4 spectra. Low is smooth. Every one of them lands in the smoothest third under every spectrum, so the account that a spectrum chooses a scale survives as a statement about levels. What separates them is the length of each row's line, which is how much the answer moves when the instrument changes: one measured slendro spans 2.9 percentile points and Rast, Turkish theory spans 17.6, a factor of 6.0. The scale usually offered as the case for spectral matching is the one whose standing barely depends on the spectrum, and Rast, Turkish theory is the one that depends on it most. Scales and modes

The scale least committed to its own instrument

The essays on scales beyond twelve draw scales without spectra and spectra without scales, and none of them had put a tradition's own degrees through the roughness model. Doing it for six scales and five spectra says the account survives — every tradition lands between the eighth and the twenty-sixth percentile of random scales of its size, and under a pure tone not one of them does. But the scale whose standing moves least when the instrument changes is the gamelan's, at 2.9 percentile points, and the one that moves most is the five-limit diatonic at 17.4. The scale the argument is usually made about is the one it explains least.

 The bare bore does not care which valve is down. The distance in cents of partials 2 to 8 from the harmonic series that fits them best, at each valve combination of a B♭ trumpet, computed with the mouthpiece in place and again with the bore alone. The bare bore is flat — 0.78 cents from best to worst across the whole set — so the length changes nothing. With one cup serving all of them the same bore runs 19.53 to 24.83 cents, a spread of 5.30, because the cup's popping frequency stays at 642 hertz while the series underneath it drops. Timbre and acoustics

One cup and seven lengths

A trumpet is seven tubes with one mouthpiece serving all of them, and the harmonicity of its resonance series is a different number in every position — 19.5 cents open, 24.8 with all three valves down. The bare bore is flat across the same seven lengths to within eight-tenths of a cent, so none of it is a length problem. It is the cup, standing still while the series walks past it, and pressing all three valves does to the series exactly what fitting a mouthpiece 56 per cent of the catalogue depth would do.

 There is less to collapse the higher the note is. The same string model struck at 80 decibels on nine fundamentals, an octave apart. The heavy line is how far the spectral centroid falls between the strike and the note's death, in semitones — the whole of the colour drain, in the unit a musician has for pitch. It runs from 25.2 semitones at A0 to 6.6 at C8, a factor of 3.8, and it falls monotonically. The reason is the dashed line, which is how many partials exist at all: 727 under the audio ceiling at A0 and 4 at C8. From C7 upward a struck note has fewer partials than the ear could have resolved — 9 against 10 — so there is nothing left for a collapse to take away. Timbre and acoustics

The collapse belongs to the bass

Every envelope drawn until now has no pitch in it: the decay model counts partials rather than measuring them in hertz. Put a fundamental in and the collapse it describes shrinks monotonically up the keyboard, from 25.2 semitones of colour drain at A0 to 6.6 at C8, because a partial has to fit under twenty kilohertz to exist and the top note has four of them. Above C6 a struck note has fewer partials than the ear could have resolved, and there is nothing left for a collapse to take away.

 Where the register break falls on a tenor's page. The two measured laryngeal crossings — 330 hertz going up and 294 coming down — read as WRITTEN notes, against the pitch standard the part is performed at. The crossings are frequencies and do not move; the notation does, so the seam slides down the stave by exactly the interval the standard rises. At A392 the upward crossing is written F♯4, at A415 it is F4, at A440 E4 and at A465 E♭4 — a minor third of movement across four centuries, on a part nobody rewrote. Across the range drawn the seam passes 4 written semitones. The shaded horizontal band is the tenor's written compass, C3 to A4; the seam is inside it at 7 of the 7 documented standards drawn. Pitch and tuning

A standard moves the page, and not the seam

Every earlier essay has priced a pitch standard against something with a fixed length in it. A voice has none, so nothing about it changes at all — what changes is where the written note falls against a break in the larynx that is a frequency and stays put. At A415 that break is written F4, at A440 it is E4 and at Chorton it is E♭4: a minor third of movement across four centuries, on a part nobody rewrote.

 Every pitch standard, given the width 8 degrees gives it. Each documented standard drawn not as a point but as the band an ensemble occupies while the room warms by 8 degrees: the air columns sharpen by 23.3 cents, the steel strings flatten by 20.1, and 13.8 cents of spread inside each wind instrument's own register cannot be pulled out because it is a gradient along the bore rather than an offset. The band is 57 cents wide, and 5 of the 6 adjacent steps in the whole record are narrower than it — which is to say that 5 of the distinctions four centuries of committees argued about are smaller than the pitch spread inside one orchestra on one evening. Pitch and tuning

A standard is a point, and a performance is a band

Nine earlier essays draw every pitch standard as a single number, because none of them has a temperature in it. An air column sharpens as the room warms and a steel string flattens, at 2.95 and 2.49 cents a degree; add the 13.8 cents of spread inside one wind instrument's own register and eight degrees makes an orchestra 57 cents wide. Five of the six steps in four hundred years of pitch standards are narrower than that.

 Partials 3, 4, 12 of "hod", over one vibrato cycle. The level of three partials of a 220 hertz note on the vowel in "hod", each about its own mean, over one cycle of a vibrato of ±71 cents at 6.0 hertz. The pale curve is the frequency deviation itself, for phase reference. Partial 3 at 660 hertz swings 4.22 decibels and peaks with the frequency; Partial 4 at 880 hertz swings 0.41 decibels and peaks twice a cycle; Partial 12 at 2640 hertz swings 8.62 decibels and peaks against it. The formants of this vowel are at 730, 1090, 2440 hertz and do not move; a partial below one rises as the frequency rises and one above it falls, so the modulations of a single note run in opposite directions at the same instant. Instruments and their design

The partial that gets louder as it goes sharp

Eleven earlier essays sweep a set of partials and hold their amplitudes still, and a real tract does not move with the fundamental. Put the formant account and the sweeping account together and every partial acquires an amplitude modulation at the vibrato rate: 0.07 decibels on the fundamental and 8.6 on the twelfth partial of the same note. They are in phase below a formant and anti-phase above one — not ninety degrees apart — and the whole note swings 0.82 decibels, because they cancel.

 14 players summed, against the one at their average onset. Each thin line is one player's rising envelope, started at its own moment, with a spread of 30 milliseconds about the beat and a 90-millisecond attack. The heavy line is the section: nominally identical sources add incoherently, so their powers add and the sum is the root of the mean of their squares, drawn here as a fraction of the section's own peak. The dashed line is the single player who started at the section's average onset. The section reaches the 6 dB below peak criterion at 18.1 milliseconds and that player at 27.6, a difference of 9.5. The section is early because the players who started first are already sounding while the average one is still building, and nothing a late player does can make the sum quieter. Rhythm and metre

Twelve violins are more punctual than one

Every essay until now treats a part as one player, and an orchestral part is a dozen. Sectioning does two things at once and only one of them was expected: it pulls the part's heard moment forward, by four milliseconds against a map spanning twenty-six, and it makes the part's arrival more accurate by very nearly the root of the number of players. So the map of required leads applies to an orchestra better than it applies to a quartet, and the case where it fails is three trumpets rather than fourteen violins.

 One attack time, three shapes, 48 ms of disagreement. Three amplitude envelopes with the same 90-millisecond attack time, which is the only quantity the published tables report. a resonator from a step rises as one minus a decaying exponential; an excitation ramping rises in a straight line; a ramp through a resonator is a raised cosine. Each is normalised so that its own 10-to-90 per cent rise takes exactly 90 milliseconds, so all three are the same measurement. The horizontal rules are the three criteria a heard moment is read off. At 6 dB below peak the three shapes put the heard moment at 28.5, 56.4, 76.3 milliseconds — a spread of 48, on one attack time, from a property nothing in the table records. Perception and the listener

An attack time is not an attack

Eight earlier essays read a heard moment off an envelope, and every one of them used the same envelope shape without saying so: the source table records one curve for all nine of its families, and the map's own arithmetic does not carry the parameter at all. A published attack time fixes a ten-to-ninety time and nothing else. Under the two other shapes the same measurement admits, every millisecond computed so far doubles — and the constructed passage called inaudible earlier becomes three times a listener's threshold.

 The partials the air makes, on the series the bore actually has. The input impedance of a trumpet, with the partials wave steepening manufactures from its A4 at 428 hertz drawn on it as vertical marks. The steepening is a distortion of one periodic waveform, so it makes its partials at exact integer multiples — 856, 1285, 1713, 2141, 2569 hertz. The bore's own resonances are not at integer multiples of anything: the peaks the manufactured partials aim at sit at 886, 1346, 1812, 2278, 2741. So every manufactured partial lands flat of the peak it might have used, by 59, 81, 98, 108, 112 cents — 2.6, 2.7, 3.0, 4.0, 4.9 half-widths of the peaks in question, which is outside the half-power point of every one of them. The dashed line is where the bell stops reflecting, at 2945 hertz; above it there is no peak to land on or miss. Instruments and their design

Which notes go brassy first

Wave steepening manufactures partials at exact integer multiples of the note being played. A brass instrument's resonances are not at integer multiples of anything, and twelve earlier essays have measured how far off they are without ever putting the two on one axis. Put them there and a manufactured partial never lands on a peak — never once, at any note, by a miss that is the same number of hertz every time. So the alignment cannot be what decides which notes blare, and the thing that does turns out to be the bell.

 Four throats on one instrument, drawn to scale. 4 bores 148 centimetres long, drawn to scale in radius and length, differing only in the radius of the tube itself — 6.0, 9.6, 14.0, 22.0 millimetres across at the throat. The mouth is held at 124 millimetres, which is what a maker changing a leadpipe actually does. A consequence is that the flare ratio falls from 20.7 to 5.6 and the bell's own boundary with it, from 5906 hertz to 1158. Beside each is the fourth resonance and how sharp it is: 427 Hz at Q 23, 428 Hz at Q 34, 427 Hz at Q 44, 421 Hz at Q 51. The notes move by 27 cents across the whole sweep and the sharpness rises by a factor of 2.2. Instruments and their design

The throat that decides both

Eight earlier essays draw a tube eleven millimetres across and none of them draws another. Opened from six millimetres to twenty-two, it does exactly what was predicted to the loss in the walls — and it moves the radiation loss three times further, in the opposite direction, and monotonically, which the bell does not. So the two ends of a brass instrument do not divide the two losses between them. One end owns one loss and both ends own the other.

 Two registers 50 microseconds apart on one string. The partials of a 355-millimetre string plucked at 32 and 50 millimetres from the nut, drawn twice: once with the two quills releasing together, once with the far one releasing 0.05 milliseconds later, which is 0.026 of this string's period. A delayed release turns partial n through 2·pi·f_n·dt, so the rotation is proportional to the partial number: the fundamental is turned 9 degrees and partial 24 is turned 226. Simultaneous, the pair is missing partials 17 and 20 — holes it digs for itself where the two combs are equal and opposite. Staggered, it is missing none of them: a rotation of anything at all takes two amplitudes out of opposition. The partials neither comb can excite at all — 7, 11, 22 — are filled either way, because where one comb is zero the sum is the other one whatever its phase. The fundamental's gain over the far register alone falls from 4.36 decibels to 4.34. Instruments and their design

The interval between two quills

Two jacks on one key are voiced separately and do not let go at the same instant. That interval turns each partial of the later pluck through an angle proportional to its number — so it leaves the fundamental alone and inverts the twentieth partial, and the holes the pair digs for itself vanish at a hundredth of a period. What a regulator can tolerate turns out to be one fixed fraction of a period at every pitch, which on a five-octave instrument is a factor of sixteen in milliseconds.

 Where a bar and its pipe stop being two things. The two normal modes of a bar and a resonator tuned to it, against how strongly they are coupled, at 262 hertz. Below a threshold the pair has one frequency and two different decay rates — the pale curves, which are the damping splitting rather than the pitch — and above it the frequencies separate. The threshold is exact and it is not a matter of degree: it is where the coupling rate equals half the difference between the two damping rates, which for a bar of Q 197 against a tube of Q 80 is a coupling of 0.37 per cent. A marimba's own coupling is 0.62 per cent — 1.67 times the threshold, and not free: it is fixed by how much louder the tube makes the note, since the coupling that splits the pair is the coupling that carries the energy out. So the resonator model's assumption that the tube is a filter downstream of the bar is wrong at middle C, and it is wrong by less than a factor of two. Instruments and their design

A bar and its pipe are one object

Three earlier essays treat a marimba's resonator as a filter the bar's output passes through, and both of them said in their own caveats that the coupling was not modelled. It is here, and the debt was right: the coupling is 1.67 times the threshold at which the pair acquires two frequencies instead of two decay rates, so the tube is not downstream of anything. Every consequence of that is smaller than the peaks it would have to be seen between.

 Where a woodwind's A♭3 leaves it, below its corner and above it. The same fingering — 6 holes open on a 15-millimetre bore 567 millimetres long, sounding A♭3 at 207 hertz — drawn twice, with each opening's circle scaled by the share of the radiated power that leaves through it. At 400 hertz 78 per cent of it leaves through the first open hole, the bell takes 0 per cent, and the number of apertures really doing the radiating is 1.6; At 2600 hertz 6 per cent of it leaves through the first open hole, the bell takes 49 per cent, and the number of apertures really doing the radiating is 3.3. The lower frequency is below this fingering's corner and the higher one above it: below the corner the instrument is a short tube with one opening at the end of it, and above the corner it is the whole lattice at once. The power-weighted station — where a listener would say the sound is coming from — moves from 392 millimetres to 515. Instruments and their design

Where a woodwind actually sounds from

Every number so far is read at the mouthpiece, and the corner's whole musical meaning is at the other end. Run the same solver forwards and it gives the flow leaving every hole — from which a clarinet's radiating aperture turns out to be a function of fingering and of frequency, but not the way it was predicted to: the fingering sets how far the aperture opens, almost exactly to the number of open holes, and barely moves the frequency at which it does.

 Scored the way these figures score it, a clarinet is the worst of the six. One close triad at 70 decibels through 17 registers, drawn once for each of the 6 spectra to hand. The score is the share of every partial written, which is the quantity the register figure published. pure 100 per cent at best, string 79 per cent at best, clarinet 54 per cent at best, reed 71 per cent at best, bell 52 per cent at best, organ 72 per cent at best. A clarinet's four even partials are twenty-eight decibels below its odd ones and are inaudible beside their own neighbours before any chord is built, so counting them in the denominator makes the spectrum that survives its own masking best look like the one that survives it worst. Perception and the listener

A clarinet keeps what a string loses

Every masker, probe, chord, line and texture until now is eight partials falling as 1/n, and it was not even an option a placement could pass. Sweeping the six spectra to hand says the clarinet is the worst of them — 54 per cent of itself at best against a string's 79 — and that answer is an artefact of the score. Counted against what each note keeps on its own, the clarinet keeps 100 per cent where the string keeps 79, because its components stand a twelfth apart rather than an octave. The missing parameter was the spectrum; the second missing parameter was the denominator.

 One doubling, held down a phrase. Where a single held arrangement of 4 players on 3 notes stands among the 36 at each chord of a 5-chord passage, best at the top, with what each chord would rather have named along the bottom. The held answer is flue pipe · trumpet · clarinet+violin, and it is the chord's own first choice at 4 of 5 of them. Holding it costs 16.2 per cent of the passage's roughness against re-scoring every chord — which is 2.4 per cent of the range the choice actually spans, since the arrangements at one chord differ by a factor of 7.8 on average. The cost is not spread over the passage: 1 chord carries nearly all of it. Instruments and their design

An orchestrator doubles a line, not a chord

Three earlier essays made the objective a functional over a passage and a later one went back to holding one chord still. Put the doubling back into time and the retreat turns out to have been cheap: one arrangement held down a five-chord phrase is that phrase's own best answer at four of its five chords and costs 2.4 per cent of the range the choice spans — while the forward mask named earlier as the third temporal constant reaches for twenty milliseconds rather than two hundred, and cannot change the answer at any pace at all.

 The attack is the balance dial, turned by the clock. The level of a violin against a clarinet on one note at 392 hertz, moment by moment through the attack, with both players starting together. Two envelopes rising at different rates are a balance, so this axis is the same dial a conductor turns — and its whole travel is 6.02 decibels, which is twenty times the log of the ratio of the two attack times, 45 against 90 milliseconds, and nothing else. The pair does not begin as one player alone: both envelopes leave zero at the same slope ratio, so the dial starts at a finite offset rather than at silence. The dashed line is the balance at which the composite changes owner, -3.48 decibels — inside the travel, so the note belongs to a clarinet for its first 29 milliseconds and to a violin for the rest of its life. Timbre and acoustics

The blend arrives before the note does

Nine essays on spectrum draw a steady state, and the strongest cue that two instruments are two instruments is that they do not start together. Two envelopes rising at different rates turn out to be a balance — the same dial an earlier essay swept — so the attack is that dial moved by the clock, and its whole travel is fixed at twenty times the log of the two attack times. It is six decibels for a clarinet with a violin against a crossing twelve to twenty-two decibels out, so one pair in ten changes hands during its own attack, and which one depends on a convention rather than on the instruments.

 The setting a listener reports is not the interval they preferred. Five published preferences for an interval size, and the value a listener would have to play in a key context for that preference to be what they hear. The key pulls a heard interval back toward the scale, so a listener adjusting until it sounds right has to overshoot — and the reported setting, which is what they played, exaggerates the preference. The corrections run from 1.6 cents to 12.8 on notes of 0.25 seconds. The largest is nearly a syntonic comma, on a preference of thirteen cents, which is to say that the correction is bigger than the effect it is a correction to. Intervals and chords

The setting is not the preference

Every published number for an interval listeners prefer — the pure third a quartet is said to find, the raised leading note, the harmonic seventh — is a value somebody adjusted until it sounded right. A listener adjusting inside a key is adjusting through the posterior computed just before, so the value they stopped at is not the one they preferred: it is the one whose heard size equals it. On quarter-second notes the correction for a pure major third is 12.8 cents, which is nearly a syntonic comma and is larger than the 13.7-cent preference it corrects.

 A loudly struck note hides the beats it is being struck to reveal. The share of the fluctuation in its own auditory filter that belongs to each of the first five members of a mistuned octave's beat family, against how loud the note is. The filter's lower skirt shallows by about 38 per cent of its 51-decibel value every ten decibels, so more of the neighbouring partials get into the filter and the pedestal each member sits on grows. The mean share falls from 0.55 at 40 decibels to 0.07 at 100. Past about 90 decibels the curves are flat because the model's skirt is clamped rather than because anything stops changing — that clamp is the model's floor and not a measurement. Pitch and tuning

How hard the note was struck

The auditory filter is not a fixed shape: its lower skirt shallows by about 38 per cent of its reference value every ten decibels, so a loud note is analysed through a wider filter than a quiet one. Every share computed so far was quoted at a moderate level, and a tuner does not strike moderately. Recomputed, the mean share of a mistuned octave's filter falls from 0.55 at forty decibels to 0.07 at seventy, and the count of separable beats goes from two to none — which is a prediction too strong to be right, and the way it fails is the useful part.

