Acoustics & Music Theory
why it sounds like that
Theme

The ear is not a microphone

Loudness is not amplitude, pitch is not frequency, and a note whose fundamental is missing still has one. Perception is a model, not a measurement.
Two tones in the ratio 3 to 2. Two sine tones whose frequencies are in the ratio 3 to 2, and their sum. The combined pattern repeats every time both waves return to the start together, which for a simple ratio is soon — here after 2 cycles of the lower tone and 3 of the upper. Intervals and chords

Two notes and a ratio, which is the whole of consonance

Sound two tones together and the pair either settles or does not. What decides it is the ratio of their frequencies, and the rule is that simpler ratios settle — which is two and a half thousand years old and still not quite an explanation.

 220 Hz against 223 Hz. Two tones 3 hertz apart, added. The rapid oscillation is their average; the slow swelling is their difference, heard as 3 beats a second and used by every tuner who has ever worked by ear. Intervals and chords

Beats are arithmetic that anybody can hear

Two tones a few hertz apart swell and fade at exactly their difference. It is the most direct evidence available that the ear does sums on what reaches it, and it is how every instrument in the world gets tuned.

 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.

 The critical band, measured in semitones. The width of the ear's frequency-analysis band at each pitch, converted from hertz into semitones. Two intervals drawn as horizontal lines cross the curves: below the crossing the interval fits inside one band and its notes are not resolved from each other, and above it they are. Intervals and chords

A third is rougher in the bass

Consonance is usually presented as a property an interval has. It is not. The same major third is muddy two octaves below middle C and clean two octaves above it, the ratio never changed, and the frequency where it stops being muddy can be solved for.

 The first eight partials of a string. A string vibrating in one, two, three and more equal parts, with the frequency ratio and the nearest named note beside each. The seventh partial is a third of a semitone flat of anything on a keyboard, which is a fact about strings rather than about tuning. Timbre and acoustics

A string does everything at once

A plucked string does not vibrate at one frequency. It vibrates at all the whole-number multiples of one frequency simultaneously, and nearly everything in music theory is downstream of that fact.

 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.

 Three envelopes. How loudness changes over the life of a note, for plucked, bowed and struck. Remove the attack from a recorded piano and it stops sounding like a piano, which is the shortest demonstration that the envelope carries as much identity as the spectrum. Timbre and acoustics

The shape of a note, which is most of what an instrument is

Cut the first fifty milliseconds off a recorded piano and listeners stop calling it a piano. The attack carries more identity than the steady tone it leads into, and it is the part every spectrum plot leaves out.

 Where one interval stops being itself. Identification as a function of interval size: the probability that a listener names each category, modelled as a logistic with the boundary positions and sharpness a study reports. One category's share falls from three-quarters to one-quarter over 24 cents, against the 100 that separate adjacent categories — so the change of mind happens in 24% of the gap and the rest of it is not in doubt at all. That is what makes a mistuned third a third that is out, rather than a different interval. Intervals and chords

The ear sorts into boxes, and the boxes are the theory

Slide one note slowly upward against another and the interval between them changes continuously. What a listener reports does not. It stays a minor third, stays a minor third, and then in the space of about twenty cents becomes a major third — and nothing in the sound corresponds to the moment of the change.

 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.

 3 against 2, from a rhythm to an interval. The same 3:2 pattern at 6 rates, an event rate from 1.5 to 300 per second, on a logarithmic axis. Below about 20 events a second the two streams are counted; above about 40 they are heard as two pitches a 3:2 apart, which for 3:2 is 702 cents. Nothing in the pattern changes across that boundary. The slowest rate drawn is a tempo of 90 events a minute; the fastest is a pitch of 300 Hz against 450 Hz. Rhythm and metre

A rhythm fast enough to be a chord

Three against two is a polyrhythm. Speed the same pattern up until the events arrive faster than about twenty a second and it is a perfect fifth. The ratio never changed, the figure never changed, and the only thing that moved is the rate — which makes rhythm and pitch one continuum with a perceptual boundary across the middle of it.

 One pattern, four metres. The same 12-step onset pattern read under 4 candidate metres, each scored by a preference rule set: 3 for a strong position that carries an onset, -2 for one that does not, -1 for an onset that lands off every strong position. The scores are 4/4 -4, 3/4 12, 6/8 12, 12/8 as 3+3+3+3 12, so 3/4 and 6/8 and 12/8 as 3+3+3+3 tie and the model does not choose. Nothing about the sound differs between these readings; the bar line is supplied by the listener. Rhythm and metre

The beat is inferred, and sometimes wrongly

A bar line is not in the sound. One sequence of onsets supports a reading in four, in three and in six, and which one a listener settles on comes from a small set of preference rules rather than from anything in the signal — which is why the same recording can be heard two ways and why nobody can be talked out of either.

 Three attacks, the first 50 ms. How loudness changes over the life of a note, for plucked, bowed and struck, drawn over the first 50 milliseconds. By the right-hand edge the plucked note is at 91%, the bowed note is at 36%, the struck note is at 97% — attack times of 4 ms, 140 ms, 2 ms, a spread of 70 to one, and the part a listener uses to tell them apart. Remove the attack from a recorded piano and it stops sounding like a piano, which is the shortest demonstration that the envelope carries as much identity as the spectrum. Timbre and acoustics

The first fifty milliseconds

A spectrum is supposed to be what makes a trumpet a trumpet. Cut the first fifty milliseconds off a recorded note and listeners stop being able to name the instrument — while the spectrum they are hearing is unchanged. Identity is in the part of the sound that ends before the note has properly started.

 Every point on one curve sounds equally loud. The equal-loudness contours of ISO 226:2003, evaluated from the standard's own parameters. The lowest curve is the threshold of hearing. Because the curves are not parallel — they crowd together in the bass and spread apart in the middle — the same change in decibels is a different change in loudness at every frequency, and a spectrum that was balanced at one level is not balanced at another. Perception and the listener

The quietest thing audible, and why the volume knob is a tone control

A decibel is a fact about air. A phon is a fact about a listener, and the two do not line up — the map between them bends with frequency, and it bends differently at every level. One consequence is that turning a piece of music down does not turn all of it down equally, and the amount by which it does not is a number.

 What one tone hides, and in which direction. The masked threshold beside a tone masker: any probe below one of these curves is inaudible while the masker sounds. The frequency axis is in Bark, the scale on which the ear's filters are evenly spaced, so the pattern is a pair of straight lines. The upper slope is much shallower than the lower one and gets shallower still as the masker gets louder — masking spreads upward, not downward. Perception and the listener

One sound hides another, and it hides upward

A tone can be made completely inaudible by a second tone that is nowhere near it in frequency, and the region it disappears into is lopsided. Masking spreads up the spectrum and barely down it, and the reach grows with the masker's level — so which line in an arrangement vanishes is a prediction, not a matter of taste.

 The threshold before, during and after a burst. The level a brief probe needs in order to be heard, plotted against when it happens relative to a 70 dB burst that occupies the shaded band. To the right is forward masking, which decays over about 200 milliseconds. To the left is backward masking: the threshold is raised for a probe that has already finished before the masker begins. Perception and the listener

A sound hides what came before it

Masking does not stop when the masker does. A loud sound raises the threshold for about a fifth of a second after it ends, which is unremarkable, and for several milliseconds before it begins, which is not. The auditory present is a window rather than an instant, and inside the window the order of events is not the order they arrived.

 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.

 1000 and 1200 hertz, and what the ear adds. Two tones presented to a listener, and the frequencies a nonlinear ear generates from them. Nothing in the air is at any of the marked positions: they are products of the pair, at f₂ − f₁ = 200 Hz, 2f₁ − f₂ = 800 Hz, 3f₁ − 2f₂ = 600 Hz, 2f₂ − f₁ = 1400 Hz. The cubic difference tone sits just below the lower primary and is audible at modest levels; the quadratic one is far below both and needs a loud pair. Intervals and chords

The ear makes its own sound, and it is not the missing fundamental

Play two loud tones and a third pitch appears that is in neither of them. The ear is not a passive analyser: it is nonlinear, it generates frequencies of its own, and it emits sound back out of the ear canal. None of which explains the missing fundamental — the products land in the wrong place, and finding out where they land is the experiment that made the residue theory necessary.

 Cycles of roughness completed in 125 milliseconds. For each interval and each register, how many cycles of its own beating fit inside a note 125 milliseconds long. Cells at or above 4 cycles are the ones a listener has time to hear as rough; the pattern runs the opposite way from roughness itself, which is largest in the bass. A short dissonance low down is the case where the two disagree. Intervals and chords

A dissonance has to last

Roughness is a fluctuation, and a fluctuation needs cycles. A minor second at the bottom of a cello fluctuates thirty-three times a second, so a semiquaver holds four of them and a demisemiquaver holds two — which turns the counterpoint rule that a dissonance may pass if it is short into a number, and puts that number at about seventy milliseconds through most of the range, once the pairs beating too slowly to be roughness at all are taken out of the average.

 Four voices, placed by the arithmetic. The rules in force are: no parallel octaves; no parallel fifths; no voice crossing; no gap over an octave above the tenor; the leading note is not doubled; the leading note resolves, outer voices; no augmented melodic interval. The four parts are drawn lowest to highest — bass, tenor, alto, soprano. I – vi – ii – V – I in C major, realised in four voices by the cheapest set of voicings obeying 7 rules, at 16 semitones of motion in all. Every gap between adjacent voices narrower than the fission boundary of 5.2 semitones is marked, and below that boundary two parts cannot be heard as two however hard a listener tries. Perception and the listener

A voice is a stream, and the ear decides which

Seven earlier essays have assigned voices to notes. Whether a listener follows the assignment is a separate question with laboratory numbers attached, and the numbers are unkind to it: two parts closer than about five semitones cannot be heard as two at any speed, a third of the gaps in the cheapest four-part writing are inside that limit, and in a third of chord changes the ear's own rule for continuing a line does not recover the parts as written.

 Which harmonics of a 200 Hz note arrive one to a filter. One row per harmonic of a 200.0 Hz tone. The bar is the ear's analysis band at that harmonic on the equivalent rectangular bandwidth model, and the two small marks either side are the neighbouring harmonics. A harmonic is counted as resolved when the spacing to its neighbours, 200.0 Hz, exceeds that bandwidth, and 8 of 12 are. The third to fifth harmonics are picked out because published measurements put the pitch's dominance region there; nothing in this drawing derives that. Intervals and chords

Which harmonics carry the pitch

A missing fundamental is inferred from a pattern, and a pattern has to be legible before it can be matched. Counting how many harmonics of a note land in separate auditory filters prices the inference — and the answer at the bottom of a bass guitar's range is none of them.

 A 200-a-second click train, correlated with itself. The autocorrelation of a click train at 200 a second, smoothed by the ring of an auditory filter centred at 4000 Hz — an equivalent rectangular bandwidth of 456 Hz, so a ring of 2.2 ms. The regular train peaks at 5.0 ms, one period. With each click displaced by a standard deviation of 20 per cent of the period — 1.00 ms — the peak's contrast against the surrounding lags falls from 0.41 to 0.12. The average rate and the long-term spectrum are unchanged by the jitter; only the timing is. Perception and the listener

A pitch with nothing to match

Filter a click train into a band where no partial is separable from its neighbours and it still has a pitch at its repetition rate. Displace each click by a fraction of a millisecond, leaving the average rate and the long-term spectrum exactly where they were, and the pitch goes. The mechanism is reading the timing — which bounds the account endorsed here from the start.