 A struck octave becomes countable a second after the strike, or never. The number of separable beats a mistuned octave on A3 delivers, second by second after both notes are struck at 80 decibels, with every partial dying at its own rate (a 12-second fundamental, losses rising as frequency to the power 0.7). Read with the filter broadened by the level of the whole note, which is how the level-dependent count was first computed, the count is zero at every instant: the partials fall below audibility before the filter has narrowed enough to separate them. Read with the filter broadened by the level inside itself, which is what the published parameterisation was fitted against, the count is 0 at the strike, reaches 2 at 1.0 s and falls to nothing at 3.3 s. Pitch and tuning

Counted in the decay, or not at all

A mistuned octave struck hard delivers no countable beat at the strike, and the reconciliation offered for that was that a tuner listens to the decay. Computed through a real decay it fails on its own terms: the partials fall silent before the filter has narrowed enough to separate them. It succeeds only when the filter is broadened by the level inside it, which is what the published parameterisation was fitted against — and then the window opens at a twelfth of the note's life and shuts at a quarter.

 A string quartet's open strings are five keys of a Pythagorean keyboard. The five pitch classes a string quartet's open strings sound — C, G, D, A and E — laid out as the chain of fifths they are tuned along, outward from the A the ensemble is given, with each fifth pure. The bars give each string's departure from the same note on an equal-tempered keyboard: C −5.87 cents, G −3.91 cents, D −1.96 cents, A 0.00 cents, E +1.96 cents. Above, the strings each instrument owns: the violin G, D, A, E; the viola C, G, D, A; the cello the same four an octave lower. The cello's C2 is 0.221 hertz below the keyboard's, and the widest span of the chain, from the cello's C to the violin's E, is a Pythagorean third and two octaves, 21.5 cents wider than a just one. Pitch and tuning

The tuning a string quartet cannot change

A string quartet can put every stopped note wherever it likes, and it has five pitches it cannot move once the pegs are turned: the open strings C, G, D, A and E, tuned in pure fifths from the A it was given. That makes its only fixed tuning a five-key Pythagorean keyboard — the cello's C nearly six cents below a piano's, and every third two open strings can make a syntonic comma from just. Counted key by key, the clash is worst in G, C and F, the keys that use every open string, and absent from A and E.

 A tempered fifth is a beat a second on the violin and one in four and a half seconds on the cello. How fast each open fifth of a string quartet beats when it is narrowed by 1.955 cents, the narrowing that meets an equal-tempered keyboard. The beat is the lower string's third partial against the upper string's second, so it is proportional to the lower string's frequency. Cello C2–G2: 0.22 a second, one beat every 4.5 seconds; cello G2–D3: 0.33 a second, one beat every 3.0 seconds; cello D3–A3: 0.50 a second, one beat every 2.0 seconds; viola C3–G3: 0.44 a second, one beat every 2.3 seconds; viola G3–D4: 0.66 a second, one beat every 1.5 seconds; violin G3–D4: 0.66 a second, one beat every 1.5 seconds; violin D4–A4: 1.00 a second, one beat every 1.0 seconds; violin A4–E5: 1.49 a second, one beat every 0.7 seconds. The slowest, the cello's C2–G2, is 6.7 times slower than the violin's A4–E5. Pitch and tuning

The cello cannot hear its own tempering

Narrowing a quartet's fifths to meet a piano is one number, 1.96 cents a fifth, and it is a different beat on every string: once every two thirds of a second on the violin's A–E and once every four and a half seconds on the cello's C–G. Set by ear for two seconds a fifth, the violin's E lands within two thirds of a cent and the cello's C within 5.7 — which is as large as the Pythagorean error the tempering was meant to remove. The string whose tuning is most wrong is the string whose tuning is least certain, and a cellist tuning down the chain cannot tell pure from tempered.

 A doubled pizzicato gives its note away while it is still the louder. The power of a violin plucked, against a flue pipe holding the same note at 392 hertz, through the first 600 milliseconds of the pluck, with the pluck starting 12 decibels up and its fundamental decaying over 1 second. With each partial losing level in proportion to its number, the composite stops resembling the pluck at 70 ms, when the pluck is still 5.2 decibels the louder. With every partial fading together it would keep the note until 543 ms. The dashed line is the balance at which the steady-state doubling changes owner, minus 20.6 decibels: the release crosses the owner long before its balance gets there, because what hands the note over is the pluck's upper partials going, not its level. Timbre and acoustics

A doubled pizzicato gives its note away early

The attack turns the balance between two players on one note by a few decibels and stops. A pluck does not stop — every partial of it decays, so a pizzicato doubled by a held instrument walks the balance for the whole note, and the expectation was a handover as slow as the decay. It is fast. A one-second pizzicato over a flute loses its note in 70 milliseconds, while it is still five decibels the louder, because what hands the note over is its upper partials going first. A uniform fade would have kept it eight times as long.

 The ninety per cent was a ceiling. The share of 188 scheme bars read right on both key and degree with a bass note worth 1.5, for three bass lines — every chord's root, a line moving to the nearest chord tone, every chord's fifth — and two ways of using the bass: rewarding the triad rooted on it, or any triad containing it. Wide bars are with no profile in the emission, narrow bars with the profile alone. root line, root rule: 89% and 86%; root line, member rule: 60% and 29%; smooth line, root rule: 68% and 46%; smooth line, member rule: 61% and 47%; sixfour line, root rule: 3% and 17%; sixfour line, member rule: 52% and 30%. The reading with no bass at all is 47%. Scales and modes

A bass line is not a list of roots

Every bass note the key-finder has been given was its chord's root, and under that line a bass cue reads 89 per cent of scheme bars on the right degree. Give the same chords an economical bass that moves to the nearest chord tone, as a keyboard reduction would, and nearly half of them are inverted. The cue that rewards the triad rooted on the bass then reads 68 per cent at best and worse as it is trusted more; the cue that rewards any triad containing the bass cannot be fooled and stops at 61. The same inverted line does one thing the roots never did: it puts the leading note of each new key at the bottom, and finds the rondo's modulations.

 Weighted by the pairs a melody sounds next to each other, every seven-note scale is rougher than most. Each scale placed by how smooth it is against 2000 random scales of the same size, under 4 spectra, with each pair of its degrees weighted by how often the plainest melody its ascent and descent permit sounds the two — here the pairs a melody sounds next to each other. Low is smooth. Raga Bhupali: 40.3 to 56.8, spread 16.5; Raga Deshkar: 25.1 to 37.5, spread 12.4; one measured slendro: 31.8 to 44.5, spread 12.7; Rast, Arabic theory: 92.0 to 98.6, spread 6.6; Rast, Turkish theory: 91.8 to 97.8, spread 6.0; the diatonic major, tempered: 90.7 to 97.0, spread 6.3; the diatonic major, five-limit just: 91.8 to 97.8, spread 6.0. Scales and modes

The smoothness is in the skips

Traditional scales come out smoother than random scales of their size because every pair of their degrees is counted once. Count only the pairs a melody on the scale actually sounds next to each other, and every seven-note scale here is rougher than ninety per cent of random ones, under every spectrum, and under a pure tone as well. The smoothness is carried by the pairs a melody reaches by skipping — its thirds and above all its fifths, which are smoother than all but two random scales in two thousand. Counted by their own ascents and descents, the two ragas that share one set of notes stand in different places at last: Deshkar's skipped Re buys a smoother ascent and costs it the fifths.

 An unaccompanied quartet settles where its open strings put it. The average pitch of a quartet correcting toward itself over 480 corrections, in cents from the note it was given, averaged over 24 runs. With no pull from the open strings the ensemble random-walks, and the shaded band is how far: 3.7 cents root-mean-square by the end. With each open string pulling the notes that share its pitch class at a weight of 0.05, the ensemble settles at −0.97 cents in A major, against −1.01 from the open strings' weighted mean; −2.46 cents in C major, against −2.42 from the open strings' weighted mean; −2.53 cents in E♭ major, against −2.54 from the open strings' weighted mean. Pitch and tuning

An open string pulls the quartet flat

Once the tuning note has stopped, a quartet corrects toward itself and nothing holds its pitch. But four of its pitches do not move: the open strings, on a Pythagorean chain from C 5.9 cents flat to E 2.0 sharp, each ringing when a stopped note shares its pitch class. Give that sympathy a weight of a hundredth of a correction and it beats the random walk within a movement. The quartet settles flat in every major key — by 0.9 cents in E, 2.5 in C and 2.6 in A♭ — and in A♭ major the cellist's tuning scatter moves the whole ensemble by 1.7 cents.

 Four pure fourths and a pure third leave a comma on the top strings. Each of a guitar's six open strings, in cents from the note its own frets make, when the strings are tuned against each other three ways. fretted unisons: E2 +0.00, A2 +0.00, D3 +0.00, G3 +0.00, B3 +0.00, E4 +0.00. harmonics, pure third on G–B: E2 +0.00, A2 −1.96, D3 −3.91, G3 −5.87, B3 −19.55, E4 −21.51. harmonics, B from the low E's twelfth: E2 +0.00, A2 −1.96, D3 −3.91, G3 −5.87, B3 +1.96, E4 +0.00. Tuned with pure fourths and a pure third the high E closes two octaves 21.51 cents flat, a syntonic comma; taking the B from the low E's twelfth lands the high E exactly and moves the error into the third between the G and B strings. Pitch and tuning

A guitar tuned by harmonics hides a comma

A quartet's stopped notes go wherever its players put them, so its open strings can sit on a pure chain. A guitar's stopped notes are fixed by frets that are already an equal temperament, and its six open strings — four fourths and a third — close two octaves exactly a syntonic comma short when tuned pure, since (4/3)⁴·(5/4) is 4 × 80/81. Tuning by harmonics decides where the comma goes, not whether: a pure third puts it in the octaves of an open E major chord, 17.6 cents narrow; a B from the low E's twelfth puts it in the G–B third, 21.5 cents wide.

 A string that decays twice opens its count at once and shuts it early. The count of separable beats a mistuned octave on A3 delivers after both notes are struck at 80 decibels, with each filter read at the level inside it, under one exponential decay of 12 seconds and under two stages — a prompt sound of 1.5 seconds carrying all but the last 20 decibels, and an aftersound of 12 seconds. One exponential: open from 1.00 s to 3.30 s, 11.8 beats. Two stages: open from 0.15 s to 1.77 s, 8.8 beats — and the single exponential struck 20 decibels softer closes at 1.77 s. Pitch and tuning

A string that decays twice is counted early

A mistuned octave's beats were found countable only between a twelfth and a quarter of a note's life, on a note decaying once. A piano string decays twice, a fast prompt sound over a slow aftersound, and the prediction was that this would open the count sooner and close it later. It opens sooner — at a seventh of a second rather than a second — and closes exactly where a single decay struck twenty decibels softer closes, so at 80 dB it holds 8.8 beats instead of 11.8. The count now rises with the strike to 90 dB, and a tuner who strikes hard is right.

 Weighted by the ring of the note before, a note every 0.6 seconds, Raga Deshkar moves least. Each scale placed by how smooth it is against 2000 random scales of its size under 4 instruments, each with its own spectrum and its own ring, with every pair of notes its plainest melody sounds weighted by how much of the earlier note is still sounding when the later one begins, a note every 0.6 seconds. Low is smooth. Raga Bhupali: plucked string, 6 s 53.6, long-ringing string, 12 s 35.8, blown note, 2 s hall 41.3, free bar, 4 s 58.5; spread 22.8; Raga Deshkar: plucked string, 6 s 39.0, long-ringing string, 12 s 25.9, blown note, 2 s hall 26.1, free bar, 4 s 42.0; spread 16.1; one measured slendro: plucked string, 6 s 42.9, long-ringing string, 12 s 27.8, blown note, 2 s hall 32.6, free bar, 4 s 46.4; spread 18.6; Rast, Arabic theory: plucked string, 6 s 92.0, long-ringing string, 12 s 65.2, blown note, 2 s hall 92.3, free bar, 4 s 96.1; spread 30.9; Rast, Turkish theory: plucked string, 6 s 88.0, long-ringing string, 12 s 58.1, blown note, 2 s hall 90.8, free bar, 4 s 93.6; spread 35.5; the diatonic major, tempered: plucked string, 6 s 84.8, long-ringing string, 12 s 54.2, blown note, 2 s hall 89.5, free bar, 4 s 91.3; spread 37.0; the diatonic major, five-limit just: plucked string, 6 s 87.7, long-ringing string, 12 s 56.7, blown note, 2 s hall 90.3, free bar, 4 s 92.9; spread 36.2. Scales and modes

A scale is committed to how long its instrument rings

A traditional scale's standing on the roughness model moves when the instrument changes, and that movement was read as a commitment to the instrument's spectrum. Weight every pair of notes a melody sounds by how much of the earlier note is still ringing when the later one begins, and the movement grows — the tempered diatonic's by 37 percentile points at a note every 0.6 seconds — but almost none of it is the spectrum. Put four instruments' rings on one spectrum and the scale moves 38.6 points; put four spectra under one ring and it moves 9.4. And the partials that make a fifth smooth are the first to stop sounding.

 How long a doubled pizzicato keeps its note, seat by seat, in two rooms. How long a doubled violin pizzicato on 392 hertz keeps its note against the metres from the players, the pluck starting 12 dB up and decaying over 1 s with a loss exponent of 1. a concert hall, a flue pipe: 1 → 86 ms, 1.5 → 123 ms, 2 → 226 ms, 3 → 359 ms, 5 → 445 ms, 7 → 481 ms, 10 → 506 ms, 15 → 522 ms, 20 → 528 ms, 30 → 532 ms; a concert hall, an oboe: 1 → 52 ms, 1.5 → 55 ms, 2 → 59 ms, 3 → 77 ms, 5 → 149 ms, 7 → 195 ms, 10 → 224 ms, 15 → 242 ms, 20 → 248 ms, 30 → 254 ms; a concert hall, a clarinet: 1 → 44 ms, 1.5 → 45 ms, 2 → 45 ms, 3 → 47 ms, 5 → 53 ms, 7 → 65 ms, 10 → 86 ms, 15 → 105 ms, 20 → 112 ms, 30 → 118 ms; a large stone church, a flue pipe: 1 → 440 ms, 1.5 → 578 ms, 2 → 651 ms, 3 → 728 ms, 5 → 784 ms, 7 → 803 ms, 10 → 814 ms, 15 → 820 ms, 20 → 822 ms, 30 → 824 ms; a large stone church, an oboe: 1 → 56 ms, 1.5 → 65 ms, 2 → 89 ms, 3 → 207 ms, 5 → 281 ms, 7 → 303 ms, 10 → 315 ms, 15 → 322 ms, 20 → 325 ms, 30 → 326 ms; a large stone church, a clarinet: 1 → 44 ms, 1.5 → 43 ms, 2 → 43 ms, 3 → 44 ms, 5 → 53 ms, 7 → 67 ms, 10 → 80 ms, 15 → 89 ms, 20 → 92 ms, 30 → 94 ms. The mid-band critical distance is 5.3 m in a concert hall and 2.3 m in a large stone church. In none of the 60 cases does the note return to the pluck once it has left. Timbre and acoustics

A room keeps a pizzicato from giving its note away

Doubled by a flute, a one-second pizzicato loses its note in 70 milliseconds dry, because its upper partials go first. The question left open was whether a room, whose reverberation keeps those partials alive, gives the note back afterwards. It does not give it back. It stops the note going: ten metres into a concert hall the pluck keeps it for 506 milliseconds, in a stone church for 814, and the room's own uneven decay takes back between a quarter and two fifths of that. In a room the loss law that decided everything dry matters a tenth as much, because the room's decay has become the clock.

 A bow shows the loss law a blow conceals. The same string on 130.8 hertz under three loss laws, drawn twice each: struck, and held by a continuous drive. The pale marks are the spectrum a blow produces, and they are identical in all three rows — a strike is the source spectrum and has no loss in it yet, which is why both endpoints of a struck note were found to carry nothing about the law. The solid marks are where each partial settles when a drive balances its own loss, at drive over loss, so the steady spectrum rolls off as the source's roll-off plus the exponent. At an exponent of 0.5 the held spectrum's centroid sits at 167 hertz, 5.7 semitones under the strike's 232; At an exponent of 1 the held spectrum's centroid sits at 144 hertz, 8.2 semitones under the strike's 232; At an exponent of 2 the held spectrum's centroid sits at 133 hertz, 9.6 semitones under the strike's 232. The quantity that is invisible at both ends of a struck note is the slope of a bowed one, for as long as the bow moves. Timbre and acoustics

A bow holds the number a blow hides

Two struck notes with different loss laws are identical at the strike and identical at the end, which is why separating them at all meant looking in the middle. Drive the same two strings continuously and the loss law stops being a rate and becomes a slope: each partial settles at its drive over its own loss, so the exponent adds to the source's roll-off and sits in the spectrum for as long as the bow moves. It is 18.7 decibels of separation available from the first instant, against 41.3 that a blow delivers after four tenths of a second and then takes away.

 From partial 3 the room is the slower of the two. Decay rates in nepers a second for each partial of a note on 130.8 hertz, in a concert hall. The rising curve is the string's own loss, which grows as the partial number to the power 1. The flat-ish curve is the room's, from its reverberation time at that partial's frequency. A reverberant field is the source convolved with the room, so a partial's tail falls at the SLOWER of the two — the heavy line — and the room keeps returning energy the string has stopped making. From partial 3, at 392 hertz, the room is in charge: 6 of the note's 8 partials are held up by the room rather than let go by the string. Those are exactly the partials the string was losing fastest, which is why the room does not merely lengthen the note. Timbre and acoustics

The room is the slower of the two

A reverberant field is the source convolved with the room, so a partial's tail falls at the slower of the two rates rather than at their sum — and the room is slower for exactly the partials the string is losing fastest. Half a note's colour is gone in 0.163 seconds in no room at all, 0.313 in a concert hall and 1.441 in a stone church. The destination is identical in all three, because a room cannot hold a partial up above the fundamental it is also holding. What a hall takes away is the rate, and the rate was the whole of the identity cue.

 The drain does not stop when the key does. The note does.. Semitones of colour gone, against time, for a note on 130.8 hertz left to ring and for the same note released after 0.4 seconds onto a damper of 0.15 seconds. The two curves lie on each other until the key comes up, and the damped one then ends: the note is inaudible at 0.52 seconds with 8.7 of the free note's 9.9 semitones delivered. A damper adds one loss to every partial alike, so it adds the same number to every decay rate and leaves every DIFFERENCE between rates exactly as it was — the spectrum at each instant is the ringing spectrum shifted bodily down by 400 decibels a second. The colour goes on draining at its own rate the whole time. What the damper takes away is not the drain but the seconds. Timbre and acoustics

A damper changes the clock, not the colour

A damper is an extra loss on the string rather than a second decay, so it adds the same number of nepers a second to every partial — and adding a constant to every rate leaves every difference between rates exactly where it was. The damped spectrum at any instant is the ringing spectrum at that instant shifted bodily down, to machine precision. The colour goes on draining at its own rate; the note simply runs out of seconds, and how many it gets is written on the page as a note value and a tempo.