 Roughness and loudness of one interval against level. A minor third on C4 evaluated at every level from 20 to 100 dB SPL per note, with both quantities drawn relative to their own value at 60 dB. Roughness is the Plomp–Levelt sum over the partials that are above ISO 226's threshold at that level; loudness is the same partials in sones. Between 50 and 95 dB the interval grows 3.2 × 10⁴ times rougher and 24.9 times louder, so roughness grows like loudness raised to the power 3.2. Intervals and chords

The same chord is harsher when it is louder

Every roughness number so far was computed at a level nobody stated. Roughness is the product of two partial amplitudes, so it is quadratic in pressure, while loudness is compressive — which makes a minor third at middle C thirty-two thousand times rougher at fortissimo than at pianissimo and only twenty-five times louder. A chord has no single consonance to report.

 The swing ratio, against the categories it passes through. The same swing curve read against the boundaries between duration categories rather than against notated values. A category's centre is a simple ratio — 1:1, 2:1, 3:1 — and the boundary between two of them is the midpoint, which is arithmetic. The curve crosses 2 of them: out of 2:1 and into 1:1 at 240 beats a minute, out of 3:1 and into 2:1 at 171 beats a minute. So the same notated figure is, by the categorical criterion, a different rhythm at each end of an ordinary tempo range, and the notation says triplet feel throughout. Perception and the listener

How late is a different note

A deviation of thirty milliseconds is expression and a deviation of two hundred is a wrong note, so there is an edge. The edges in time are arithmetic — the midpoints between the simple ratios — and the swing ratio crosses two of them as the tempo rises, at 171 and at 240 beats a minute, while the notation says triplet feel throughout.

 One decay, two verdicts, and the line is the listener's. The early-to-late energy ratio against reverberation time, for 50 millisecond and 80 millisecond windows. Nothing about the room differs between the curves; only where the line is drawn across its decay. Zero comes at 1.00 seconds for the 50 ms window and 1.59 seconds for the 80 ms window — which are, to two figures, the published rules of thumb for a room for speech and a room for music. The design targets were not put in; they came out. Timbre and acoustics

The first eighty milliseconds are a different room

Draw a line across a room's decay and the energy on either side is two opposite verdicts about one building — clarity before it, reverberation after. The line is a property of the ear, not of the room, and putting it at 50 milliseconds and at 80 makes the two published design targets fall out — a room for speech at one second, a room for music at 1.6.

 Ode to Joy as a path. Ode to Joy plotted as 30 notes against the 8 scale degrees it uses, one column per note. Its largest melodic interval is 2 semitones and it spans 7; the mean absolute step is 1.24 semitones. Beethoven, Ninth Symphony, finale, 1824 — the theme as first stated, eight bars. Scales and modes

A melody is a walk, not a set

Nine essays here are about which seven of the twelve a scale takes, and every one of them describes a set. A tune is not a set; it is a path across one, and the path is nearly all small steps. That is not a matter of taste. Above about eight notes a second the ear stops being able to hold a large interval and a small one in the same line, and at sixteen the choice disappears altogether — so a fast passage is scalar because a fast passage that leaps is two pieces of music.

 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.

 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. Intervals and chords

The same distance, under two names

Four hundred cents is a major third or a diminished fourth, and on a keyboard nothing in the sound distinguishes them. An earlier essay was about the boundary between two categories; this is about two categories at one acoustic value, and the surprise is where the ambiguity comes from. In quarter-comma meantone a major third is 386 cents and a diminished fourth is 427 — two names, two pitches, forty-one cents apart. Equal temperament collapsed them, and what a listener now supplies from context used to be in the sound.

 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.

 The pitch moves and the repetition rate does not. Three partials around harmonic 10 of 200 hertz, shifted together by up to 200 hertz, with three curves. The flat line is the envelope repetition rate, which the shift cannot move at all. The rising line is the shift divided by the harmonic number — 200 hertz becoming 220.0 — which is what a harmonic template predicts and is what listeners report. The third curve is the next-best template, which overtakes the first partway along: the pitch is ambiguous, and it drops back rather than rising indefinitely. Perception and the listener

The pitch that moves the wrong distance

Take three partials two hundred hertz apart and move every one of them up by forty. The spacing has not changed, so anything reading the pitch off how often the waveform repeats must give the same answer as before. The pitch moves to 204 — the shift divided by the harmonic number — which is what a harmonic template predicts and what listeners report. Push the shift to a hundred and a second reading overtakes the first, so there are two pitches and neither is the spacing. This is the measurement that closes the question, and it closes it by ruling out one mechanism rather than by choosing between the two that are left.

 The series has three tops. Where the harmonic series stops, asked three ways, at four fundamentals. Consecutive partials stop being separately resolvable around partial 8 — that one depends on the fundamental, since a critical band is a fixed width in hertz and the spacing is not. They stop being a semitone apart at partial 17 at every fundamental, because the ratio (n+1)/n does not know what n is measured in. And they stop being distinguishable in pitch at all between partials 34 and 140. Every claim about how far up the series something happens is a claim about which of these three was meant. Intervals and chords

The series has three tops

How far up the harmonic series can an ear go? The question has three answers and they are an order of magnitude apart. Consecutive partials stop being separately resolvable somewhere around the eighth, and where exactly depends on the fundamental. They stop being a semitone apart at the seventeenth, at every fundamental, because the ratio does not know what it is measured in. And they stop being distinguishable in pitch at all between the thirty-fourth and the hundred and fortieth. Every claim about where the series runs out is a claim about which of the three was meant.

 How long a note has to be before its pitch is worth arguing about. The smallest audible frequency difference at 440 Hz, against how long the note lasts. The flat line is the steady-tone difference limen of 4.0 cents that every tuning argument on this site rests on. The falling line is the bound a finite duration imposes on its own frequency, 1/2T in cents, which no listener can beat. They cross at 486 milliseconds: below that the note is the limit and above it the listener is. A tenth of a second gives 19.6 cents and a quarter gives 7.9, against the commas drawn across the figure. Perception and the listener

How long a note has to be

Every difference limen quoted so far is for a tone that lasts as long as the listener needs, and no note in music does. A tone of duration T occupies a band about 1/2T wide whatever the ear does with it, so at 440 hertz the quoted five-cent limen is the right number only for notes longer than 486 milliseconds. A tenth of a second gives 19.6 cents, which does not clear the syntonic comma. Most of the tuning arguments in this collection are about a quantity that only exists in long notes, and the essays that made them said so about the listener and not about the note.

 500 Hz in one ear, 504 in the other. Two tones 4 hertz apart, one to each ear. They never meet in the air, so neither eardrum sees any modulation at all and there is no acoustic beat to hear. What changes is the phase between the ears, which advances a whole cycle every 250 milliseconds — and the direction that phase implies sweeps with it, drawn here as azimuth against time. The sweep is clipped at the edges, because the implied delay leaves the range a head can produce. A head 17.5 cm across gives at most 656 microseconds, so the phase stops naming a direction above 762 Hz. Perception and the listener

The beat that is not in the air

Every sound this site synthesises reaches both ears identically, and that is the assumption none of its figures ever varied. Put 500 hertz in one ear and 504 in the other and nothing sums anywhere: each eardrum sees a steady sinusoid with no modulation on it at all. A listener still hears a four-per-second beat, which means the arithmetic is being done behind the ears rather than in the room. And it stops working above about a kilohertz — not where phase locking gives out at five, but where a head 17.5 centimetres across stops being able to name a direction, which is 762 hertz.

 3 tones, one power, and the interval between them. 3 tones of fixed total power, spread symmetrically about 440 hertz, drawn against the interval between neighbours. Piled on one pitch they are one sound of that power; separated by more than a critical band — 4.5 semitones here — they are 3 sounds whose loudnesses add, and the same power reaches 2.08 times the loudness at 5 semitones. The two lines are two models of the same rule and they disagree about how abrupt the change is, not about where it goes. Perception and the listener

A chord is not as loud as its notes

The first essay on loudness said what a tone's loudness is and recorded that it had said nothing about a chord's. Here is the missing rule, and it has a musical consequence nobody would predict from it: the same three notes, at the same power, are twice as loud in the treble as in the bass — because the critical band that makes a low triad five times rougher also makes it one sound instead of three.

 One envelope, and the three places a listener might be said to hear it. The amplitude envelope of a note with a 90 millisecond exponential attack, with the three criteria the literature offers drawn across it. The heard moment is 8.0 ms at the detection criterion, 28 ms at the perceptual-onset criterion and 94 ms at the perceptual-attack criterion. The physical onset is at zero on this axis and no criterion puts the heard moment there. The buttons play this attack against a two-millisecond one, started at the same instant. Rhythm and metre

A note is heard after it starts

Every rhythm essay until now has treated a note's onset as the moment it happens. It is not: the instant a listener aligns a note with a beat is later than its physical start by an amount the note's own attack decides, and for a sung or bowed note that amount is about thirty milliseconds — the size of the whole quantity six essays on microtiming set out to measure.

 How much earlier an accent is heard, by mechanism. An accented note on an instrument with a 90 millisecond attack, drawn against how many decibels louder it is, with the three ways it can arrive early separated. A criterion tied to the note's own peak on an unchanging envelope gives exactly nothing. The same criterion on the shorter rise a harder-driven instrument has gives 5.3 milliseconds at 12 decibels. A criterion at a fixed level gives 23.0. Both together give 24.0, and the rise at that dynamic is 73 milliseconds rather than 90. The rise-shortening exponent is stipulated at 0.15 rather than measured, and the two upper curves would separate further if it were smaller. Perception and the listener

Playing louder is playing earlier

An accent has two effects on when its note is heard and neither is a timing decision. A harder-driven instrument has a shorter attack, and a criterion set by the surrounding music is crossed sooner by a bigger rise — so a twelve-decibel accent on a bowed note is heard twenty-four milliseconds early with no change whatever in when the bow was put down. It is also the measurement that tells the two competing models apart.

 Crescendo, and what the impression does. A crescendo of 20 dB over 8 seconds, drawn as three loudnesses in sones. The pale line is what is physically sounding, the middle line is the short-term loudness of the moment and the heavy line is the long-term loudness, which is the passage's loudness as a listener would report it. The gap between the last two is the whole of the effect: at its widest the moment is 1.02 times the running impression, and the impression takes 2 seconds to come down against 99 milliseconds to go up. Form and structure

Loud is relative, and it comes down slowly

The account of loudness had a model of a moment and the account of closure asked it for a model of a form. The published one exists and its content is a pair of numbers that are not the same: a listener's running impression of how loud the music is rises to meet a step in a fifth of a second and takes seven seconds to come back down. A twenty-decibel crescendo spread over eight seconds therefore buys almost no contrast at all, and the same twenty decibels taken as a step buys a factor of two.

 One pair, 12 beat rates. Two notes at 220 hertz, 15 cents apart, drawn as the beat rate between each pair of partials. The fundamentals beat at 1.91 hertz and the k-th partials at k times that, so the series of rates is a straight line and it crosses 15 hertz at the 8th partial. Below the line the fluctuation is counted; above it the same physical fluctuation is heard as roughness. 1 per cent of this spectrum's energy is on the roughness side. The bars are drawn at each partial's own amplitude, so a spectrum with little energy high up crosses the line where nothing is listening. Pitch and tuning

Every partial beats at its own rate

Five earlier essays have drawn one beat rate per figure, and every one of them is the rate between two fundamentals. Two real notes beat between all of their partials at once, the k-th pair beats k times as fast, and somewhere up the spectrum the rate passes the point at which a beat stops being a beat — so a chorused note is a beat at the bottom of itself and a roughness at the top, simultaneously, with a crossover partial that is arithmetic.