 A room pulls the compass apart rather than evening it out. How long a pizzicato entering 6 decibels above a held note keeps the composite spectrum, at eight pitches across two and a half octaves, heard 15 metres from the stage. no room: 0.07, 0.05, 0.06, 0.06, 0.02, 0.05, 0.05, 0.04 seconds; a concert hall: 0.61, 0.27, 0.50, 0.38, 0.01, 0.25, 0.27, 0.19 seconds; a large stone church: 1.03, 0.39, 0.79, 0.56, never, 0.33, 0.35, 0.23 seconds. Dry the figures barely move — a spread of 3.0 across the whole compass — because a room is the thing that varies with frequency and there is none. In a hall the spread is 44. The room does not scale the dry answer by a constant: it multiplies it by between four and nine times depending on the note, and at C6 it makes the pluck's position worse rather than better, because the two instruments' spectra nearly coincide there and the pluck starts only 4.0 decibels ahead instead of twelve. Timbre and acoustics

One note in the compass loses its pizzicato

Dry, how long a pluck keeps the composite spectrum barely depends on which note it plays: three-hundredths of a second at the worst pitch and seven at the best, a spread of three. In a concert hall the same eight notes spread by a factor of forty-three, and in a stone church one of them never gets the note at all. The room does not scale the dry answer by a constant — it multiplies it by between four and nine times depending on the pitch, and at the one note where the two instruments' spectra nearly coincide it makes the pluck's position worse instead of better.

 The breath is the looser ceiling nearly everywhere. How long a trained singer can hold a phrase on one breath, across a compass and at four dynamics, against the 8-second ceiling the psychological present puts on the same phrase. The flow through the folds rises with pitch and with loudness, so the breath ceiling falls both ways: at 60 decibels it runs 32.6 seconds at the bottom of the compass to 21.2 at the top; at 70 decibels it runs 23.1 seconds at the bottom of the compass to 15.0 at the top; at 80 decibels it runs 16.4 seconds at the bottom of the compass to 10.6 at the top; at 90 decibels it runs 11.6 seconds at the bottom of the compass to 7.5 at the top. The shaded line is the listener's ceiling and it does not move. The breath binds only where the two lines cross — 1 of the 40 cells drawn, all of them loud and high. So the constraint everybody names when asked why a phrase is the length it is, is almost never the constraint that decides it. Form and structure

The ceiling everybody names is the loose one

Ask why phrases are the length they are and the answer given is the breath. It is arithmetic — usable lung volume over the air a note costs per second — and it comes out between fifteen and twenty-three seconds at a comfortable dynamic and between seven and twelve at a loud one. The ceiling the present moment imposes, the two-to-eight seconds inside which a stretch is heard as one thing rather than as a series, is two to three times tighter at almost every note and dynamic. A singer in an adagio is not running out of breath at the phrase end. They are running out of present.

 Three G strings, and they are not one pitch. Each instrument's four open strings, at the Pythagorean position its own chain of fifths puts them, with the uncertainty its own tuning leaves drawn as a band. The A is given and carries no error; every other string is reached from it one fifth at a time, and a fifth set by ear is set by nulling a beat whose rate falls with frequency — so the error accumulates down the chain and is worst at the bottom. violin: G3 -3.9 ± 1.77, D4 -2.0 ± 0.98, A4 0.0 ± 0.00, E5 2.0 ± 0.44; viola: C3 -5.9 ± 2.83, G3 -3.9 ± 1.77, D4 -2.0 ± 0.98, A4 0.0 ± 0.00; cello: C2 -5.9 ± 2.83, G2 -3.9 ± 1.77, D3 -2.0 ± 0.98, A3 0.0 ± 0.00. The three G strings share a pitch class and are expected to sit 2.5 cents apart; the two C strings 4.0. Pitch and tuning

The quartet settles at two pitches, not four

The quartet's open strings have been treated as five fixed pitches on one chain, and they are not: the violin, the viola and the cello each tuned a G string by ear and the three are expected to sit two and a half cents apart. Giving each player their own strings, with their own scatter, and pulling each toward only their own, changes the ensemble's settled pitch by a hundredth of a cent. What it does change is systematic rather than random: a violin has an E string and no C, the lower instruments have a C and no E, so the quartet splits by section by a tenth of a cent in every key.

 The tempo moves it further than the touch does. One measured slendro, scored among random scales of its size under a free bar, 4 s, at five tempi and under each touch. Left to ring it runs from 29 at 0.15 seconds a note to 83 at 2.4 — a span of 54 percentile points, where the two touches differ by at most 17. So the scale is smoother than most of its size when the music is fast and rougher than most when it is slow, and how the bar is damped is the smaller decision. The two touches converge at the slow end because a bar that has died before its successor is sounding against nothing whatever the player does. Scales and modes

The tempo moves a scale further than the touch

A gamelan is played two ways on the same bars: a saron's are damped as the next is struck and a gendèr's ring over their resonators. That decision moves a slendro's standing among random scales of its size by up to seventeen percentile points, which is real. Over the tempo levels a piece actually moves through it moves by fifty-four — from the twenty-ninth percentile at a fast elaboration to the eighty-third at a slow one. The same five pitches on the same bars are a smoother-than-average scale and a rougher-than-average one, and which depends on how fast they are played.

 The scale's standing belongs to the ringing instrument. Where a measured slendro sits among random five-note scales, at three density ratios, scored three ways: the two instruments together, the fast ringing part on its own, and the slow damped part on its own. At one slow note to 2 fast ones the ensemble is at the 7th percentile, the ringing part alone at the 6th, and the damped part alone at the 95th; At one slow note to 4 fast ones the ensemble is at the 5th percentile, the ringing part alone at the 7th, and the damped part alone at the 95th; At one slow note to 8 fast ones the ensemble is at the 5th percentile, the ringing part alone at the 5th, and the damped part alone at the 95th. A low percentile is a scale smoother than most of its size. The damped part on its own is rougher than nineteen random scales in twenty, because it has almost no simultaneity for its intervals to be smooth in; the ensemble is at the ringing part's figure throughout. And the cross pairs, which are a quarter of the roughness, do not make the ensemble worse than either part alone. Scales and modes

The scale belongs to the ringing instrument

A gamelan plays two instruments on one scale at once — a saron damped at every stroke and a gendèr several times faster with its bars left to ring — and a quarter of the roughness a listener receives falls on pairs that cross between them, which no figure had computed. The cross pairs turn out to be no worse than either stream's own. What decides the scale's standing is the fast part: the ensemble sits at the sixth percentile among random five-note scales and so does the ringing instrument alone, while the damped one alone sits at the ninety-sixth.

 A damper is a loss on the string, so a room can overrule it. Decay rate in nepers a second against partial number, for a note on 130.8 hertz in a room of 2 seconds. The rising line is the string's own loss, 1.15 nepers a second at the fundamental and growing as the partial number to the power 1. The line above it is that plus the damper's 46.1, which is what the string does once the key comes up. The flat line is the room. What a listener receives is the SLOWER of the damped string and the room, because a hall goes on radiating what the string has already given it — and here the room is slower on 8 of 8 partials, from the fundamental upward. The composition proposed earlier — take the slower of the string and the room, then add the damper to whichever won — would put the damper outside the minimum, where nothing can overrule it, and would predict a note 2.54 seconds shorter than ringing where the arithmetic here predicts 0.90. Timbre and acoustics

A damper cannot reach into the room

The essay before this one proposed the arithmetic for a damped note in a hall: take the slower of the string's rate and the room's, then add the damper's to whichever won. The composition is wrong, and it is wrong in the one place that decides the answer. A damper is a loss on the string, so it belongs inside the minimum where a room can overrule it — and past about three seconds of reverberation it is overruled on every partial, so the damper removes no audible seconds of note at all.

 A harder strike buys beats only if the drop does not deepen with it. The beats a mistuned octave on A3 delivers on a string that decays in two stages, against how hard both notes are struck, with the aftersound's drop below the strike 20 dB at 80 dB and changing by a stated number of decibels for each decibel of strike. -0.5 dB per dB: 70 dB 2.5, 75 dB 5.7, 80 dB 8.9, 85 dB 11.9, 90 dB 12.2, 95 dB 11.6, 100 dB 10.9; 0 dB per dB: 70 dB 4.7, 75 dB 6.9, 80 dB 8.9, 85 dB 11.0, 90 dB 12.3, 95 dB 12.2, 100 dB 11.8; +0.5 dB per dB: 70 dB 7.2, 75 dB 8.1, 80 dB 8.9, 85 dB 9.8, 90 dB 10.7, 95 dB 11.2, 100 dB 11.9; +1 dB per dB: 70 dB 9.6, 75 dB 9.2, 80 dB 8.9, 85 dB 8.4, 90 dB 8.1, 95 dB 7.8, 100 dB 7.5. Where the drop stays put, a harder strike keeps buying beats to about 90 dB; where it deepens by a decibel per decibel, every strike past 70 dB costs beats. Intervals and chords

A firm touch buys beats until the aftersound sinks with it

A piano string decays twice, and a mistuned octave's countable beats were found to rise with how hard the note is struck — on the assumption that the aftersound always starts twenty decibels below the strike. It need not. The moment the count shuts is set by the level the aftersound starts at and nothing else, so a harder strike buys beats only if its aftersound does not sink with it. If the drop deepens by more than 0.85 decibels for each decibel of strike, the firm touch costs beats instead.

 No seventh chord can be spaced to last as long as a triad. Every inversion and spacing within 2 octaves over C3, struck at 80 dB, for two triads and five seventh chords: the bar is the longest any spacing keeps every pair's partial coincidence, and the tick is the bound set by the chord's worst pitch-class distance — the longest any presentation of that distance lasts. major triad: 1.26 s over 12 voicings, bound 1.26 set by the minor third; minor triad: 1.26 s over 12 voicings, bound 1.26 set by the minor third; dominant seventh: 0.66 s over 32 voicings, bound 0.64 set by the tone; major seventh: 0.42 s over 32 voicings, bound 0.38 set by the semitone; minor seventh: 0.69 s over 32 voicings, bound 0.64 set by the tone; half-diminished seventh: 0.69 s over 32 voicings, bound 0.64 set by the tone; diminished seventh: 0.86 s over 32 voicings, bound 0.87 set by the tritone. Every seventh chord contains a distance worse than any a triad contains, except the diminished seventh, whose distances are only minor thirds and tritones. Timbre and acoustics

A seventh chord cannot be spaced to last like a triad

A struck triad keeps the partial coincidences of all its intervals for at most 1.26 seconds, whichever way it is spaced. Run the same census over every inversion and spacing of five kinds of seventh chord and none gets near: the dominant, minor and half-diminished sevenths top out at about two thirds of a second, the major seventh at 0.42. The diminished seventh, which theory calls the least stable of them, lasts longest at 0.86 — because it is the only one with no tone or semitone among its pitch-class distances, and the worst distance a chord contains sets a ceiling no spacing can lift.

 A staccato is the direct sound's, and the room takes it within a fifth of the critical distance. A note on 130.8 Hz held 0.4 s and damped, in a room of 2 s reverberation, heard at distances from 0.02 to 5 times the critical distance: how long after the release the note takes to fall 10 dB and 20 dB. To fall 10 dB: 24 ms at the source, 333 ms far away; 0.02: 24 ms, 0.05: 25 ms, 0.1: 25 ms, 0.15: 26 ms, 0.2: 28 ms, 0.3: 35 ms, 0.5: 103 ms, 0.75: 187 ms, 1: 234 ms, 1.5: 281 ms, 2: 302 ms, 3: 318 ms, 5: 328 ms; doubled by 0.38 of the critical distance. To fall 20 dB: 49 ms at the source, 667 ms far away; 0.02: 49 ms, 0.05: 51 ms, 0.1: 60 ms, 0.15: 117 ms, 0.2: 198 ms, 0.3: 308 ms, 0.5: 436 ms, 0.75: 520 ms, 1: 568 ms, 1.5: 614 ms, 2: 635 ms, 3: 652 ms, 5: 661 ms; doubled by 0.14 of the critical distance. Where the direct sound and the room are equal, the damper's work is already hidden: the room's copy is only 20 dB below the direct sound at a tenth of the critical distance, and a 20 dB fall reaches it there. Timbre and acoustics

Only the player hears a staccato end

A damper stops a string in a seventh of a second, and in a hall the room goes on for two. A listener hears both, mixed in proportion to how close they sit, and the question was at what distance the short part stops mattering. The answer is closer than any seat. A damped note's twenty-decibel fall has doubled in length by a seventh of a hall's critical distance — 77 centimetres in a two-second concert hall — and by a quarter of it in a jazz club. The end of a staccato is something the pianist hears and the front row does not.

 The arch belongs to hearing, and the spacing only moves it. The share of a close major triad's twenty-four components that stand above what the rest of the chord masks, at 70 dB, with the root from C1 to C7, for three spectra given the same amplitude law and different frequencies: the harmonic series, a founder's bell, and a stiff string with B = 0.01. harmonic series: 0.04 at C1, peaking at 0.79 on E3, 0.42 at C7; a founder's bell: 0.04 at C1, peaking at 0.75 on C4, 0.38 at C7; a stiff string: 0.04 at C1, peaking at 0.71 on E3, 0.46 at C7. Only one of the three is a harmonic series, and all three rise out of the bass, peak in the middle of the compass and fall in the treble. Perception and the listener

The arch belongs to hearing, not to the series

A chord delivers most of its partials in the middle of the compass and loses them in the bass and the treble, and every spectrum that showed that arch was built on whole multiples of a fundamental. Give the same amplitudes to a bell's eight modes and to a stiff string's stretched partials and the arch is still there, peaking within a major third of where the harmonic series peaks. What the spacing changes is the detail: a bell crowds its tierce and quint into a quarter of a critical band in the bass and loses them, and a stiff string's stretch buys the bass back.

 A wrong bar costs the same whichever instrument it is on. What moving one degree of a measured slendro by 10 cents, flat or sharp, adds to the roughness per second of a two-instrument texture — a ringing part at 0.15 s a note over a damped one four times slower — on the ringing instrument and on the damped one. The ensemble in tune scores 7.47. Degree 1 (0¢): ringing 0.142 flat and 0.202 sharp, damped 0.163 and 0.185, of which beating 0.178; Degree 2 (231¢): ringing 0.119 flat and 0.068 sharp, damped 0.096 and 0.090, of which beating 0.093; Degree 3 (474¢): ringing 0.106 flat and 0.021 sharp, damped 0.069 and 0.057, of which beating 0.063; Degree 5 (717¢): ringing 0.079 flat and 0.072 sharp, damped 0.075 and 0.078, of which beating 0.077; Degree 6 (955¢): ringing 0.023 flat and 0.018 sharp, damped 0.019 and 0.022, of which beating 0.020. Over all ten errors the ringing instrument's cost 0.85 and the damped one's 0.85, and the wrong bar beating against the other instrument's right one comes to 0.86 on either — as much as the whole, because the intervals the error changes add as often as they save. One error costs about 1.1% of the texture's roughness. Scales and modes

A wrong bar beats the same on either instrument

The ringing instrument carries a gamelan scale's standing, so a tuning error on it was predicted to cost more than the same error on the damped instrument. Put in the beating the model lacked, and the prediction fails: moved ten cents, a degree costs the same 0.85 summed over the scale on either instrument, because almost all of the cost is the wrong bar beating against the right one on the other instrument, and a beat belongs to both of the bars that make it. Moving a whole instrument costs exactly what its five degrees cost separately, and it takes nothing from the scale's standing.

 The leap at the barline is not in the harmony: it is in which octave the root is played in. How often the size of the bass's motion places the barline, at one chord a bar over eight bars, against how often the bass moves inside the chord, under two realisations of the same pitch classes in a register. Roots at the bottom of the compass: 100% at 0%, 100% at 13%, 100% at 25%, 99% at 38%, 96% at 50%, 95% at 63%, 93% at 75%, 92% at 88%, 90% at 100%. Least motion: 100%, 100%, 96%, 91%, 83%, 69%, 52%, 32%, 20%. Also drawn, the reading of whether the bass moved at all: 100%, 100%, 99%, 99%, 93%, 89%, 70%, 54%, 20%. The two realisations use the same notes in the same order and differ only in which octave each is played in. Harmony and voice leading

The leap at the barline is a fact about register

A bass leaps to a root at the barline and steps inside the bar, and no reading of pitch classes can see it — so the essay that named it recorded it as a limit. Given octaves and the size of every motion, it is worth nothing: realised at least motion the reading tracks whether the bass moved at all and ends 0.87 semitones on the wrong side of zero, because a line that always takes the nearest octave has no leaps to take. Play a stated root at the bottom of the compass instead, and the same sixteen notes in the same order turn a margin of minus 1.67 semitones into plus 1.67 and hold at 90 per cent where the move reading has fallen to 20. The barline is not where the bass moves furthest. It is where the bass goes back down.

 Every pair has a worst note, and for most of them it is at one end of the compass. The margin at the strike — how much nearer the composite spectrum sits to the decaying instrument's own spectrum than to the held instrument's, in decibels of log-spectral distance — for every ordered pair of six radiators at every semitone from C3 to C7, with the decaying instrument entering 6 dB above. violin over voice: worst −34.9 dB at E♭3, best 14.9 dB at C♯6, behind at 35 of 49 pitches; violin over clarinet: worst −2.2 dB at B♭4, best 19.6 dB at F♯6, behind at 3 of 49 pitches; violin over oboe: worst −0.3 dB at B♭4, best 11.4 dB at E3, behind at 1 of 49 pitches; violin over flue pipe: worst 2.7 dB at C7, best 16.9 dB at E3; violin over trumpet: worst −13.8 dB at B♭4, best −3.4 dB at D3, behind at 49 of 49 pitches; voice over violin: worst −9.2 dB at C♯6, best 41.8 dB at E3, behind at 8 of 49 pitches; voice over clarinet: worst 1.4 dB at C♯6, best 43.1 dB at E3; voice over oboe: worst −8.2 dB at C♯6, best 44.5 dB at E3, behind at 6 of 49 pitches; voice over flue pipe: worst −14.7 dB at C7, best 47.5 dB at E♭3, behind at 8 of 49 pitches; voice over trumpet: worst −24.2 dB at C7, best 39.5 dB at E3, behind at 20 of 49 pitches; clarinet over violin: worst −14.0 dB at F♯6, best 7.0 dB at B♭4, behind at 26 of 49 pitches; clarinet over voice: worst −36.6 dB at E3, best 3.5 dB at C♯6, behind at 46 of 49 pitches; clarinet over oboe: worst −14.2 dB at F♯6, best 5.5 dB at C3, behind at 27 of 49 pitches; clarinet over flue pipe: worst −14.5 dB at G6, best 13.3 dB at C3, behind at 18 of 49 pitches; clarinet over trumpet: worst −25.5 dB at F♯6, best −6.5 dB at C3, behind at 49 of 49 pitches; oboe over violin: worst −7.5 dB at E3, best 5.8 dB at B♭4, behind at 43 of 49 pitches; oboe over voice: worst −37.7 dB at E3, best 13.8 dB at C♯6, behind at 37 of 49 pitches; oboe over clarinet: worst −0.8 dB at C3, best 19.2 dB at F♯6, behind at 10 of 49 pitches; oboe over flue pipe: worst 2.4 dB at C7, best 14.3 dB at C3; oboe over trumpet: worst −14.2 dB at F4, best −12.8 dB at F♯6, behind at 49 of 49 pitches; flue pipe over violin: worst −12.7 dB at D3, best 3.0 dB at C7, behind at 43 of 49 pitches; flue pipe over voice: worst −40.4 dB at E♭3, best 19.4 dB at C7, behind at 34 of 49 pitches; flue pipe over clarinet: worst −7.7 dB at C3, best 18.9 dB at G6, behind at 24 of 49 pitches; flue pipe over oboe: worst −10.4 dB at C3, best 2.1 dB at C7, behind at 45 of 49 pitches; flue pipe over trumpet: worst −21.0 dB at C4, best −16.3 dB at C7, behind at 49 of 49 pitches; trumpet over violin: worst 8.7 dB at C3, best 17.3 dB at B♭4; trumpet over voice: worst −29.3 dB at E3, best 29.9 dB at C7, behind at 26 of 49 pitches; trumpet over clarinet: worst 10.7 dB at C3, best 30.6 dB at F♯6; trumpet over oboe: worst 16.9 dB at D5, best 18.8 dB at C7; trumpet over flue pipe: worst 21.2 dB at C7, best 24.7 dB at B♭3. A pale cell is a pitch at which the two spectra nearly coincide and the room has almost nothing to multiply; an unfilled cell is one at which the decaying instrument never leads at all. Timbre and acoustics

A doubling's worst note belongs to one instrument, not to the pair

The essay before this one found one pitch at which a violin and a flue pipe nearly share a spectrum and asked which pitch that is for every pair. Run over six radiators, thirty ordered pairs and forty-nine semitones, the answer is that seventeen pairs choose a worst note and thirteen simply run out of compass — and that the seventeen do not choose thirty different notes. They cluster in fours and threes, and each cluster is one instrument's own filter: the violin's worst note is B flat 4, its strongest body resonance; the clarinet's is F sharp 6, its cutoff; the voice's is C sharp 6, where its second partial meets its second formant. A doubling fails at a note that belongs to one of its players.