 How far equal temperament puts each interval's coincidence out. For each interval, the pair of partials it brings together and how many cents equal temperament mistunes that coincidence by. A fifth's third-against-second is out by 2 cents and a major third's fifth-against-fourth by 14 — so the same temperament that is inaudible on a fifth produces, at 220 hertz, a beat of 8.7 per second between two sections singing a third, with every singer in both of them perfectly in tune. Timbre and acoustics

A section against another section

The choir has been treated as a unison, and no choir sings only unisons. Two sections an interval apart beat between partials rather than between fundamentals — the third brings the fifth partial of one against the fourth of the other — and equal temperament puts that coincidence fourteen cents out. So two sections singing a tempered third beat at nearly nine per second with every singer in both of them perfectly in tune, and the same temperament is inaudible on a fifth.

 Where a room stops sending the two ears the same sound. The correlation between the two ears' signals against frequency, for a seat 15 metres from the source in a 15,000 cubic metre room with a 2 second reverberation time. The pale curve is the diffuse field alone — sin(kd)/(kd) for an ear separation of 17.5 centimetres, which first crosses zero at 980 hertz. The heavy curve adds the direct sound, which is coherent and lifts the whole thing by an amount the direct-to-reverberant ratio sets. At 125 hertz the coherence is 0.98 and at 1000 it is 0.16. Perception and the listener

Where the two ears stop agreeing

A room sends both ears versions of the same sound, alike at low frequencies and increasingly unlike at high ones. Where they stop resembling each other is 980 hertz, and it is set by the 17.5 centimetres between the ears rather than by anything about the room — which is within a quarter of a frequency found earlier for a completely different reason. One minus that correlation is spaciousness, and it is computable from a room's own reverberation.

 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.

 Why a concert hall is narrow. The lateral energy fraction at the middle seat as the same hall is widened, everything else held. It peaks at 12 metres across at 0.235 and falls to 0.000 at 44. A wide hall's side walls are further away, so their reflections arrive later, weaker and — this is the part Sabine's model cannot say — from nearer the front, where the sideways weighting discounts them. The shoebox halls the orchestral repertoire was written for are all between about eighteen and twenty-five metres wide, and this is the arithmetic they are the answer to. Perception and the listener

A room with directions in it

Every room until now has been a reservoir of energy that drains at a rate. That model has no directions in it at all, so it cannot say the one thing every published measure of spaciousness is about: how much of what arrives comes from the side. Mirror the source in six walls and every reflection acquires an angle and a time — and the answer to why a concert hall is narrow falls out at eighteen metres.

 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.

 Adding parts adds power, and very little loudness. Each part is played at the same level, and the chord is realised every way its parts allow and averaged over them, so the quantity is a property of the texture rather than of one arrangement. Going from 3 parts to 8 adds 4.3 decibels of power and 0.1 decibels of loudness, because the extra parts land in bands that are already occupied — the count of occupied critical bands FALLS from 6.0 to 3.9 as the parts crowd into the same register. Form and structure

The dynamics are in the score already

Count the parts in each bar, realise them in their ranges, put every partial in its critical band, sum the loudnesses and run the result through the two smoothers built earlier. What comes out is a dynamic curve for a piece with no performance in it anywhere — and it says that doubling the number of parts inside a fixed register adds three decibels of power and about one of loudness, because the extra parts land in bands that were already occupied. Let the register widen with the parts and the same arithmetic gives eight phon, which is what a tutti actually is.

 Roughness and loudness do not rise together. One four-note chord, played at levels from 35 to 95 decibels, with both quantities drawn as multiples of what they are at the quietest. Roughness is quadratic in pressure, so 60 decibels multiply it by 1.0e+6. Loudness is compressive — about ten phons to a doubling of sones — so the same range multiplies it by 96. The gap between the two lines is the quantity: roughness per sone rises by a factor of 1.0e+4 between a pianissimo and a fortissimo of the same chord. Harmony and voice leading

The ranking survives the dynamic and the chord does not

Roughness is quadratic in pressure and loudness is compressive, so sixty decibels multiply a chord's roughness by a million and its loudness by ninety-six. Roughness per sone therefore rises ten thousandfold between a pianissimo and a fortissimo of the same four notes — and yet the ranking of which doubling is smoothest, over four hundred and eighty voicings, does not move by a single place.

 Every arrival at one seat, against the delay at which it would be an echo. The echogram: each reflection at its delay after the direct sound and its level relative to it, in a 22 by 45 by 15 metre hall with 18 per cent absorption. The line is the published echo threshold for speech — 40 milliseconds at equal level and about 3.0 more for each decibel of attenuation — so anything to the RIGHT of it is late enough and loud enough to be heard separately. The once-reflected rear wall arrives at 210 milliseconds, 25 decibels down, against a threshold of 114 — well past it. Nothing here stands clear enough of its neighbours to be heard as an echo, and 181 arrivals are fused with the direct sound instead. Perception and the listener

An echo is prevented by the crowd around it

The echogram says when every reflection arrives and how loud it is; the published echo threshold says when a reflection that late and that quiet is heard separately. Put one against the other and the rear wall of every hall anybody builds is past the threshold — a 45-metre hall puts it 210 milliseconds late and 25 decibels down against a threshold of 114. It is not heard as an echo, and what saves it is not the geometry. It is everything else arriving at the same time.

 Two fusion cues, and they do not agree about a single spectrum. Each spectrum twice. Hollow is the harmonicity census — the fraction of partials near enough a whole multiple of one fundamental to fuse, which is harmonicity. Filled is the same fraction under common fate: how many partials decay at a rate within a factor of 2 of the strongest partial's. Ranked by harmonicity the order is an ideal string, a piano string, a bell, a kettledrum, a bar; ranked by common fate it is a kettledrum, a bar, an ideal string, a piano string, a bell. The two orderings are nearly reversed. An ideal string is perfect on the first cue and 20 per cent on the second, and a kettledrum — the worst spectrum in this collection for fitting a series — is the only one whose partials all die together. Perception and the listener

The partials that do not die together

The fusion census is a still photograph: it asks whether a set of partials fits one harmonic series and has no term for time. Put time in and the ranking reverses. An ideal string is perfect on harmonicity and holds a fifth of its partials together by decay; a kettledrum — the worst spectrum in this collection for fitting a series — is the only one whose modes all die at one rate.

 A vibrato flattens the dissonance curve. Each interval twice: hollow is its roughness computed at the two notes' nominal frequencies, filled is the average of its roughness over a vibrato cycle of 50 cents at 6 hertz. They are not the same number, because roughness is a curved function of the frequency difference and the average of a curve is not the curve of the average. The largest effect is at octave, where the moving average is 19.3 times the still value — an interval sitting in a deep narrow minimum is smeared out of it. The smallest is at major seventh, where it is 0.95: an interval near a maximum is smeared out of that too. Vibrato pushes every interval toward the middle, and what it takes away from the consonances is much more than what it takes away from the dissonances. Intervals and chords

A roughness with a rate of its own

Every roughness figure so far computes one number for a steady spectrum. Evaluate the same sum at every instant of a vibrato and there are three numbers instead — a mean, a depth and a rate — and the mean is not the roughness of the mean frequency. On an octave it is nineteen times it, because an octave sits in a deep narrow minimum and a vibrato smears it out of 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.

 The same roughness, before and after the window it has to be heard through. The instantaneous roughness of an interval under a vibrato, and the same quantity after a running average of 59 milliseconds — the time a dissonance has to last to be heard as one, which is 4 cycles of this interval's own 68-hertz fluctuation rather than a number chosen for the figure. The mean is identical to every digit, 0.1487 against 0.1487, because a running average cannot change an average — so the earlier Jensen factor of 1.0 survives the window untouched and its prediction that the window would shrink it is wrong. What the window destroys is the depth: 0.30 of the mean becomes 0.23, which is 77 per cent. The roughness a vibrato adds is heard; the fact that it is moving is mostly not. Intervals and chords

The mean survives the window

A roughness that moves has a mean, a depth and a rate — all three of which a listener could only have through a temporal window. Applying the window already to hand settles which of the three survives, and the answer refutes the guess: a running average cannot change an average, so the octave's factor of nineteen stands and the movement is what goes.

 Where the sound is, and how wide. An earlier essay sorted a hall's arrivals into echoes and everything else. Everything else is not nothing: a reflection too early to be heard as a separate event still moves the apparent source, widens it and colours it, and all three come out of the same list of times, levels and angles. At this seat the direct sound arrives from 26.6 degrees off the front and the image is heard 8.5 degrees left of it, pulled by the near side wall. The apparent width is 33 degrees, from a lateral energy fraction of 0.23. And the strongest early reflection arrives 0.6 milliseconds behind off 1× the floor, which is a comb filter with notches every 1608 hertz and 25 decibels deep. The trading ratio and the discount on late arrivals are stipulated rather than measured, so the degrees are ordinal: what the figure claims is the direction and the shape, not the number. Perception and the listener

A position and a width

Sorting a hall's arrivals into echoes and everything else settled that fusion is not a yes or a no. Everything else is not nothing: a reflection too early to be heard separately still moves the apparent source, widens it and colours it. All three come out of the same list of times, levels and angles, and none of them needed a new input.

 What a competition decides when the two cues do not agree. Each spectrum with its two cue readings and the grouping the competition chooses. Harmonicity asks whether a partial is near enough a whole multiple to belong; common fate asks whether it decays at the same rate as the rest. Where they disagree there is no rule in this collection, so the published apparatus is used instead: every way of splitting the partials into one stream or two is scored for the partials each cue says it has wrongly grouped and wrongly separated, and the cheapest wins. an ideal string — harmonicity 100 per cent, common fate 20, and the competition says one stream; a piano string — harmonicity 90 per cent, common fate 20, and the competition says a cut after partial 2; a bell — harmonicity 88 per cent, common fate 13, and the competition says a cut after partial 1; a bar — harmonicity 33 per cent, common fate 67, and the competition says a cut after partial 4; a kettledrum — harmonicity 60 per cent, common fate 100, and the competition says one stream. The exchange rate between the two cues is the number nobody here can supply, so what is reported beside each is how many decades of it leave the answer unchanged. Perception and the listener

The exchange rate nobody has

There are now two cues that disagree about how a spectrum divides, and every figure so far reports them separately because there is no principled way here to weigh one against the other. The published apparatus is a competition between grouping hypotheses with a cost per cue — and the useful thing it produces is not the winner but how much of the exchange rate the winner survives.

 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.

 The period is still there, and it is wider. The autocorrelation of a 12-partial complex on 220 hertz, drawn twice: steady, and averaged over one cycle of a 71-cent vibrato. A vibrato moves every partial by the same number of cents, so the complex is exactly harmonic at every instant and nothing is mistuned — what moves is the period the extractor is looking for. The peak survives. It loses 6 per cent of its height above the surrounding lags and gains 11 per cent in width, because the vibrato swings the period by 0.37 milliseconds against a peak 0.90 wide. Its maximum also moves, to 2.4 cents sharp of the still tone's, which is a prediction with a sign in it. Instruments and their design

The pitch that does not wobble

Three earlier essays have treated a vibrato as a modulation of roughness. The reason singers use one is what it does to the note, and there is an extractor here that turns a set of partials into a pitch and has never been asked what it does with partials that will not hold still. The period survives, at a cost that rises with the extent — and the practice stops within a hair of where the cost becomes total.