 A bell's tierce does not arrive late; it leaves early. The tierce's margin over what the bell's other modes put across it, in decibels, against time since the strike, for four bells, with the hum's sixty-decibel time set to 40 seconds and each mode's decay time that divided by its frequency ratio to the hum raised to 1. G1: 0 s -3.5 dB, 2 s -5.3 dB, 3 s -7.4 dB, 5 s -9.8 dB, 6 s -12.4 dB, 8 s -15.2 dB; C2: 0 s -2.2 dB, 2 s -3.7 dB, 3 s -5.6 dB, 5 s -7.7 dB, 6 s -10.1 dB, 8 s -12.6 dB; G2: 0 s 0.1 dB, 2 s -1.0 dB, 3 s -2.4 dB, 5 s -3.9 dB, 6 s -5.8 dB, 8 s -7.9 dB; C3: 0 s 1.9 dB, 2 s 1.1 dB, 3 s 0.2 dB, 5 s -0.9 dB, 6 s -2.2 dB, 8 s -3.8 dB. Above the line the tierce is heard. The earlier prediction was that the quint's faster decay would lift the tierce clear of its masking a second into the note; every curve here falls instead. Perception and the listener

A bell's tierce does not arrive late, it leaves early

The essay before this one found a low bell's tierce masked at the strike and predicted it would emerge a second into the note, because the quint above it decays faster than the prime and tierce below. It does not emerge, and the reason is that the quint was never masking it: removing the quint moves the tierce's threshold by a fifth of a decibel and removing the prime moves it by three to five and a half. The tierce is masked from below, by two modes that outlive it under any law that falls with frequency — so a bell that has a tierce at the strike loses it within a few seconds, and a bell that has none never gains one. The minor third a carillon's bass is known for is a property of its attack.

 Inside the compasses the instruments have, 8 of the map's seventeen worst notes cannot be played by both. The margin at the strike for every ordered pair of six radiators from C3 upward, with every note outside the pair's shared compass greyed: the violin from G3 to E7, a soprano from C4 to C6, a B♭ clarinet sounding D3 to B♭6, the oboe B♭3 to A6, a B♭ trumpet sounding E3 to B♭5, a flue pipe throughout. violin over voice: shared C4 to C6, worst inside it −30.6 dB at C4; violin over clarinet: shared G3 to B♭6, worst inside it −2.2 dB at B♭4; violin over oboe: shared B♭3 to A6, worst inside it −0.3 dB at B♭4; violin over flue pipe: shared G3 to C7, worst inside it 2.7 dB at C7; violin over trumpet: shared G3 to B♭5, worst inside it −13.8 dB at B♭4; voice over violin: shared C4 to C6, worst inside it −5.9 dB at C6; the full map's worst, C♯6, is outside; voice over clarinet: shared C4 to C6, worst inside it 6.9 dB at F♯5; the full map's worst, C♯6, is outside; voice over oboe: shared C4 to C6, worst inside it −3.4 dB at C6; the full map's worst, C♯6, is outside; voice over flue pipe: shared C4 to C6, worst inside it −2.5 dB at C6; voice over trumpet: shared C4 to B♭5, worst inside it −12.6 dB at F♯5; clarinet over violin: shared G3 to B♭6, worst inside it −14.0 dB at F♯6; clarinet over voice: shared C4 to C6, worst inside it −31.3 dB at C4; the full map's worst, E3, is outside; clarinet over oboe: shared B♭3 to A6, worst inside it −14.2 dB at F♯6; clarinet over flue pipe: shared D3 to B♭6, worst inside it −14.5 dB at G6; clarinet over trumpet: shared E3 to B♭5, worst inside it −22.7 dB at A♭5; the full map's worst, F♯6, is outside; oboe over violin: shared B♭3 to A6, worst inside it −6.0 dB at E4; the full map's worst, E3, is outside; oboe over voice: shared C4 to C6, worst inside it −33.0 dB at C4; the full map's worst, E3, is outside; oboe over clarinet: shared B♭3 to A6, worst inside it 0.2 dB at B♭3; oboe over flue pipe: shared B♭3 to A6, worst inside it 4.2 dB at A6; oboe over trumpet: shared B♭3 to B♭5, worst inside it −14.2 dB at F4; flue pipe over violin: shared G3 to C7, worst inside it −11.3 dB at G3; flue pipe over voice: shared C4 to C6, worst inside it −34.8 dB at C4; flue pipe over clarinet: shared D3 to B♭6, worst inside it −7.7 dB at D3; flue pipe over oboe: shared B♭3 to A6, worst inside it −10.4 dB at B♭3; flue pipe over trumpet: shared E3 to B♭5, worst inside it −21.0 dB at C4; trumpet over violin: shared G3 to B♭5, worst inside it 9.1 dB at F4; trumpet over voice: shared C4 to B♭5, worst inside it −22.4 dB at C4; the full map's worst, E3, is outside; trumpet over clarinet: shared E3 to B♭5, worst inside it 11.0 dB at E3; trumpet over oboe: shared B♭3 to B♭5, worst inside it 16.9 dB at D5; trumpet over flue pipe: shared E3 to B♭5, worst inside it 22.7 dB at B♭5. Of the seventeen worst notes the full map chose, 8 lie outside a compass and 9 inside. Timbre and acoustics

Half the worst notes of a doubling are notes nobody can play

A map of where two instruments' spectra come closest found seventeen pairs with a worst note of their own, gathered into four clusters, each belonging to one instrument's filter. It drew every instrument over the same five octaves. Read inside the compass each pair can actually play, eight of the seventeen worst notes are out of reach — the voice's C♯6 lies a semitone above a soprano's top C, and its E3 below her lowest note — and two pairs that seemed only to run downhill turn out to have a worst note after all. Of the four clusters, only the violin's B♭4 is advice every pair in it can use.

 Re-struck under a chord that has been decaying, a buried tenor gets its partials back. How many of the tenor's eight partials stand above the masking of the other three voices, for the first chord of I–vi–IV–V–I balanced for loudness in four registers, on a struck instrument whose held voices decay while the tenor is re-struck a step away at a stated time after the chord; beside it, the same tenor held from the start. Two octaves down: 0.25 s 1 moving, 0 held; 0.5 s 3 moving, 0 held; 1 s 6 moving, 0 held; 2 s 8 moving, 0 held; 4 s 8 moving, 0 held. An octave down: 0.25 s 6 moving, 1 held; 0.5 s 8 moving, 1 held; 1 s 8 moving, 1 held; 2 s 8 moving, 1 held; 4 s 8 moving, 0 held. As written: 0.25 s 8 moving, 4 held; 0.5 s 8 moving, 5 held; 1 s 8 moving, 5 held; 2 s 8 moving, 2 held; 4 s 8 moving, 1 held. An octave up: 0.25 s 8 moving, 3 held; 0.5 s 8 moving, 4 held; 1 s 8 moving, 3 held; 2 s 8 moving, 1 held; 4 s 8 moving, 0 held. Two octaves down, where the held tenor keeps nothing at any time, the moving one has six partials back by a second and all eight by two. Perception and the listener

A moving tenor is heard because it is struck again

A bass-heavy chord balanced for loudness hides its tenor under the bass's upper skirt, and arrangers keep inner voices moving in the low register as though motion were the cure. Counted as masking, motion alone is worth nothing: the note the tenor leaves stops masking the new one within ten milliseconds, and on a sustained instrument the moving tenor keeps exactly what the held one keeps. On a struck instrument everything changes, and for a reason that is not the motion. The held voices have been decaying since they were struck and the tenor's new note is struck now, so an octave down a tenor re-struck half a second into the chord has all eight of its partials back where the held tenor has one.

 Pointed, four partials above the fundamental keep their staccato further out than it does, and the highest never catch up. A note on 130.8 Hz held 0.4 s and damped, in a room of 2 s: for each of its first eight partials, the distance along the source's axis, in critical distances of an omnidirectional source, at which that partial's own 20 dB fall after the release has taken twice as long as at the source — with the source radiating equally everywhere, and with it beamed like a piston of 35 cm radius. Partial 1 at 131 Hz: 0.141 omnidirectional, 0.164 beamed; partial 2 at 262 Hz: 0.140 omnidirectional, 0.217 beamed; partial 3 at 392 Hz: 0.139 omnidirectional, 0.283 beamed; partial 4 at 523 Hz: 0.087 omnidirectional, 0.223 beamed; partial 5 at 654 Hz: 0.054 omnidirectional, 0.170 beamed; partial 6 at 785 Hz: 0.034 omnidirectional, 0.126 beamed; partial 7 at 916 Hz: 0.021 omnidirectional, 0.091 beamed; partial 8 at 1047 Hz: 0.013 omnidirectional, 0.065 beamed. Beaming multiplies each partial's reach by about the square root of its directivity factor, but the upper partials start so far behind that only partials 2, 3, 4, 5 pass the fundamental's 0.164. Timbre and acoustics

A pointing source's staccato ends dark, not bright

A damped note's release is heard as the player plays it only close to the instrument: a little way out, the room's copy takes over and the staccato is gone. A source that points, as a grand piano with its lid raised does, beams its upper partials along its axis, so their direct sound reaches further, and the prediction was that the release of a note's colour would survive into the hall where the release of the note does not — heard as a brief brightening before a room-length tail. Beaming does carry the second to fifth partials' articulation beyond the fundamental's. The highest never catch up, the whole note gains a fifth of its reach, and the release is heard not as a brightening but as a darkening.

 A moving tenor on a sustained instrument can hand over only the partials the chord leaves it. A four-voice arrival balanced for loudness, played on a sustained instrument, with the tenor moving a second after the chord: how many of the tenor's partials stand above the masking of the other three voices, on average over four arrivals, against how far the chord is transposed, for a string, an organ and a clarinet spectrum. The held tenor keeps exactly the same partials. string: 24 semitones down 0.0, 21 semitones down 1.0, 18 semitones down 1.0, 15 semitones down 1.0, 12 semitones down 1.0, 9 semitones down 2.5, 6 semitones down 1.8, 3 semitones down 2.3, as written 3.5, 3 semitones up 4.3, 6 semitones up 2.5, 9 semitones up 3.8, 12 semitones up 3.8; organ: 24 semitones down 0.0, 21 semitones down 1.0, 18 semitones down 1.0, 15 semitones down 1.8, 12 semitones down 2.3, 9 semitones down 2.3, 6 semitones down 1.0, 3 semitones down 2.8, as written 3.0, 3 semitones up 3.3, 6 semitones up 1.3, 9 semitones up 3.0, 12 semitones up 3.0; clarinet: 24 semitones down 0.5, 21 semitones down 2.5, 18 semitones down 1.0, 15 semitones down 2.5, 12 semitones down 3.5, 9 semitones down 3.0, 6 semitones down 3.3, 3 semitones down 3.3, as written 3.8, 3 semitones up 3.3, 6 semitones up 3.5, 9 semitones up 3.5, 12 semitones up 3.5. Whatever the onset does, it can only do it to these. Perception and the listener

An onset can only hand over what the chord leaves

On an organ, a tenor that moves while the chord holds keeps exactly the partials a held tenor keeps, so the masking count cannot tell them apart. What can is the onset: a component that starts thirty milliseconds after the sound around it is heard out as something new. A moving tenor starts hundreds of milliseconds after the chord and gets that for every partial it has left — and what it has left, an octave down, is one partial on a string and three on an organ, which still name the tenor's own pitch. Two octaves down there is nothing to hand over. And a tenor that moves at the chord change, with the other voices, gets nothing, which is why the motion that carries an inner voice is the motion between the harmony's changes.

 The C♯4 tube speaks in C4's attack. A marimba bar at C4, 262 hertz, struck, with the bar a semitone above it next to it in the same row. One curve is the struck note as the tubes radiate it; the other is what radiates at the neighbour's own pitch, C♯4; both in decibels against the struck note's peak. The dashed curve is where the neighbour would have to be to modulate the note by the beat-detection threshold at 15.6 beats a second, 3.1 per cent. The neighbour is above it for the first 90 milliseconds — 1.4 beats — and then drops away, while the note itself is near its loudest. Over the whole note the energy at the neighbour's pitch is 33.8 decibels below the struck note's. It comes out of the neighbour's tube, which the bar reaches at -13.5 decibels, and it leaves because that tube hands it to its own bar, which cannot radiate it. Instruments and their design

The second row of resonators keeps the neighbours out of the attack

A marimba bar reaches the mouth of the tube under the next bar at a fifth of what it puts on its own, and the next bar is tuned a semitone away. Solved as a row of coupled pairs, the neighbour's pitch speaks in every attack for about a beat and a half and is then handed to its own bar, which cannot radiate it. Laying the accidentals out in a second bank halves that for eight notes of the twelve, and leaves C, E, F and B exactly where they were.

 Where in the hall a raised lid makes a C3 staccato darker, and where brighter. A floor in front of a 35 cm piston radiator pointing up the page, in units of the critical distance of a source radiating everywhere, three wide and one and a half deep, in a 2-second room, divided into 160 seats. Each seat is marked by what a damped C3 does to its colour over the 400 ms after its release: darkens (42 seats), brightens but by no more than with no lid (26), or brightens more than with no lid (92). The darkening seats form a narrow wedge on the axis; everywhere else a raised lid makes the end of a note brighter, and over 58 per cent of the floor brighter than it would be with the lid closed. Timbre and acoustics

A lid darkens a staccato only inside a narrowing cone

Along its axis, a raised lid makes a damped note's release darker, because the brightness it beams at the listener is the part the damper stops. Most of an audience does not sit on the axis. Off it, the same beaming does the reverse, and a little way off it does the reverse more strongly than no lid at all would. The region where the release darkens is a cone whose half-width obeys one law — the fundamental times the radiator's radius times the sine of the angle is 23.6 hertz-metres — so it narrows from 28 degrees at C3 to 8 at C5.

 How wide the unison is left decides how far down the aftersound starts. A two-string unison on A3, each string decaying in 20 seconds alone and the pair's bridge-driving mode in 1.5, struck and heard at the bridge, against how many cents apart its strings are tuned: the drop from the prompt sound to the aftersound, for a hammer that strikes both strings alike and for one that strikes the second 20 and 50 per cent softer. Even strike: 0.25 cents 65.2 dB, aftersound 19.9 s; 0.5 cents 53.1 dB, aftersound 19.5 s; 1 cent 41.0 dB, aftersound 18.0 s; 1.5 cents 33.7 dB, aftersound 16.0 s; 2 cents 28.4 dB, aftersound 13.8 s; 2.5 cents 24.2 dB, aftersound 11.6 s; 3 cents 20.5 dB, aftersound 9.7 s; 4 cents 13.8 dB, aftersound 6.5 s; 5 cents 6.3 dB, aftersound 4.0 s. The earlier count assumed 20 dB; the unison gives that at about three cents. At a perfect unison the aftersound is not excited at all, and past 5.33 cents the two modes split and the unison beats instead. A strike's force scales both modes alike and appears nowhere in the drop. Intervals and chords

The octave is counted best over a unison left two and a half cents wide

A firm touch buys a mistuned octave's beats only if the aftersound's drop does not deepen with the strike, and the drop was left as a number to be measured. It has a mechanism, and the mechanism has been known since the essay on three strings: the unison's two coupled modes. Computed from them, the drop does not depend on the strike at all. It depends on how wide the tuner left the unison — sixty-five decibels at a quarter of a cent, twenty at three — and the octave's beats peak, at 6.8, over the same two-and-a-half-cent unison that makes the longest note.

 The bar a tuner can least afford to leave wrong is the one the skeleton lands on. What a 10-cent error on each degree of a measured slendro adds to the roughness per second of a two-instrument texture over 128 bars, averaged over both instruments and both directions, with the elaborating part meeting the skeleton on the last stroke of every four-stroke unit. Top row: the skeleton's degrees drawn at random. Each row below: the skeleton ending every unit on one degree. Random: 1 0.219, 2 0.054, 3 0.083, 5 0.095, 6 0.104. Ending on 1: 1 0.348, 2 0.028, 3 0.049, 5 0.062, 6 0.068. Ending on 2: 1 0.171, 2 0.228, 3 0.049, 5 0.064, 6 0.067. Ending on 3: 1 0.170, 2 0.028, 3 0.247, 5 0.064, 6 0.068. Ending on 5: 1 0.170, 2 0.028, 3 0.049, 5 0.285, 6 0.069. Ending on 6: 1 0.171, 2 0.027, 3 0.049, 5 0.064, 6 0.305. The tonic's error moves its octave too, so it is two bars. In every row the degree the units end on costs the most, three and a half to five times what the dearest other single bar costs. Scales and modes

The bar a gamelan can least afford wrong is the one its skeleton lands on

A wrong bar in a gamelan costs what its beating against the other instrument costs, and that depends on how often the music doubles the degree. The obvious place for doubling is where an elaborating part meets the skeleton at the end of each four-stroke unit, so the prediction was that the meeting rule would rank the degrees. It barely does: a part left to ring doubles a degree whenever it sounds it while the other instrument's bar still sounds, and the meetings add three per cent. What ranks the degrees is where the skeleton lands. A skeleton that ends its units on one degree makes that bar three and a half to five times as costly to leave wrong as any other single bar.