 Intonation is a unison problem and nothing else. The roughness between two instruments on one note, against how far apart they are in cents, drawn for a unison and for the intervals beside it. A perfect unison is 0.0007 — the partials coincide and there is nothing to beat. Five cents apart it is 0.0465, 65 times as rough, and ten cents apart it is rougher than a major third played exactly. The mechanism is that partial n of a note mistuned by c cents is mistuned by c cents as well, which is n times as many hertz — so the top of the spectrum enters the critical band long before the fundamental does. The other curves are flat, because a third's roughness is set by which partials nearly coincide and a few cents does not change which. Timbre and acoustics

Two players on one note

Six essays have put one instrument on each note of a chord, and the commonest thing an orchestrator actually does is put two on the same note. Two independent sources add in power, so the composite is neither of them — except that it nearly always is one of them, because the level at which ownership changes hands is rarely at zero. And a unison ten cents out is rougher than a major third dead in tune.

 One contrast survives every tempo anybody plays and the other does not. How much of each quantity's contrast between chords a listener still has at the end of each chord, against how long a chord lasts. The roughness curve is flat at one down to 45 milliseconds a chord and then falls off a cliff, because its window is 37 milliseconds and a boxcar either fits inside a chord or does not. The loudness curve is already losing at a second a chord and keeps 83 per cent at the slowest pace here, 39 at the fastest. Nothing in music is faster than the roughness window and a great deal of music is faster than the loudness one, so a passage delivers its dissonance and averages its dynamics. Form and structure

The dissonance arrives and the dynamic does not

A scoring decides two things at once and both of them have to be integrated by a listener before they exist. The loudness smoother's release is two seconds and the roughness window is thirty-seven milliseconds, and that ratio of fifty decides which of the two survives at the pace music is actually played. Nothing anybody performs is fast enough to blur a dissonance, and a great deal of it is fast enough to average a dynamic.

 The hall, as two numbers a listener has. Every reflection at this seat, placed by the interaural delay it produces rather than by the direction it came from. Time runs down; the dot's size is its energy. The direct sound is at 0 microseconds and the reverberation spreads over the whole available range, with a root-mean-square width of 322 against a geometric maximum of 656. That is the position and width computed earlier, in the units a listener has instead of the vectors used until now. 10 of the 56 reflections arrive from behind and carry 9 per cent of the energy — and they are drawn where they are because the interaural delay of a reflection from 120 degrees is identical to one from 60. Perception and the listener

The hall through a head

Every direction computed so far is a vector from a seat to an image source, and a listener has no vectors. Run the echogram through the head computed six essays ago and two things happen: the position and width become microseconds, and a third of the room disappears — because both cues fold at ninety degrees and a reflection from behind is identical to one in front.

 Nothing at all until fifteen decibels, and then it depends on the tempo. The fraction of a line's partials that stay above threshold, over how fast the line moves and how much louder everything before each note is. Darker is more lost. The whole left-hand side is white: at equal levels a note cannot be masked by its predecessor at any tempo, and that is a proof rather than a measurement — forward masking leaves a threshold at most ten decibels below the masker, and a note's own partials mask each other from the same components at full level. The boundary is between twelve and eighteen decibels, and beyond it the loss grows with the tempo: at 280 to the crotchet and 36 decibels of contrast, 26 per cent of the line's partials are gone. Fifteen decibels is about the gap between a forte and a piano. Perception and the listener

An equal note cannot be masked

Three earlier essays are about one instant, and forward masking lasts two hundred milliseconds — longer than a note at any brisk tempo. So a fast line should be a sequence of events hiding each other, and it is not: a note masks itself ten decibels harder than its predecessor can, at any speed. What does hide a line is dynamic contrast, and the boundary is fifteen decibels.

 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.

 A page has two decibels and a player has sixty. Across, parts added to a final chord one at a time, each at the same level; up, the loudness that results, on a logarithmic scale. Going from one part to eight moves the total by 1.8 decibels and does not move it monotonically — four parts are louder than five and than eight. The faint line is what a naive power sum would give: 9.0 decibels. The band down the right is the same chord played by people, from forty to a hundred decibels, which spans 62. So a texture that thins from eight parts to one is not a diminuendo. It is a change of colour at constant loudness, and everything the closure figures call a dynamic belongs to the performance. Perception and the listener

A page has two decibels

The account of closure called its dynamic component a corpus debt: it had no model of dynamics in a form. Loudness supplies one, and applying it answers the debt by refusing it. Adding parts to a final chord one at a time, each at the same level, moves the loudness by under two decibels and not monotonically — while a player has sixty. A texture that thins is not a diminuendo.

 A turn of 0.99 degrees tells front from back. A source 45 degrees off centre and its mirror image 135 degrees off, which produce the same interaural delay and are therefore the same signal to a listener who does not move. As the head turns the two predictions separate: the front source's delay falls and the rear source's rises, because the fold at ninety degrees puts them on opposite branches of the same curve. They differ by the 15-microsecond threshold after 0.99 degrees of turn — which is exactly half the 1.97 degrees a source would have to move for the same listener to notice it moving, and it is half for a reason: a turn displaces the two hypotheses from each other by twice what it displaces either of them from where it started. Perception and the listener

The turn is half the angle

A stationary head cannot tell a sound in front from the same sound behind, and an earlier essay said so at length. The turn that breaks the confusion is 0.84 degrees — exactly half the angle a source would have to move for the same listener to notice it moving, and half for a reason. In a hall the same turn does something else: the source swings at 8.9 microseconds a degree and the room swings at 2.7, so a listener who moves is separating the soloist from the reverberation as well as the front from the back.