 A violin's worst doubling note follows its own body resonance. A violin plucked over a clarinet, an oboe and a trumpet holding the same note: the note within the pair's shared compass at which the pluck's margin at the strike is smallest, against how far the violin's B1− corpus mode is moved from its modelled 460 Hz, in semitones. The diagonal is the note nearest the mode itself. -3: mode near G4, worst notes clarinet G4, oboe C♯5, brass G4; -2: mode near A♭4, worst notes clarinet A♭4, oboe A♭4, brass A♭4; -1: mode near A4, worst notes clarinet A4, oboe A4, brass A4; 0: mode near B♭4, worst notes clarinet B♭4, oboe B♭4, brass B♭4; +1: mode near B4, worst notes clarinet B4, oboe B4, brass B4; +2: mode near C5, worst notes clarinet C5, oboe C5, brass C5; +3: mode near C♯5, worst notes clarinet C♯5, oboe C♯5, brass C♯5. As modelled, all three pairs' worst note is B♭4. Moved, it moves with the mode, a semitone for a semitone, within a semitone at 36 of 39 settings. Timbre and acoustics

A violin's worst doubling note is wherever its own body resonates

A pizzicato violin doubled by a held clarinet, oboe or trumpet has one note in the shared compass at which its pluck comes nearest to never owning the note, and for the modelled violin all three partners choose B♭4. That is where the violin's lower corpus mode sits, and real violins place that mode a semitone or two either way. Move it and the worst note moves with it, a semitone for a semitone, at 36 of 39 settings — so every violin has its own worst doubling note, and it can be found by knowing where the instrument resonates. Held rather than plucked, the violin's body decides nothing.

 An elaboration that arrives early makes all three goal tones the dearest bars. What a 10-cent error on each degree costs, for a skeleton whose four-stroke units end on 6, 2 or 3, with the elaborating part meeting the goal on the unit's last stroke and arriving on it 0, 3 or 7 of its own strokes earlier. 0 early: 1 0.171, 2 0.097, 3 0.082, 5 0.064, 6 0.161. 3 early: 1 0.139, 2 0.115, 3 0.106, 5 0.056, 6 0.221. 7 early: 1 0.100, 2 0.159, 3 0.136, 5 0.043, 6 0.295. Meeting on the stroke, the goal on 3 costs no more than degree 3 does with the skeleton at random; arriving seven strokes early, the cheapest goal costs 3.2 times the dearest single bar that is not a goal. Scales and modes

An elaboration that arrives early makes every goal tone the bar to tune first

A gamelan's bars are costliest to leave out of tune where the skeleton lands, but a mode with three goal tones spread the cost so thinly that a low goal could end up cheaper than a high bar that is never a goal. The rule tested there put the elaborating part on the goal only at the goal stroke. Elaborations in the style reach their goal early and stay on it. Let them arrive seven strokes early and every goal tone costs more than three times the dearest bar that is not a goal; the goal's own cost rises by about half for every few strokes of anticipation, and a whole instrument out of tune becomes dearer too.

 A cone opens its aperture as far as a cylinder does only if its holes grow with it. For every fingering of four woodwinds with twelve holes each, each tuned to its own semitones — a clarinet's cylinder with graded holes and with holes all 9 millimetres across, a conical bore of an oboe's proportions with holes 0.6 of the local bore radius, and the same cone with every hole 7 millimetres across — the largest number of openings really radiating together at any frequency above 700 hertz, as a participation number counting the bell. The dotted line is every open hole and the bell sharing equally. clarinet: a cylinder: 1 open 2.0, 2 open 3.0, 3 open 3.9, 4 open 4.8, 5 open 5.6, 6 open 6.5, 7 open 7.2, 8 open 7.8, 9 open 8.8, 10 open 9.6, 11 open 10.1, 12 open 10.5. cone, holes growing with the bore: 1 open 2.0, 2 open 2.9, 3 open 3.9, 4 open 4.9, 5 open 5.9, 6 open 6.7, 7 open 6.6, 8 open 7.2, 9 open 7.7, 10 open 7.4, 11 open 8.4, 12 open 8.9. cone, every hole 7 mm: 1 open 2.0, 2 open 2.8, 3 open 3.7, 4 open 4.5, 5 open 5.4, 6 open 5.6, 7 open 6.1, 8 open 5.6, 9 open 6.5, 10 open 6.7, 11 open 7.1, 12 open 6.8. clarinet, every hole 9 mm: 1 open 2.0, 2 open 2.9, 3 open 3.9, 4 open 4.9, 5 open 5.9, 6 open 6.8, 7 open 7.7, 8 open 8.6, 9 open 9.6, 10 open 10.1, 11 open 9.2, 12 open 10.2. Instruments and their design

A cone radiates like a cylinder only if its holes grow

The prediction was that a conical woodwind radiates from fewer of its holes than a clarinet, because a hole near the wide end of a cone sits on a tube that is already easy to push air along, and so pulls less of the passing flow out. The mechanism is right and the conclusion is wrong for the instruments that exist. With holes that grow with the bore, as an oboe's and a saxophone's do, a cone opens its aperture as far as a cylinder up to six open holes and falls short above that by a hole or two. Only a cone whose holes are all one size tops out far below — near seven apertures with thirteen open. What a cone really changes is where and how much: its aperture opens nearly a major third lower, its bell carries twice the share, and it gets out at best under half of what reaches it, where the clarinet gets out seven tenths.

 A bell's tierce is best placed at the first instant of the strike, and only for an instant. The tierce's margin above its masked threshold, in decibels, through the first 400 milliseconds after the strike, for bells on G1, C2, E2, G2, C3, with every mode starting at full amplitude and built up in the listener's own auditory filters. G1: -1.3 at 0.5 ms, -3.9 at 400 ms, never above threshold; C2: 0.3 at 0.5 ms, -2.5 at 400 ms, above threshold from 0.5 to 0.7 ms; E2: 1.6 at 0.5 ms, -1.2 at 400 ms, above threshold from 0.5 to 4.3 ms; G2: 2.7 at 0.5 ms, -0.2 at 400 ms, above threshold from 0.5 to 104.2 ms; C3: 4.7 at 0.5 ms, 1.7 at 400 ms, above threshold from 0.5 to 400.0 ms. Every curve falls from its first moment: there is no peak inside the window. Perception and the listener

A bell's strike gives its tierce a millisecond

A low bell's tierce is masked from below by its prime and hum, so it is present at the strike or never — and the strike was the one moment the count treated as a steady state. Give the modes a rise as well as a fall and the prediction was that the tierce's margin would peak somewhere inside the first two hundred milliseconds. It peaks at the very first instant and falls from there, because the shadow a masker casts upward grows faster than the masker. On a C2 bell the tierce is above its threshold for less than a millisecond, and under the most generous rise anybody could propose for twenty-five: ten to thirty decibels too brief to be heard. What the onset visits for an instant, a softer stroke keeps for the whole note.

 A section of twelve violins nearly fills in the note one violin never owns. The margin at the strike at the worst note in the shared compass for a section of violins plucked together over a held clarinet, against the number of players, each player's two corpus modes moved by its own draw from a spread of one semitone; the line is the median of six sections and the bar the range. The modelled violin alone: −2.2 dB at B♭4. air mode shared: 1 −1.7 dB (-2.1 to -1.0), 2 −1.4 dB (-2.0 to -0.6), 4 −0.7 dB (-1.1 to -0.1), 8 −0.5 dB (-0.8 to -0.2), 12 −0.6 dB (-0.8 to -0.4), 16 −0.6 dB (-0.7 to -0.3). air mode varied too: 1 −1.7 dB (-2.3 to -1.0), 2 −1.4 dB (-2.0 to -0.3), 4 −0.5 dB (-1.0 to -0.1), 8 −0.1 dB (-0.8 to 0.3), 12 0.0 dB (-0.5 to 0.5), 16 0.2 dB (-0.6 to 0.5). Timbre and acoustics

A section of violins fills in the note one violin gives away

A pizzicato violin doubled by a held clarinet has one note — the note whose fundamental sits on its own lower corpus resonance — at which the pluck never owns the sound, and real violins put that resonance a semitone or two either way. A section plucks with a dozen different bodies at once. Summed, their resonances smear into a broad shelf, and the section's worst note over the clarinet rises from 2.2 decibels below zero to about zero: the note one violin gives away, twelve nearly keep. Over an oboe the section owns every note. And if every member shares the same air resonance, that shared resonance becomes the section's worst note instead — a section is weakest where its instruments agree, not where any one of them is extreme.

 A set in twelve keys keeps its shape through any number of rock songs and loses it to two minor ones. The I–vi–IV–V ground in all twelve keys, with songs outside it added one key at a time — minor-key songs on A, E, B, F♯ and on round the circle, or rock songs on G, D, A, E and on — at five frets a cent: the set's mean cost under its one best tuning, with the shape that tuning is exact for above each point. minor-key songs, i–iv–V–i: 0 added 5.57 (D), 1 added 5.77 (D), 2 added 5.97 (C), 3 added 6.01 (C), 4 added 6.04 (C), 6 added 6.08 (C), 8 added 6.12 (C), 12 added 6.03 (E minor). rock songs, I–♭VII–IV–I: 0 added 5.57 (D), 1 added 5.49 (D), 2 added 5.42 (D), 3 added 5.36 (D), 4 added 5.22 (D), 6 added 5.14 (D), 8 added 5.09 (D), 12 added 4.96 (D). Pitch and tuning

Two minor songs retune a guitar set for their dominant

A guitar tuned once for an evening of pop songs in every key is tuned exact for the D shape, because every key's route through the ground passes through it. Add songs that are not that ground and the tuning holds or breaks by a rule nobody would guess. Rock songs on I, ♭VII and IV never move it off the D shape, however many are added: they retune only the two strings the D shape leaves silent. One minor-key song leaves it alone too. Two move it to the C shape's tuning, and the chord they move it for is not one of their minor chords but their one major chord — the dominant, fingered in the C shape. Twelve move it again, to the E-minor shape's.

 A gamelan's goal-tone beating is a clock hand only if each goal beats at its own rate. Bits of position unknown over the first cycle of a sixteen-step cycle of gong, kempul and kenong, against the cycle's duration, when the elaborating parts arrive early on each goal tone — kenong and gong on degrees 2 3 2 6 — and a pair of instruments tuned apart beats on it for a step before the stroke: with the pairs tuned together (no swell), tuned apart at one beat rate for every degree, tuned apart by 25 cents so that each degree beats at its own rate (told apart when the rates differ by 15 per cent), and with every swell told apart. tuned together: no swell: 40 s 0.112, 60 s 0.203, 80 s 0.329, 120 s 0.417, 160 s 0.646, 240 s 0.870. tuned apart at one beat rate: 40 s 0.104, 60 s 0.192, 80 s 0.301, 120 s 0.376, 160 s 0.404, 240 s 0.576. tuned apart by 25 cents: 40 s 0.076, 60 s 0.114, 80 s 0.125, 120 s 0.137, 160 s 0.146, 240 s 0.285. every swell told apart: 40 s 0.000, 60 s 0.000, 80 s 0.000, 120 s 0.000, 160 s 0.003, 240 s 0.132. Scales and modes

A gamelan's beating names its goal only if each goal beats at its own rate

When a gamelan's elaborating parts reach a goal tone early, a pair of instruments tuned slightly apart beats on that degree for the strokes before the goal — a swell of shimmer just before each structural stroke, which is exactly where a recognisable event does a slow cycle's clock most good. Whether it does that good depends on whether one goal's swell can be told from another's. Tune every pair to the same beat rate, the practice that keeps a gamelan's shimmer even, and every swell is the same event: it buys a third of what it could. Tune every pair the same number of cents apart and each degree beats at its own rate, fifteen per cent faster than the degree below, and the swells start naming which goal is coming.

 A leadpipe, and a cylinder of its own mean bore. The first 60 centimetres of a 148-centimetre trumpet bore, with the radius drawn above a floor of 2.5 millimetres and magnified, so that a millimetre can be seen. The top row is the bore every earlier figure drew: 11.0 millimetres across from the mouthpiece to the flare. Below it are three leadpipes opening from 8.8 millimetres at the receiver to the bore, over 10, 25, 40 centimetres, each drawn over the dashed outline of a cylinder of the taper's own mean radius. At the right is how far each set of resonances sits from a harmonic series, in cents: no leadpipe 31; 10 cm, taper 22 against cylinder 21; 25 cm, taper 15 against cylinder 24; 40 cm, taper 14 against cylinder 33. Instruments and their design

A leadpipe tunes the series, and its average does not

Every trumpet drawn in the essays before this one runs at full bore from the mouthpiece to the flare, and no trumpet does. Put a real taper in front of it and the wall losses it was predicted to cost come to a fraction of a per cent. What it does instead is pull the resonances onto a harmonic series, halving their average mistuning, while a cylinder of the same mean bore makes them worse.

 An ending built up part by part, counted two ways. A final chord built up one part at a time from middle C to eight parts, every part a string tone at 62 decibels, with the loudness of each stage as a loudness level: the level of a 1 kHz tone judged equally loud. Grouped into critical bands one after another, as the earlier essays counted it, the chord runs 76.2, 80.9, 79.4, 81.9, 76.4, 77.9, 74.2, 74.5 phons — a span of -1.8, and not monotone. Integrated as an excitation pattern it runs 73.2, 77.9, 79.5, 81.6, 82.5, 83.9, 84.5, 85.6 — a span of 12.4, rising at every step. The dashed line is the power sum, 9.0 decibels from one part to eight; spread across the compass, the parts are louder than their power. Perception and the listener

A page has twelve phons, and its entries always said so

An ending built up one part at a time was found to move its loudness by under two decibels, and five essays have leaned on that number since. It came from a way of grouping partials into critical bands that sorts an eight-part chord's sixty-four partials into two bands. Counted along the ear instead, the same ending gains twelve phons at every step, which is what the collection's own price for a single entry always implied.

 How far under its mask a passenger is, and how much it still changes. Every pairing of two parts entering together on any chord of the passage in which the quieter part — the passenger — sits under its mask on average: 71 of them. Across, the passenger's mean margin, the measure every earlier walk used; up, how far its partials together change the arrival above the threshold of detection. Every one of the 71 is above zero: the lowest is 0.11 dB. 35 of the 41 buried more than 3 dB by the mean still change the arrival by more than 3 dB. The few near the zero line are the clarinet on the bracketing dominant, where every one of its partials is just under. Form and structure

A passenger under its mask is still heard in the arrival

The best way to bring parts in together was a loud part carrying a quiet one, and the quiet one was scored as buried — ten decibels under its mask, on average over its partials. An average is the wrong question. A masked threshold is the level at which a sound becomes a detectable change, and nearly every buried passenger has partials well clear of it. Scored as a change to the arrival rather than as a line of its own, every one of seventy-one passengers is heard, and the carriers' walk passes almost untouched.

 Where the tilt of the blow matters, and where it does not. The octave's beats at 80 dB against the tilt of the blow from the vertical: one string, others muted, 0° 2.81, 1° 2.81, 2° 2.81, 3° 2.81, 5° 2.81, 8° 2.87, 12° 3.19, 16° 3.62, 20° 4.27; a unison half a cent wide, 0° 1.51, 1° 1.51, 2° 1.51, 3° 1.51, 5° 1.57, 8° 1.95, 12° 2.60, 16° 4.00, 20° 4.87; a unison 2.5 cents wide, 0° 6.81, 1° 6.81, 2° 6.81, 3° 6.81, 5° 6.87, 8° 6.87, 12° 6.70, 16° 6.97, 20° 7.25. One string alone and a clean unison both get their aftersound from the second plane and both gain beats as the blow tilts. The wide unison already has an aftersound twenty-odd decibels down from its own detuning, and the second plane, at forty, adds nothing it can hear. Intervals and chords

The plane of the blow matters only where the unison is clean

A piano string vibrates in two planes, and the one parallel to the soundboard gives even a single string a second, slower decay. A hammer tilted off the vertical feeds it. That was the last way left for a firm touch to cost a tuner beats: if harder blows came down straighter. They would have to straighten by half for every seven decibels, and even then only on a single string or a unison tuned clean. Over a unison left two and a half cents wide, no tilt of the blow reaches the count.

 What each minor-key song gains from the C shape's tuning. For a minor-key song, i–iv–V–i, in each of the twelve keys, the saving in cents plus frets over five from playing it under the C shape's tuning rather than the twelve-key ground's own, the D shape's: C minor 1.69, C♯ minor 1.69, D minor 1.69, E♭ minor 1.69, E minor 1.69, F minor 1.69, F♯ minor 1.69, G minor 0.33, A♭ minor 1.69, A minor 1.69, B♭ minor 1.69, B minor 1.61. Switching the tuning costs the twelve ground songs 3.90 between them, so the switch pays when the added songs save more than that: the lines mark a half and a third of it, what each of two or three songs would have to save. Ten keys save 1.69 each — less than half, more than a third. G minor saves 0.33, because under the ground's tuning its dominant is the open D chord, already exact. Pitch and tuning

A guitar set keeps its tuning until the third minor song

Two minor-key songs were found to move a pop set's guitar tuning from the D shape to the C shape. Priced as a vote — what the move costs the twelve ground songs, against what it saves each minor song — two songs do not pay for it and three do. The earlier search had never tried keeping the tuning the set already had. And the key a minor song is in matters only at the nut: ten keys vote alike, and G minor, whose dominant is the open D chord, hardly votes at all.

 The minor songs' tonic takes the tuning only at the ninth minor song. The twelve-key ground with minor-key songs added on A, E, B, F♯ and on round the circle: the set's mean cost, in cents plus frets over five, under three fixed tunings — the ground's own (the D shape's), the C shape's (the minor songs' dominant exact) and the E-minor shape's (their tonic exact). 0: 5.567, 5.892, 6.499; 1: 5.767, 5.937, 6.436; 2: 5.938, 5.975, 6.327; 3: 6.081, 6.008, 6.285; 4: 6.211, 6.037, 6.247; 5: 6.326, 6.062, 6.214; 6: 6.428, 6.085, 6.185; 7: 6.520, 6.105, 6.159; 8: 6.602, 6.123, 6.135; 9: 6.638, 6.102, 6.076; 10: 6.707, 6.118, 6.058; 11: 6.711, 6.134, 6.042; 12: 6.772, 6.148, 6.028. The ground's tuning is cheapest to two songs, the C shape's from three to eight, and the E-minor shape's from nine — when minor songs are nine of twenty-one. Pitch and tuning

A guitar set tunes for its minor chords only when nearly half its songs are minor

Add a third candidate to the vote — the tuning that makes the minor songs' own tonic chord exact — and a pop set's guitar passes through three tunings as minor songs are added: the ground's to the second, the dominant's from the third, and the tonic's only from the tenth, when ten songs in twenty-two are minor. Nothing in between is ever chosen: the two minor-song tunings differ on two strings, and every compromise between them costs the ground more than either end. Two keys break the pattern at the nut, and one earlier figure turns out to be wrong by a song.

 Naming the goals costs no roughness: it saves a little. Pairs tuned apart so that each slendro degree's pair beats at a rate proportional to its pitch raised to a power — nought for one rate on every degree, one for a constant interval, more than one for a rate rising faster than the pitch — with the average rate held at 5.01 beats a second. Line: the roughness the detuned pairs add to the goal-tone passage, as a share of what one rate adds — 0 100.0%, 0.5 99.7%, 0.75 99.4%, 1 99.0%, 1.25 98.5%, 1.5 98.1%, 2 97.3%, 2.5 96.4%, 3 95.6%. Steps: the bits of position the swells leave unknown at 160 seconds a cycle, rates told apart at 15 per cent — 0 0.404 (AAAA), 0.5 0.404 (AAAA), 0.75 0.226 (AAAB), 1 0.146 (ABAC), 1.25 0.146 (ABAC), 1.5 0.146 (ABAC), 2 0.146 (ABAC), 2.5 0.146 (ABAC), 3 0.146 (ABAC). Every steeper rise costs less roughness, and from a constant interval upward the swells name three different goals. Scales and modes

A gamelan tuner can name the goals for nothing

A pair tuned so that each goal beats at its own rate lets the swells before a gamelan's structural strokes say which goal is coming, and the question left was what that costs in roughness. Held at the same average shimmer, nothing: a constant interval adds one per cent less roughness than an even beat rate, and a rate rising as the square of the pitch adds three per cent less while naming the goals for a far coarser ear. The even shimmer puts its widest pairs on the low degrees, where a cent of detuning costs most.

 In a hall a section's shallow shelf goes under all at once. The margin at the strike for a pizzicato over a held clarinet, note by note from C4 to F♯5, dry (thin) and 15 metres into a hall (thick): the modelled violin, and the median of six sections of twelve with corpus modes spread by a semitone and air modes by half. C4 2.1/2.1 dry, -1.9/-1.9 in the hall; C♯4 -0.4/1.2 dry, -4.3/-2.8 in the hall; D4 2.6/3.3 dry, -1.6/-0.8 in the hall; E♭4 4.2/4.5 dry, 0.0/0.3 in the hall; E4 4.9/5.2 dry, 0.7/0.9 in the hall; F4 5.1/5.3 dry, 1.0/1.1 in the hall; F♯4 5.0/5.1 dry, 0.9/1.0 in the hall; G4 4.4/4.6 dry, 0.4/0.6 in the hall; A♭4 3.3/3.7 dry, -0.7/-0.3 in the hall; A4 0.5/2.2 dry, -3.3/-1.7 in the hall; B♭4 -2.2/0.4 dry, -5.9/-3.4 in the hall; B4 1.2/0.2 dry, -2.6/-3.5 in the hall; C5 -0.1/0.5 dry, -3.9/-3.3 in the hall; C♯5 0.3/2.1 dry, -3.5/-1.7 in the hall; D5 5.0/5.0 dry, 1.1/1.2 in the hall; E♭5 11.1/11.0 dry, 7.3/7.2 in the hall; E5 12.2/12.1 dry, 8.5/8.4 in the hall; F5 12.7/12.6 dry, 9.1/9.0 in the hall; F♯5 13.0/12.9 dry, 9.5/9.4 in the hall. Dry, the section fills the violin's notch at B♭4 into a shelf a few tenths above zero. In the hall the room lowers every note by the same few decibels for both, and the shelf goes under with the notch: the section owns 0 of the six notes from A♭4 to C♯5, the violin 0. Timbre and acoustics

In a hall a section of violins loses the notes it saved

Dry, twelve violins with different bodies fill in the note one violin's pizzicato gives away to a held clarinet, leaving a shallow shelf a few tenths of a decibel above zero. In a hall that shelf goes under all at once. The room takes the same few decibels from each note whether it comes from one violin or twelve, so a section's smoothing survives only to the front of the hall — about two metres over a clarinet, six over an oboe — and from ten metres back a section loses exactly as many notes as one violin does.

 A bell is never minor alone. For bells whose hum is on C1, E1, G1, C2, E2, G2, C3, E3, G3, C4, struck so that the hum is at 20 to 110 decibels: whether the masked-threshold count delivers the tierce, the minor third above the prime, and the deciem, the major third above the nominal. C1: ···MMMM············; E1: ··MMMMMMMM·········; G1: ··MMMMMMMMMM·······; C2: ·MBBBMMMMMMMMMM····; E2: ·BBBBBBBBMMMMMMM···; G2: ·BBBBBBBBBBMMMMMM··; C3: BBBBBBBBBBBBBMMMMM·; E3: BBBBBBBBBBBBBBBMMM·; G3: BBBBBBBBBBBBBBBBMMM; C4: BBBBBBBBBBBBBBBBBMM (B both thirds, M the major only, m the minor only, a dot neither; levels in steps of five from 20). Of 190 cells, 82 deliver both thirds, 59 the major third only, 49 neither, and 0 the minor third alone. A dot in a cell marks a stroke at which fewer than two of the three modes that imply the strike note survive. Perception and the listener

A bell is never minor alone

A tuned bell carries two thirds: the tierce, a minor third above the prime, which is why a bell is called minor, and the deciem, a major third above the nominal. Run the masking count across every register and every stroke and the minor one is never delivered without the major one. Struck softly a bell delivers both; struck harder, only the major; and below C2 the tierce is never delivered at all. The deciem survives because it is masked by one mode barely louder than itself, the tierce by two that outweigh it.

 A stiff string quickens the narrow intervals' beats and slows the wide ones'. Each equal-tempered interval above A3 on a piano whose strings are those of the grand modelled here, each note with its own string's inharmonicity: the beat rate of its coincident partials as a share of the rate a bearing plan prints, which assumes whole-number partials. fifth (narrow): printed 0.74, true 0.96, a share of 1.29; fourth (wide): printed 0.99, true 0.63, a share of 0.64; major third (wide): printed 8.73, true 8.27, a share of 0.95; minor third (narrow): printed 11.87, true 12.88, a share of 1.08; major sixth (wide): printed 9.98, true 8.86, a share of 0.89. In every coincidence the lower note supplies the higher-numbered partial and stiffness raises it more, so a narrow interval, whose lower partial is already on top, is pushed further apart and a wide one is pulled together. Intervals and chords

A stiff string slows the fourths and quickens the fifths

Every rate in a bearing plan is computed from whole-number partials, and a piano string's partials are not whole numbers. The correction is usually summarised as 'every real beat is faster than the table'. It is not: stiffness quickens the beat of every narrow interval and slows the beat of every wide one, and on a small upright the tempered fourth stops beating altogether. Both errors push the same way, so a tuner who counts the printed table leaves every interval wide, and a fifth and a fourth counted that way make an octave stretched to the 4:2 type or a little past it.

 One retuning is worth what the pop songs pay for the minor tuning, and no more. The twelve-key ground with minor-key songs on A added, and what the evening saves, in cents plus frets over five summed over its songs, if the guitarist may retune once between two songs, against tuning once for the whole set. With the minor songs together after the ground: 1 songs 2.48, 2 songs 4.95, 3 songs 6.25, 5 songs 7.81, 9 songs 10.94, 10 songs 11.18, 14 songs 11.18, 20 songs 11.18. From ten songs on it is 11.18 exactly — what the ground's twelve songs pay for the E-minor shape's tuning, which the one-tuning guitar has by then moved to — and it never grows again. With the same songs spread one at a time through the ground, one retuning saves 1.33 at 3, 1.07 at 5, 2.30 at 10, 1.30 at 20. The dashed line is what any number of retunings could save, every song in its own cheapest tuning: 12.04 from ten songs on. Pitch and tuning

One retuning is worth what the pop songs pay for the minor tuning

A guitarist allowed to retune once in an evening plays the pop songs in their own tuning and the minor songs in theirs. What that saves rises with every minor song until the tenth, and then stops: from there on it is exactly what the twelve pop songs would have paid to be played in the minor songs' tuning, and adding minor songs cannot raise it. The C shape's tuning, which a one-tuning guitar spends most of the range in, is never used. And the whole saving belongs to the running order. Put the same ten minor songs in two runs instead of one and a single retuning is worth three-fifths as much; spread them through the evening and it is worth a fifth.

 A bell whose deciem is the tierce's octave is minor at every stroke. For bells whose hum is on C1, E1, G1, C2, E2, G2, C3, E3, G3, C4, struck so that the hum is at 20 to 110 decibels, with the deciem tuned from 2.5 times the prime to 2.4 — a minor third above the nominal and exactly an octave above the tierce: whether the masked-threshold count delivers a third at all. C1: ···mm··············; E1: ···mmmmmm··········; G1: ··mmmmmmmmm········; C2: ·mmmmmmmmmmmmm·····; E2: ·mmmmmmmmmmmmmm····; G2: ·mmmmmmmmmmmmmmm···; C3: mmmmmmmmmmmmmmmmm··; E3: mmmmmmmmmmmmmmmmmm·; G3: mmmmmmmmmmmmmmmmmm·; C4: mmmmmmmmmmmmmmmmmmm (m a minor third, a dot none; levels in steps of five from 20). 131 of 190 cells read minor and none major. Of the 59 cells in which the bell as cast delivered only its major third, 49 now deliver a minor one. Perception and the listener

A bell can be made minor by the third nobody tunes

A low bell struck hard delivers its major third, the deciem, and loses its minor one, the tierce, to masking. A founder could flatten the deciem from a major to a minor third above the nominal, which puts it exactly an octave above the tierce. The masking count says that move costs the deciem only one or two decibels of stroke at the top of its window, so the flattened deciem still outlasts the tierce in every register. A bell tuned that way reads minor in every cell of register and stroke that delivers a third at all. In-between tunings read as neutral thirds, so the move only works if it goes all the way.

 The fourth below a note and the fifth above it differ by the same amount wherever the note is put. The fourth A3–D4 and the fifth D4–A4 on the grand modelled here, each note with its own string's stiffness, with the octave 1.9 cents wider than 2:1 in its fundamentals: the beat rate of each, signed so that a wide fourth and a narrow fifth are both positive, as the middle note is moved three cents either side of its tempered place. Both lines move together, and the fourth always beats 0.15 a second faster than the fifth — the difference between the upper note's second partial and the lower note's fourth, which the middle note does not touch. The two beat alike only when the octave is 1.60 cents wide, the octave whose 4:2 member is silent. Intervals and chords

The fourth–fifth check cannot hear the note it checks

A tuner tests a temperament by sounding the fourth below a note and the fifth above it, inside one octave, and comparing their beats. Both beats run to the same partial of the middle note, so moving that note speeds or slows both together and leaves their difference untouched. The check is a test of the octave, and of one particular octave: the 4:2. Measured with each note's own stiffness, the printed table laid fourth first builds exactly that octave on any piano; laid fifth first it overshoots, and the check pulls it back by a third of a cent.

 Each family of checks builds its own octave, and the thirds build one twice as wide as the fourths and fifths. On each lower note from F3 to F4, on the grand modelled here: how many cents wider than 2:1 the octave lands when it is filled by a chain of intervals each counted to the rate a bearing plan prints — a fourth then a fifth, three major thirds, a major sixth then a minor third, four minor thirds — against the 4:2, 6:3 and 8:4 octave types, each note with its own stiffness. At A3: fourth, then fifth 1.60, three major thirds 3.20, minor third, then major sixth 3.57, four minor thirds 8.09; types 1.60, 3.56 and 6.22. The fourth and fifth build the 4:2 octave exactly and the minor third and major sixth the 6:3 within a few hundredths of a cent; three major thirds build an octave 2.00 times as wide as the fourth and fifth's, just short of the 6:3. Intervals and chords

The thirds and the fourths check different octaves

Every interval counted to its printed beat rate on a stiff string is left wide, by an amount set by the squares of the partial numbers that meet in it. Add the links of a chain and the sum lands on an octave type: a fourth and a fifth make the 4:2, a minor third and a major sixth make the 6:3, and three major thirds come just short of the 6:3. So the two most familiar checks in a bearing are exact tests of two different octaves, and on a stiff piano no octave passes both. Each fails on the other's octave by a beat a second at A3.

 Only three minor keys want a tuning of their own, and each is a key with open strings in its chords. A minor-key song, i–iv–V–i, in each of the twelve keys, in cents plus frets over five: under the E-minor shape's tuning, the best of the one-tuning vote's three for every minor key, and under the tuning a search finds for that song alone. C minor 5.69 and 5.69; G minor 5.69 and 5.69; D minor 5.69 and 5.35; A minor 5.69 and 5.09; E minor 4.91 and 4.91; B minor 5.69 and 4.53; F♯ minor 5.69 and 5.69; C♯ minor 5.69 and 5.69; A♭ minor 5.69 and 5.69; E♭ minor 5.69 and 5.69; B♭ minor 5.69 and 5.69; F minor 4.89 and 4.89. Nine keys keep the E-minor shape's tuning. D, A and B minor each find a tuning of their own and save 0.34, 0.60, 1.16; A minor's makes its tonic and subdominant one shape, the open A-minor chord and the same shape at the fifth fret, both exact. Pitch and tuning

The minor half's own tuning belongs to one key

A guitarist who retunes once between the pop songs and the minor songs has been choosing among three tunings found for whole evenings. Searched for the minor songs alone, the best tuning is none of them — but only in A, B and D minor, where it gives the tonic and subdominant one shape at the nut and both are in tune. In A it saves six-tenths a song, and the refund a retuning buys stops having a ceiling. In the other nine keys, and in any minor half that mixes keys, the E-minor shape's tuning was already the answer.

 A bell with a major tierce is major wherever it delivers a third. For bells whose hum is on C1, E1, G1, C2, E2, G2, C3, E3, G3, C4, struck with the hum at 20 to 110 decibels, the deciem as cast at 2.5 times the prime and the tierce at 1.25, 386 cents above the prime: which third the masked-threshold count delivers. C1: ···MMMM············; E1: ··MMMMMMMM·········; G1: ··MMMMMMMMMM·······; C2: ·MMMMMMMMMMMMMM····; E2: ·MMMMMMMMMMMMMMM···; G2: ·MMMMMMMMMMMMMMMM··; C3: MMMMMMMMMMMMMMMMMM·; E3: MMMMMMMMMMMMMMMMMM·; G3: MMMMMMMMMMMMMMMMMMM; C4: MMMMMMMMMMMMMMMMMMM (M major only, b a minor tierce with the major deciem, n a neutral tierce, a dot none; levels in steps of five from 20). 141 cells major, 0 with thirds that disagree, 49 with none. Dots mark the 10 cells in which this bell delivers a third and the minor bell with its deciem flattened to the tierce's octave delivers none. Perception and the listener

The major-third bell has no mirror problem

A minor bell struck hard loses its tierce to masking and delivers its major deciem instead. Raise the tierce to a major third and the two thirds agree — but the raised tierce sits closer to the quint, and it was fair to fear that the quint would bury it as the prime buried the minor one. It does not. The prime is what masks a tierce, and moving away from it gains four to twelve decibels of stroke while the quint stays nine decibels too weak to matter. Tuned to agree, the major bell delivers its colour in 141 of 190 cells; the minor bell, made consistent, in 131.

 Heard a little late, two vibratos lock in step; heard a third of a cycle late, they lock in opposition. Two singers' vibratos at 6.0 and 6.2 a second, each pulled towards the phase it hears from the other with a strength of 10 radians a second, the hearing delayed by a stated time: how far apart the two vibratos' peaks settle, in milliseconds, after twenty seconds. 0 ms: 1.6 apart (4°); 20 ms: 2.4 apart (5°); 40 ms: 61.9 apart (-169°); 60 ms: 70.7 apart (-176°); 80 ms: 79.5 apart (-176°); 100 ms: 88.1 apart (-176°); 120 ms: 95.7 apart (-175°); 140 ms: 1.6 apart (4°); 160 ms: 1.6 apart (4°); 180 ms: 1.8 apart (4°); 200 ms: 69.6 apart (-176°). Under about forty milliseconds the lock is near in phase; from about forty to a hundred and twenty it is near antiphase, the peaks most of half a cycle apart; from about a hundred and forty, nearly a whole cycle late, it is near in phase again. The shaded band is the two milliseconds the formant asked for. Instruments and their design

Singers who hear each other late lock their vibratos in opposition

A choir's vibratos give back the section's tuning at the singers' formant only if they are locked to within about two milliseconds. Singers who lock do it by hearing each other, and they hear each other late. Two vibratos pulled towards what each hears of the other settle near in step while the delay is under about forty milliseconds, and even then come within two milliseconds only if each singer corrects a phase error in about thirty. From forty to a hundred and twenty milliseconds late — a third to two-thirds of a vibrato cycle — they lock in opposition, peaks seventy to ninety milliseconds apart, and hide the section's tuning more completely than vibratos that never locked at all.

 Weighed as a share of the beats they compare, the two checks fail equally a ninth of the way from the 4:2 to the 6:3. On A3, on the grand modelled here, each note with its own stiffness: how far apart the fourth and fifth of the fourth–fifth check beat, and the minor third and major sixth of the third–sixth check, each as a share of the rates the table prints for them (0.99 and 11.87 a second), against how many cents wider than 2:1 the octave is. The 4:2 octave is 1.60 cents and the 6:3 3.56. The failures are equal at 1.82 cents, 11.2 per cent of the way from the 4:2 to the 6:3, where both checks fail by 11.3 per cent; balanced in beats a second instead, the octave is 2.78 cents and the fourth–fifth check fails by 60.7 per cent. Both checks are within the 15.0 per cent line on octaves from 1.31 to 1.89 cents. Intervals and chords

The octave that fails both checks least

A tuner who cannot pass both the fourth–fifth check and the third–sixth check settles on an octave that fails them equally, and a listener compares beat rates by ratio. Weighed that way, the compromise sits a ninth of the way from the 4:2 octave to the 6:3 on every note of every piano, because the fourth beats twelve times slower than the minor third and that ratio belongs to equal temperament, not to the strings. On a grand both checks then pass to a fifteen-per-cent ear up to middle C, and above it no octave passes both.

 With a capo up to fret 7, 8 of the twelve minor keys are played as B minor in one tuning. A minor-key song, i–iv–V–i, in each of the twelve keys, in cents plus frets over five, under the tuning searched for the song on A (the B string raised 13.7 cents): at the nut, and at the capo position up to fret 7 that suits it best, with the E-minor shape's tuning at its best capo for comparison. C minor 5.72 → 4.53 (capo 1); G minor 5.72 → 5.72; D minor 5.72 → 4.53 (capo 3); A minor 5.09 → 5.09; E minor 5.72 → 4.53 (capo 5); B minor 4.53 → 4.53; F♯ minor 5.72 → 4.53 (capo 7); C♯ minor 5.72 → 4.53 (capo 2); A♭ minor 5.72 → 5.72; E♭ minor 5.72 → 4.53 (capo 4); B♭ minor 5.72 → 5.09 (capo 1); F minor 5.72 → 4.53 (capo 6). 8 keys reach 4.53, B minor's price, by putting the capo where B minor's open-string chords become theirs. E minor costs 4.53 at capo 5, against 4.91 in the E-minor shape's tuning. Pitch and tuning

A capo is a transposition the tuning cannot see

Tempered frets make a chord shape's error the same wherever it sits, so a song with a capo at the fifth fret is, to the tuning, the song five semitones lower on a shorter neck. That lets every minor key borrow B minor's open-string fingerings, and eight of the twelve take the raised-B tuning's best price with a capo no higher than the seventh fret. A minor half in mixed keys, which gave a retuning nothing, now nearly doubles its refund — but the capo pays the single-tuned evening as well, and a minor half all in A gets less from its retuning than before.

 With the tierce at 1.25 times the prime, the lower modes point only at the strike note's own name. For bells whose hum is on C1, E1, G1, C2, E2, G2, C3, E3, G3, C4, struck with the hum at 20 to 110 decibels, the tierce at 1.25 times the prime, 386 cents: the series the delivered lower modes — hum, prime, tierce and quint — fit best, within thirty cents. C1: ···················; E1: ·······SSS·········; G1: ·····SSSSSSS·······; C2: ··SSSSSSSSSSSS·····; E2: ·SSSSSSSSSSSSSSS···; G2: ·SSSSSSSSSSSSSSS···; C3: SSSSSSSSSSSSSSSSS··; E3: SSSSSSSSSSSSSSSSSS·; G3: SSSSSSSSSSSSSSSSSS·; C4: SSSSSSSSSSSSSSSSSSS (S the strike note's name, 3 the note a major third below it, a dot fewer than two lower modes delivered; levels in steps of five from 20). 124 cells point at the strike's name, 0 a major third below, 66 at nothing. Perception and the listener

The raised tierce points the bell at its own note

A bell's four lowest modes — hum, prime, tierce and quint — can imply a pitch of their own. With a major tierce they are the second, fourth, fifth and sixth harmonics of the prime two octaves down: the strike note's own name, with one empty slot. It was the minor tierce that pointed elsewhere all along. Its lower modes are the fifth, tenth, twelfth and fifteenth harmonics of the note a major third below the strike, and in 82 of 190 struck bells that is the series the delivered modes fit. Raising the tierce removes a competitor rather than adding one.

 A drum that is always felt is worth no more than one that is always heard, until the crowd drowns the hearing. Ten metres known, entered at a random step, recognised within the 16-step present, the crowd covering half the time at a stated level above the band: the points marking the downbeat adds, for each way of marking it heard through the crowd, and for a low drum that is felt as well and so always arrives. a louder melody note: +0 28.9, +5 20.4, +10 14.7, +15 11.6, +20 11.3, +25 11.3, +30 11.3; a bass note an octave below: +0 28.9, +5 22.1, +10 19.8, +15 18.0, +20 13.7, +25 11.4, +30 11.3; a low drum: +0 28.9, +5 22.1, +10 19.8, +15 19.8, +20 17.3, +25 12.9, +30 11.4; a high bell: +0 28.9, +5 22.1, +10 19.8, +15 18.2, +20 14.3, +25 11.5, +30 11.3; a low drum, felt as well: +0 28.9, +5 22.1, +10 19.8, +15 19.8, +20 19.8, +25 19.8, +30 19.8. A quiet room's mark is worth 23.4. Rhythm and metre

A felt drum reaches only where the heard one already does

A low drum near a stage is felt in the body as well as heard, and talk cannot mask a vibration. Given that second channel, a drum-marked downbeat always arrives — and is then worth 19.8 points to a late dancer, exactly what an always-heard drum was worth, because the crowd has already taken the soft strokes that made a mark worth more. The felt channel pays only under crowds twenty decibels over the band. And it is driven by the drum's own level, which in a reverberant room is high enough to be felt only where the drum is already loud enough to be heard.

 In an equal bearing the progression of thirds pulls the octave back onto the 4:2. On A3 on the grand modelled here, with the bearing set equal: the failures of the fourth–fifth check, the minor third–major sixth check and the progression of thirds across the octave's edge, each as a share of what it compares, against how many cents wider than 2:1 the octave is. The progression passes exactly on an octave 0.13 cents wide; the 2:1 type is 0.40, the 4:2 1.60, the 6:3 3.56. The two checks alone balance at 1.82 cents, failing by 11.3 per cent; with the progression as a third vote the octave that fails all three least is 1.63 cents, where the worst failure is 12.6 per cent (fourth–fifth 1.6 per cent, third–sixth 12.6 per cent, progression 12.6 per cent). All three pass at fifteen per cent on octaves from 1.31 to 1.88 cents. Intervals and chords

The progression of thirds votes for the bearing, not the octave

Added as a third vote beside the two contiguous checks, the progression of thirds was expected to push a stiff piano's octave wider. It pulls it narrower, because the octave it asks for is not set by the strings at all: it is whatever stretch the bearing already carries, plus an eighth of a cent. In an equal bearing that drags the compromise back onto the 4:2 and costs the top of the bearing its pass; give the bearing a fifth of the octave's own stretch and the third vote falls silent.

 With the capo left at fret 5, E minor becomes B minor and the other keys pay for it. Twelve minor-key songs, i–iv–V–i, one in each key round the circle, in the raised-B tuning, in cents plus frets over five: at the nut, with the capo left at fret 5 for every song, and at each song's own best capo up to fret 7. C minor 5.72, 8.62 at fret 5, 4.53 at its own; G minor 5.72, 5.72 at fret 5, 5.72 at its own; D minor 5.72, 5.09 at fret 5, 4.53 at its own; A minor 5.09, 6.13 at fret 5, 5.09 at its own; E minor 5.72, 4.53 at fret 5, 4.53 at its own; B minor 4.53, 8.62 at fret 5, 4.53 at its own; F♯ minor 5.72, 5.72 at fret 5, 4.53 at its own; C♯ minor 5.72, 8.62 at fret 5, 4.53 at its own; A♭ minor 5.72, 5.72 at fret 5, 5.72 at its own; E♭ minor 5.72, 5.72 at fret 5, 4.53 at its own; B♭ minor 5.72, 7.82 at fret 5, 5.09 at its own; F minor 5.72, 5.72 at fret 5, 4.53 at its own. The twelve cost 66.82 at the nut, 78.04 with the capo at fret 5 and 57.87 with it moved for every song. At fret 5 the song on E takes B minor's fingerings and the song on D A minor's; every other key is moved to a key the tuning does not favour, on a neck 5 frets shorter. Pitch and tuning

A capo left in place serves one key

Moved for every song, a capo lets nearly every minor key borrow B minor's open-string fingerings. Left at one fret for a whole run of minor songs, as most guitarists leave it, it does something much weaker: it renames the keys. Over a run in every key that is a reshuffle on a shorter neck and can never beat the nut; over the four keys guitarists use most it gains nothing either, because two of them are already the tuning's best. A fixed capo pays only a run in very few keys, spaced the way the tuning's cheapest keys are.

 Two parts entering together mask each other unequally: the carrier loses a partial or two, the passenger up to eleven. Every pairing of two parts entering together on a chord of the passage in which the quieter part sits under its mask on average, 71 of them, sorted. Up: how many of the carrier's partials that are clear of the mask with the carrier alone over the chord are pushed under it by the passenger. Down: how many of the passenger's are pushed under by the carrier. In all, carriers lose 19 partials and passengers 135. 54 carriers lose none, 15 lose one and 2 lose two; passengers lose 0 in 25, 1 in 25, 2 in 6, and three or more in 15, the most being 11. Form and structure

A carrier gives up room, not partials

A passenger masks its carrier as the carrier masks it, so the arrival might be a new spectrum belonging to neither part. Across the seventy-one pairings of the passage it almost never is. Carriers lose nineteen partials in all to their passengers and passengers lose a hundred and thirty-five to their carriers; a voice beside a violin takes thirteen decibels of the violin's room and not one of its partials. The nine pairings that leave neither part a majority are mostly on one chord, and there the chord had already done it.

 A venturi four centimetres long moves the middle of the series as much as the top. The trumpet with a 25-centimetre leadpipe, and the same leadpipe with a venturi four centimetres long narrowing its radius by 20 per cent at its centre, three centimetres from the receiver: how far each of the first ten resonances moves, in cents. The written middle is shaded. 1: +7.3, 2: +6.1, 3: +2.4, 4: -2.4, 5: -7.9, 6: -8.5, 7: -4.2, 8: +1.8, 9: +6.1, 10: +6.7. The largest shift in the written middle is 8.5 cents and above it 6.7; the series' mistuning goes from 15.0 to 14.2 cents rms. The lowest resonances rise, the middle ones fall, and the top rises again. Instruments and their design

The venturi is heard by every resonance

The narrowing makers put just inside a trumpet's leadpipe was predicted to be invisible to the written middle and to act only at the top of the compass, because short tapers had acted only there. It is not a taper. To every resonance in the compass a venturi a few centimetres long is one lumped object, described by a single number — its length times its depth — and by where it sits. Four centimetres at a fifth of the radius moves the middle notes by up to eight and a half cents, raising some and lowering others, while the series as a whole stays as well tuned as before.

 From the page alone, a thinning ending is a diminuendo and a fuller final chord is barely louder. Short-term loudness through the closing bars of a thirty-two-bar scheme whose texture is 6 parts until the ending, computed from each bar's part count with no performance in it: the parts realised every way their ranges allow, each bar's partials summed as an excitation pattern, and the result passed through the two loudness smoothers. The dashed line is the running impression under the thinning ending. thins to one: final bar 73.2 phons, arriving at 0.690 of the running impression (-4.46 phons); full final chord: final bar 84.7 phons, arriving at 1.004 of the running impression (+0.05 phons); unchanged: final bar 83.3 phons, arriving at 1.000 of the running impression (+0.00 phons); thins then full: final bar 84.2 phons, arriving at 1.014 of the running impression (+0.17 phons). Perception and the listener

The page can write a diminuendo and not a climax

Counted on the excitation pattern rather than in grouped bands, the four ending textures the closure essays compare come out the right way round at last: a thinning ending is a ten-phon diminuendo written into the score, and an ending that adds parts arrives louder than what came before rather than quieter. But the gain is tiny. Three more parts on a six-part final chord buy under a phon and arrive a twentieth of a phon above the listener's running impression; only a full chord straight after a thinning stands out, and only at a quick tempo. A page can take loudness away. It cannot make a climax.

 A third part takes from the carrier what it would have taken alone; the room the passenger spent adds almost nothing. Every pairing of carrier and buried passenger, with each other part of the roster entering beside them as a third: 284 arrivals. For each third part, the carrier's partials — clear of the mask over the chord alone — that are under it with all three sounding, split into those the passenger alone already took, those the third part would take on its own, and those lost only to the two together. clarinet: 46 arrivals, 19 partials (14 to the passenger, 3 to the third alone, 2 only together), the third alone costing 1.5 dB of room; oboe: 50 arrivals, 20 partials (9 to the passenger, 9 to the third alone, 2 only together), the third alone costing 1.3 dB of room; brass: 46 arrivals, 23 partials (8 to the passenger, 14 to the third alone, 1 only together), the third alone costing 1.0 dB of room; voice: 47 arrivals, 40 partials (14 to the passenger, 20 to the third alone, 6 only together), the third alone costing 7.3 dB of room; violin: 47 arrivals, 127 partials (17 to the passenger, 108 to the third alone, 2 only together), the third alone costing 8.8 dB of room; flue: 48 arrivals, 209 partials (14 to the passenger, 187 to the third alone, 8 only together), the third alone costing 14.1 dB of room. In all 438 partials: 76 to the passengers, 341 to the thirds on their own, 21 only to the combination. Form and structure

A third part takes what it would have taken alone

A passenger costs its carrier room rather than partials, and the obvious worry is that the room compounds: that a third part, entering beside a carrier already ten decibels nearer its limit, finds it fragile. Across 284 three-part arrivals it does not. Of the 438 carrier partials lost with three parts sounding, 21 are lost only to the combination; the rest are taken by one of the two parts alone. Masking adds in power, so a threshold the passenger has raised rises less when a third part adds to it. What decides the loss is which part enters third, and two parts — the flue and the violin — take five sixths of it.

 A bearing counted to the printed rates leaves its thirds beating in a zigzag, and which thirds zig depends on the plan. A bearing from F3 to E4 laid out from A3 on the modelled grand, each note set so that the fourth or fifth from a note already set beats at its printed rate: the true beat rate of each major third whose notes are both in the bearing, by two plans, against an equal bearing on the same strings and the printed table. counted: fourths up, fifths down: F3 6.46, F♯3 6.80, G3 8.41, A♭3 7.63, A3 9.45, B♭3 8.57, B3 9.01, C4 11.17; counted: the sharps by fifths up: F3 6.46, F♯3 7.99, G3 7.14, A♭3 8.97, A3 8.02, B♭3 8.57, B3 9.00, C4 9.46; equal temperament: F3 6.67, F♯3 7.04, G3 7.43, A♭3 7.84, A3 8.27, B♭3 8.71, B3 9.17, C4 9.66. Intervals and chords

A bearing counted correctly is stretched by its route, not its strings

A tuner who sets a temperament octave by counting each fourth and fifth to its printed beat rate leaves every one of them a little wide on a stiff string, and the bearing ends up carrying a stretch nobody chose. Whether that stretch is enough to quiet the progression of thirds across the octave's edge turns out to be the wrong question. The stretch belongs to the order of the plan, not to the piano: one ordinary plan carries five times what the progression needs, another a fraction. And either plan leaves the thirds inside the bearing beating in a zigzag, faster then slower then faster, by up to eighteen per cent a step on a grand — which is why the thirds have to set the bearing rather than check it.

 In a ring of minor bells, five of the eight hidden pitches fall off the scale the ring rings. A ring of eight bells with minor tierces tuned to a major scale, the tenor's strike note C4: for each bell and each stroke with the hum at 40 to 90 decibels, what the delivered lower modes — hum, prime, tierce and quint — imply. C4 (hidden A♭, off it): 333333333SS; D4 (hidden B♭, off it): 3333333333S; E4 (hidden C, on the scale): 33333333333; F4 (hidden D♭, off it): 33333333333; G4 (hidden E♭, off it): 33333333333; A4 (hidden F, on the scale): 33333333333; B4 (hidden G, on the scale): 33333333333; C5 (hidden A♭, off it): 33333333333 (3 the note a major third below the strike, S the strike's own name, a dot too few lower modes delivered). 33 strokes imply a note on the ring's scale and 52 a note off it. Perception and the listener

A ring of minor bells rings a second scale a major third below

A minor-tierce bell's four lowest modes, read as a residue, imply a note a major third below the bell's strike note. Put eight such bells in a ring tuned to a major scale and the implied notes are that scale moved down a major third — a ring in C carries the scale of A♭ major underneath it. Only three of the eight land on notes the ring rings, from the mediant, the submediant and the leading note; the tonic, the octave, the supertonic, the subdominant and the dominant all imply notes the ring never sounds, each a semitone from two that it does. Above a tenor on C3 the lower modes deliver the implied note at nearly every stroke. A reading that weights the whole spectrum still hears none of it.

 Felt on every landmark, the stream recovers most of a quiet room's recognition even when no accent is felt. Ten dances known, entered at a random step, the crowd covering half the time at the level on the axis above the band: the share of entries in which the dance is recognised within the present, for four ways of carrying the landmarks — a drum on the downbeat and the melody's softer strokes between, heard; the same with the downbeat felt as well; a kick drum on every landmark eight decibels softer than the downbeat, heard; and the same felt, with a felt accent recognised 0 per cent of the time. A quiet room gives 80.9. drum downbeat, melody between: 0 dB 58.4, 5 dB 41.0, 10 dB 37.4, 15 dB 37.3, 20 dB 34.9, 25 dB 30.4, 30 dB 28.9; the downbeat felt: 0 dB 58.4, 5 dB 41.0, 10 dB 37.4, 15 dB 37.3, 20 dB 37.3, 25 dB 37.3, 30 dB 37.3; a kick on every landmark, heard: 0 dB 80.9, 5 dB 80.9, 10 dB 68.1, 15 dB 50.2, 20 dB 35.7, 25 dB 30.4, 30 dB 28.9; a kick on every landmark, felt: 0 dB 80.9, 5 dB 80.9, 10 dB 80.9, 15 dB 80.9, 20 dB 75.8, 25 dB 71.6, 30 dB 70.1. Rhythm and metre

The felt channel carries the inner beats, not the downbeat

A low drum felt in the body gets the downbeat through any crowd, and is then worth less than it was in a quiet room, because the crowd has taken the soft strokes between. Put a kick drum on every landmark of the dance and let all of it be felt, and in a crowd thirty decibels over the band a late dancer recognises the dance in seventy per cent of entries against thirty-seven for the felt downbeat — with no felt accent at all. The accent, felt or not, is worth about ten points on top. The worry that a stream felt alike has no downbeat turns out to be the small half of the arithmetic.

 Ordered so that neighbouring songs are a fifth apart, a set moves the capo most and recovers least. Twelve minor-key songs, three each in E, A, D and B minor, i–iv–V–i, in the best of the four tunings, the capo allowed to change fret 3 times between songs: the share of what moving the capo for every song saves over playing at the nut. grouped by key: 100%; a different key each song, at random: 51% on average over 40 orders, from 35 to 70; neighbours a fifth apart: 35%. The fifths order is B E B D A E B E D A D A. Pitch and tuning

A set ordered by key relation moves the capo most

Grouped by the fret each song wants, a set of twelve minor songs in E, A, D and B gets the whole of what a moving capo saves with three capo changes. A real set is not grouped by fret; it is ordered for the music, and the commonest rule for that order is key relation — neighbouring songs a fifth apart. That order is the worst the capo can face: no two neighbours want the same fret, and three changes recover thirty-five per cent of the saving against fifty-one for a random order. The capo groups keys a tone apart, and a set built on fifths is the order that keeps them furthest from each other.

 Marked down, the violin gives its carriers back their partials while it is still heard; the flue does not. Every arrival in which a carrier and its buried passenger enter with a third part beside them (284 arrivals), the third part marked 12 dB below the others: for each third part, the carrier partials clear beside the passenger that the third still pushes under the mask (filled bar; the outline is its cost at full level), and the share of its arrivals in which the third itself still changes the sound audibly (printed). flue: 114 of 195, heard in 100%; violin: 8 of 110, heard in 66%; voice: 1 of 26, heard in 28%; brass: 0 of 15, heard in 0%; oboe: 0 of 11, heard in 2%; clarinet: 1 of 5, heard in 2%. In all 124 of 362. Form and structure

A quieter third part gives back only as deep as it took

A high part entering third costs its carrier partials, and the masking rule suggests that marking it down should return them all at once. It does not. Each partial comes back when the third part has been lowered by about as far as it had pushed that partial under, so the return is as gradual as the depths are spread. The violin's are shallow: twelve decibels down it gives back nine-tenths and is still heard in two arrivals of three. The flue's run to thirty decibels, and there is no level at which it is both heard and harmless.

 Two strings a few cents apart carry the pair through every interval the regulator feared, and its brightness breathes. A harpsichord's two eight-foot registers, plucking at 32 and 50 millimetres from the nut of a 355-millimetre string, each on its own string of a unison at C5, the second tuned 2 cents above the first: the spectral centroid of the pair, in partial numbers, through the first three seconds of a note (both strings decaying alike). At the pluck it is the aligned pair's 2.44; through the note it ranges from 2.29 to 3.67, repeating every 1.65 seconds. The detuning carries the pair past the regulation tolerance of a decibel at the fundamental — 0.153 of a period — after 254 milliseconds (the line). Every half second: 0.0 s 2.44, 0.5 s 3.12, 1.0 s 3.34, 1.5 s 2.53, 2.0 s 2.90, 2.5 s 3.66, 3.0 s 2.89. Instruments and their design

Two strings undo what two quills were regulated for

A harpsichord's two eight-foot registers each pluck their own string of a unison, and the unison is never exact. Two cents apart, the second string's partials turn against the first's at a rate proportional to their number, so within a quarter of a second at C5 the pair has drifted past the regulation tolerance the jacks were voiced to, and through the rest of the note it passes again and again through every interval a regulator could have set. The pair's fixed spectrum becomes one that breathes, its own holes become flickers a few per cent of a cycle long, and averaged over the note the second register is worth a decibel and a half at the fundamental, not four.

 Set by its own thirds, the bearing bends smoothly upward and carries five eighths of its octave's stretch. The bearing F3–E4 on the modelled grand, A3 at equal temperament: each note's cents from equal temperament when every note is solved at once so that each major third beats faster than the one below by exactly the printed step and the fourths and fifths come as near their printed rates as that allows, against the two plans counted by fourths and fifths. set by its thirds: F3 -0.19, F♯3 -0.17, G3 -0.13, A♭3 -0.07, A3 +0.00, B♭3 +0.07, B3 +0.16, C4 +0.27, C♯4 +0.40, D4 +0.53, E♭4 +0.69, E4 +0.88; stretch 1.15 cents an octave. counted: fourths up: F3 +0.43, F♯3 +1.39, G3 +0.23, A♭3 +1.17, A3 +0.00, B♭3 +0.92, B3 +1.93, C4 +0.82, C♯4 +1.82, D4 +0.71, E♭4 +1.69, E4 +2.78; stretch 1.62 cents an octave. counted: sharps by fifths: F3 +0.43, F♯3 -0.82, G3 +0.23, A♭3 -1.03, A3 +0.00, B♭3 +0.92, B3 -0.28, C4 +0.82, C♯4 -0.38, D4 +0.71, E♭4 -0.51, E4 +0.57; stretch 0.51 cents an octave. The octave's own stretch by the two contiguous checks is 1.82 cents. Intervals and chords

A bearing set by its thirds carries five eighths of its octave

A bearing counted from the printed fourths and fifths leaves its thirds zigzagging round the printed progression and takes whatever stretch its route gives it. Set the other way — every note solved at once so that each major third beats faster than the one below by exactly the printed step — the bearing bends smoothly upward and carries 63 per cent of its octave's own stretch, on a grand, on an upright and on everything between. That is more than three times what the octave's three checks need in order to agree, so following the thirds reconciles them with room to spare. The price is paid by the fourths and fifths, every one of which then beats slow.

 A straight tone's slow wander drifts a chord's notes out of step and leaves them there. The tuned close seventh over C4 held without vibrato, each note's pitch wandering on its own by 0.5 cent rms with its spectrum falling above two hertz: how far ahead of the bass each upper note's timing has run, in milliseconds, over eight seconds of one draw, with this register's onset tolerance of ±0.070 ms shaded. Over eight draws the chord is in step 4.3 per cent of the time and first leaves the tolerance after 0.16 s on average; a three-cent vibrato on every note, each its own, keeps it in step 48.2 per cent. Timbre and acoustics

A straight tone drifts out of step and stays there

A tuned close seventh binds — its filters flutter together — only while its notes keep their timing within a fraction of a millisecond, and a narrow vibrato keeps them there because a sinusoid's timing offset swings inside a fixed range. A straight-tone singer's pitch is never quite straight: it wanders slowly, a cent or less, at random. That wander's timing offset is a random walk. Over C4 a wander of half a cent takes the chord out of step within a fifth of a second and leaves it in step four per cent of the time, where a three-cent vibrato holds it nearly half. What matters is the wander's slow part, and adding a vibrato on top never helps.

 Searched over every order, a set can step by fifths at each capo change for free, and pays about seven points for each fifth beyond. Twelve minor songs, three each in E, A, D and B minor, no key twice running: among all 41,304 orders, for each number of neighbouring songs a fifth apart, the largest share of a moving capo's saving recovered with 3 capo changes, with the mean over all orders at that count. 0: best 84%, mean 84% (2 orders); 1: best 84%, mean 81% (66 orders); 2: best 84%, mean 70% (590 orders); 3: best 84%, mean 62% (2630 orders); 4: best 78%, mean 55% (6738 orders); 5: best 70%, mean 50% (10626 orders); 6: best 63%, mean 46% (10626 orders); 7: best 61%, mean 42% (6738 orders); 8: best 54%, mean 39% (2630 orders); 9: best 46%, mean 37% (590 orders); 10: best 40%, mean 36% (66 orders); 11: best 37%, mean 37% (2 orders). The earlier essay's fifths order (9 fifths) recovers 35%, and a random order 51% on average. Pitch and tuning

A set steps by fifths where the capo moves

A set of twelve minor songs in E, A, D and B can be ordered for the capo or for key relation, and the two orders pull apart. Searched over every one of its 41,304 orders, the trade between them is exact and simple: a set can put a fifth between neighbouring songs at each capo change for nothing, because a fifth always crosses frets and a capo change is where crossing is paid for. Beyond that each extra fifth costs about seven points of the capo's saving. The tone steps the earlier essay suspected of carrying the compromise are where the capo stays still.

 Struck, a suspension's roughness falls whether or not it resolves, and the step itself sends it back up. One 4–3: C5 prepared over a C chord, tied over G3, D4 and G4, and stepping to B4, each moment 1 s. Roughness, Plomp–Levelt over eight partials of every pair weighted by the product of their amplitudes. Sustained: suspension 0.327, resolution 0.325. Struck, with a T60 of 12 s at middle C: suspension 0.149, resolution 0.054, and with the note left hanging 0.024 at the same moment. Harmony and voice leading

A tied note is the oldest thing in its own dissonance

A suspension is taught as a dissonance that resolves, and the sum is done on a chord symbol with no time in it. Put the time back. On voices or an organ, a 4–3 is no smoother after its step than before: of eight spacings, two fall and four rise. On a piano the tied note was struck a chord ago, so it supplies under a third of the roughness it is blamed for, the fall to the resolution is the clock's, and the step itself makes the sound about three times rougher than leaving the note where it was.

 The twelve fourths and fifths of a bearing are one closed circle, and on stiff strings they cannot all beat at their printed rates. A bearing from F3 to E4 holds seven fourths and five fifths that join all twelve notes in one closed circle: A3, E4, B3, F♯3, C♯4, A♭3, E♭4, B♭3, F3, C4, G3, D4 and back. On the modelled grand, counting all twelve at their printed rates misses closing by 2.22 cents. Shared equally, every interval beats 9.4 per cent slower than the table says, and the bearing carries 1.14 cents an octave. Intervals and chords

A stiff string gives the bearing a comma of its own

The twelve fourths and fifths inside a piano's temperament octave are not twelve checks. Laid end to end they are one closed circle, and on stiff strings their printed rates fail to close it by 2.22 cents. Every disagreement between a bearing's tests follows from that one number: which interval never gets counted decides the stretch, and the zigzag in the thirds is the seam. Spread it round the circle and every fourth and fifth beats nine per cent slow of the table, and the thirds come right without being asked.

 At speed each note of a clarinet's rising scale is cut off nearer its steady level than the last, which is a crescendo nobody wrote. A two-octave scale from D3 on a modelled clarinet at 16 notes a second, 62.5 ms a note, each fingering change handing the air column a new resonance that starts from nothing and rises with a time constant set by its Q. The level each note reaches before the next begins: -3.7 dB, -3.4 dB, -3.1 dB, -3.0 dB, -2.7 dB, -2.5 dB, -2.2 dB, -2.1 dB, -1.9 dB, -1.7 dB, -1.6 dB, -1.4 dB, -1.2 dB, -1.0 dB, -0.9 dB. From bottom to top that is a rise of 2.8 dB. Instruments and their design

A fast run is a crescendo nobody wrote

A wind note takes tens of milliseconds to build, and a fast run gives each note about sixty. The question sounds like whether a note can arrive at all, and the settling time makes it sound worse than it is, because a note cut off at nine tenths has lost under a decibel. What speed really costs is evenness: the settling time falls up the compass, so at sixteen notes a second a rising two-octave scale gains between two and four and a half decibels with no change in the player's breath. A repeated note, which keeps its resonance, loses only a quarter as much.

 A loud, slow tune spends a singer's whole breath before each phrase ends, and the extra breaths have to go inside the phrases. Ode to Joy from D4 at crotchet 40 and 90 dB, sung by a trained singer with 2.8 litres of usable air and a flow set by each note's pitch and level. The line is the air left. The plan is the cheapest that reaches the end, with breaths at notated phrase ends costing nothing: 6 breaths, 4 of them inside a phrase. The tune lasts 48 s. Form and structure

A singer needs a quarter of the breaths the page offers

A breath in a sung phrase looks like evidence of a boundary, and also like a physical necessity, and it cannot be both at once. Solved exactly over three tunes, a trained singer is forced to breathe inside a phrase on 8 settings of 147, all of them slow and loud; at a comfortable level the air asks for about a quarter of the phrase ends a tune offers. So nearly every breath is a choice, which is what makes it a cue. Where the air does force one, it lands on a listener's boundary no more often than chance.

 Letting songs move a tone adds between five and twenty points at any count of fifths, and never makes the trade disappear. Twelve minor songs, three each in E, A, D and B minor, with three capo changes and no key twice running, searched exactly. For each number of neighbouring songs a fifth apart, the largest share of a moving capo's saving, with every song in its own key and with up to 12 moved up or down a tone. 0: 84% and 92%; 1: 84% and 92%; 2: 84% and 92%; 3: 84% and 92%; 4: 78% and 83%; 5: 70% and 75%; 6: 63% and 75%; 7: 61% and 75%; 8: 54% and 66%; 9: 46% and 58%; 10: 40% and 57%; 11: 37% and 57%. Pitch and tuning

A set free to change key becomes a set in two

A set list orders its songs for the music and a capo wants them ordered by fret, and the trade between the two looked as if it might disappear if a singer would move a song a tone. It does not. Searched exactly, moving songs adds five to twenty points of the capo's saving at any count of fifths and stops improving at six moved; at its limit the best set rocks between B minor and E minor, because a count of fifths rewards a pendulum. A set that really walks the chain of fifths has three to spend, not eleven.

 Up every wind instrument's compass the partials draw together, so whether a note parts depends mostly on how high it is. Five modelled bores, every semitone of a playing compass: how far apart the first and last of a note's partials reach 50 per cent of their steady amplitude, against a threshold of twenty milliseconds for hearing a partial out. trumpet: 165 Hz 26 ms, 262 Hz 21 ms, 416 Hz 16 ms, 660 Hz 10 ms; horn: 87 Hz 36 ms, 138 Hz 29 ms, 219 Hz 24 ms, 348 Hz 15 ms, 552 Hz 9 ms; trombone: 82 Hz 33 ms, 130 Hz 36 ms, 207 Hz 30 ms, 328 Hz 21 ms, 521 Hz 14 ms; clarinet: 147 Hz 33 ms, 233 Hz 24 ms, 370 Hz 16 ms, 588 Hz 11 ms, 933 Hz 0 ms, 1482 Hz 0 ms; alto saxophone: 139 Hz 43 ms, 221 Hz 45 ms, 350 Hz 26 ms, 556 Hz 13 ms. Below twenty milliseconds from: trumpet 311 Hz, horn 293 Hz, trombone 368 Hz, clarinet 311 Hz, alto saxophone 441 Hz. Perception and the listener

A blown note parts by its register, not its family

A wind note's partials reach their steady levels at different rates, and one ranking of five instruments had the saxophone's parting soonest and the horn's latest. Run at every note rather than one, that ranking dissolves: it was read at pitches from 126 to 428 hertz, and pitch was doing most of the sorting. Moving up one instrument shifts the point of parting three times as far as changing instrument at one pitch, and read at half of steady level, all five come together within a fifth of each other, between about 290 and 440 hertz.

 A section whose upper partials beam keeps its pizzicato deep into a hall on its axis, and loses it faster than an unpointing one a little way off. The margin at the strike for a pizzicato section of twelve over a held clarinet, 15 m into a hall, from C4 to F♯5, with each partial's direct sound scaled as from a piston of 10 cm radius. Above zero the pluck owns the note. no beam: A♭4 to C♯5 -0.3, -1.7, -3.4, -3.5, -3.3, -1.7; on the axis: A♭4 to C♯5 5.2, 4.1, 2.8, 3.0, 3.5, 5.3; 15° off: A♭4 to C♯5 4.4, 3.2, 1.8, 1.8, 2.2, 3.9; 30° off: A♭4 to C♯5 1.9, 0.3, -1.6, -2.2, -5.1, -2.1. Timbre and acoustics

A section keeps its pizzicato only inside a cone

In a hall a section's pizzicato loses its corpus notes to a held clarinet within two metres, on the assumption that a violin radiates equally in every direction. It does not, and twelve players sharing a beam cannot average it away. Directly on the players' axis, the same section keeps every note for twenty-five metres. Thirty degrees off, with the beam's upper partials thrown elsewhere, it is worse than if it had never pointed. A question about how far back a pizzicato carries turns out to be a question about which way the listener's seat faces.

 A trill between a suspension and its resolution stays rougher than the chord struck once, all the way through. One 4–3 on a struck instrument, T60 12 s at middle C: C5 over G3, D4 and G4 for the one second of the suspended chord. Mean roughness over the chord: tied 0.077, struck once 0.119, trill to the upper neighbour D5 at 8 notes a second 0.108, trill to the resolution B4 0.141. Trills are read per whole cycle of two notes. Harmony and voice leading

A trill is half a consonance unless it borrows its own resolution

A struck suspension loses roughness because its tied note is old, and the obvious repairs are all things a player already has. Three of them fail in ways nobody would predict. A re-strike under about nine decibels below the chord is smoother than the tie it replaces. No rate of trill stops the fading, because the three notes under the trill are what is fading. And a trill on the upper neighbour spends half its time on a chord note: only the trill that alternates a suspension with its own resolution gives back more than a single strike.

 One walk round the circle takes most of the error out of the interval nobody counted. One seeded tuner on the modelled grand, each count 5.0 per cent out at random, the closing unison read to 0.02 Hz. Each interval's true rate against the rate that closes the circle exactly. From the table alone the twelfth interval, which is never counted, is -133 per cent off; after one walk and a correction of -11.6 per cent it is 36 per cent off. Stretch 1.38 against 1.07 cents an octave. Intervals and chords

One walk round the bearing measures the piano and the tuner together

A tuner who counts twelve fourths and fifths to the printed table and comes home flat has measured something. Divide the closing beat's rate by three and correct the whole table by that share, and the worst interval improves from a hundred per cent off to about twenty, on any piano. It keeps the table with one number. And the number cannot tell a stiff string from a slow count: a tuner counting nine per cent slow gets the right bearing from the wrong table, and one correction removes that luck along with the error.

 A vibraphone's motor makes a note quieter and far longer, with a tremolo that is shallower than the valve's two states at middle C. The power a struck bar and its tube radiate at middle C, in decibels, with the coupling switched at 6 openings a second. A tube's filling time here is 98 ms. With the bar holding the same energy, the open valve radiates 13.8 dB more than the shut one; with the motor running the tremolo's depth is 10.0 dB. Instruments and their design

A vibraphone's valve is a door on the sound and a drain on the bar

A vibraphone's motor is usually described as a duty cycle: loud while the valve is open and quiet while it is shut, with a tremolo as deep as the gap between them. Put the tube's own time back and the description comes apart. The door and the drain obey different laws. A low note's tremolo is a third shallower than a high note's at the same speed, the note rings almost three times longer than with the valve left open and well beyond any mean of the two states, and the same stroke can land eight decibels apart according to where the motor happens to be.

 Moved down an octave a third part gives back most of what it takes and is still heard; moved up it takes more. The 284 arrivals in which a carrier and its passenger enter with a third part beside them, the third moved 1 octave down. Carrier partials it pushes under the mask, against where it stood: flue 50 (was 195), heard in 100%; violin 3 (was 110), heard in 98%; voice 23 (was 26), heard in 100%; brass 7 (was 15), heard in 98%; oboe 5 (was 11), heard in 100%; clarinet 7 (was 5), heard in 100%. Altogether 95, against 362. Form and structure

An octave down a third part is heard and harmless, with nothing to spare

A flue pipe cannot be marked down, but it can be moved, and the expected answer was upward: an octave higher clears the carrier's main region and spreads over partials that are already faint. The count says the opposite. Up an octave a third part takes half as many partials again. Down an octave it takes a quarter of what it took, gives a voice back its whole upper spectrum, and is still heard in nearly every arrival. No level reaches that. The price is that the part has spent its spare level, and three decibels soft takes it out of a third of them.

 Antiphonal second violins own their worst pizzicato note almost nowhere, unless a hard wall stands close behind them. A plan of a 24 by 36 m hall with the second violins at the right of the stage, plates turned away from the audience, and a perfectly reflecting wall 2 m behind them. A seat is shaded if the section's worst note from A♭4 to C♯5 stays a pizzicato rather than a clarinet with a click. The seconds keep 30 per cent of seats, the firsts on the other side 30. Timbre and acoustics

A wall puts antiphonal seconds back beside the firsts

Second violins seated opposite the firsts have their plates turned away from the audience, and in a hall a section's pizzicato depends on its beam. With nothing behind them they keep their worst note in about one seat in a hundred. A perfectly hard wall two metres behind gives them back a cone as large as the firsts' own, but a wall that loses three decibels takes most of it away again. And the seats that cone serves turn out to be, within a seat or two, the seats the same section would serve sitting beside the firsts.

 A meantone keyboard is a stretch of the chain of fifths, and every split key lengthens it by one place at an end. 14 keys hold the positions from A♭ to D♯. That is 10 major and 10 minor triads; major keys E♭, B♭, F, C, G, D, A, E; minor keys with a raised leading note C, G, D, A, E. Pitch and tuning

A split key buys one of everything, and two buy C and E

An organ builder with money for two extra keys could have put them anywhere, and the arithmetic of triads cannot say where. On a meantone keyboard every added key buys exactly one major triad, one minor triad, one major key and one minor key, wherever it goes. So the count is no guide, and the choice was about which keys. The two accidentals most often divided, G♯/A♭ and D♯/E♭, are one on each side of the chain, and they are exactly what completes the minor keys on C and E. What they buy by the roughness sum is seven per cent.

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