 The same geometry at four sizes of head. Woodworth's interaural delay against direction, for 4 head radii from 5.8 to 9.8 centimetres. The whole range runs from 431 microseconds for a newborn to 735 for a large adult, and it scales exactly with the radius because the delay is (r/c)(θ + sin θ) and r is a multiplier. The detection threshold does not scale with the listener, so the number of distinguishable delays across the whole range falls from 98 to 57: a smaller head has the same directions in front of it and a shorter ruler to measure them with. Perception and the listener

A smaller head in the same hall

Ten earlier essays draw one head. Every parameter belonging to the room has been varied by some figure and the one belonging to the listener never has, and it is the only one whose change the detection threshold does not follow: a six-year-old in the same seat receives the same fifty-four reflections at the same instants and reads them onto an axis with seventy distinguishable positions instead of eighty-seven. The speed of sound, swept over every temperature a hall is ever at, changes nothing at all — and the reason it cannot is the reason head size can.

 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.

 Eleven partials is one partial too many. What fraction of a spectrum the harmonicity census finds fused, against how many partials it is asked to census. At ten a perfect harmonic series fuses 10 of 10 and the fundamental it finds is the right one. At eleven it fuses 5 of 11 and the fundamental jumps to exactly 2.00 — the octave above. The cause is the cap the census carries for a reason established earlier: without it a bell fuses perfectly at a fundamental nobody could hear, so the search refuses any fundamental more than about ten harmonics below the top partial. At eleven partials the first thing that cap excludes is the series' own fundamental, and the census then takes the octave and calls every odd partial inharmonic. So the number of partials and the cap are the same number, and nothing had ever said so, because every earlier figure censuses ten. Perception and the listener

Eleven partials is one too many

Six earlier essays census exactly ten partials and no figure has ever passed another number. At eleven, the harmonicity census stops finding a perfect harmonic series' own fundamental, takes the octave above it, calls every odd partial inharmonic, and the competition cuts an ideal string in two. It is not the arbitration — the cost of a second stream was swept over a factor of fifty and every verdict came back identical — it is a cap that exists for a good reason and turns out to be the same number as the count.

 An entering part is worth 0.9 phons, in the middle of its range. A texture of 5 parts at 62 decibels each, with one more part added at the same level, tried at every semitone from C2 to C7. The vertical axis is what the addition is worth in phons, and a phon is a decibel here; the shaded strip is the difference limen for loudness, so an entry inside it is not heard as a change of level. The median entry is 0.89 phons and only 29 of 61 clear the limen — the lowest of them at A♭4, 415 hertz. The best available, at B♭6, is worth 4.2. The two lines are the two loudness models to hand: they agree everywhere above the tenor register and part company below it, where the greedy critical-band grouping reports 24 entries that make the texture quieter and the excitation pattern reports none. Perception and the listener

A part entering is not a change of level

Five earlier essays measure a sonority that is already sounding. The one thing an orchestrator actually controls is the entry, and priced at every semitone it comes to 0.89 phons — under the difference limen — with only 29 of 61 available entries clearing it and none below A♭4. What an entry is instead is an object arriving, and the reason is that its own rise time is faster than the listener's.

 Articulation is worth 2.6 phons, and nobody counts it. A passage of notes at 80 decibels, 2 to the beat, at 7 tempi and 3 articulations, scored against the same level held continuously. The variable is the fraction of each inter-onset interval that is sounding — 0.95 is a legato, 0.4 a staccato — and the vertical axis is what that costs the passage's running loudness in phons. Nothing here is anybody playing harder or softer. At 40 to the beat the span from legato to staccato is 1.47 phons; at 200 it is 2.61, because a staccato note there lasts 60 milliseconds and no longer reaches its own loudness either. Perception and the listener

A staccato is a dynamic mark

Every loudness figure in this collection is of a sound that has been going on long enough, and no note in music has. Run the running-loudness model on notes with lengths in them and an articulation turns out to command 1.5 phons at a slow tempo and 8.1 at a fast one — more than the 1.8 decibels a whole texture commands, on the same page, written down in the same ink, and counted by nobody.

 The top voice arrives whole and the bottom one arrives as a sine. 4 parts sounding together, each of 8 partials, with every partial tested against the summed masked threshold of every component in the texture. A filled mark is a partial the listener receives and an open one is a partial the part would have had alone and does not have here. The bass at C3 keeps 1 of 8, the tenor at C4 keeps 2 of 8, the alto at E4 keeps 5 of 8, the soprano at G5 keeps 8 of 8, every one of them at 70 decibels. Every part is at the same level and the difference is entirely where each one sits: masking spreads upward, so the part at the top of the texture has nothing above it to be masked by and the part at the bottom has everything. Perception and the listener

The listener is given the top voice, and the bass as a sine

Four earlier essays put the masker and the probe in the same voice. Put them in different voices — a four-part texture at one level — and the soprano arrives with all eight of its partials, the alto with five, the tenor with two and the bass with one. Balancing the loudness, which is the constraint a scoring is solved under, changes none of that: equal loudness is not equal spectrum and cannot be made so.

 The roughest chord on the page is at C1 and the roughest one heard is at A♭2. One close triad at 70 decibels through 17 registers, with its roughness computed twice: over every partial in the score, and over only the partials that stand above what the chord itself masks. The written curve rises all the way down and its maximum is the lowest register drawn, C1, which is the low-interval rule as it has always been computed here. The delivered curve turns over at A♭2 and falls to nothing below A♭1: a close triad down there is not rough, because it is not arriving as a chord — 1 of its 24 partials survives at C1 and there is almost nothing left to beat against anything. Perception and the listener

A low chord stops being rough by stopping being a chord

Every count of audible partials until now is of a chord at middle C, and the three registers it did compare span C3 to C5 — a third of the range a chord is written in. Move the same triad down and the count collapses: 79 per cent of its partials arrive at E3 and 4 per cent at C1. So the roughest chord on the page is the lowest one and the roughest chord a listener receives is at G2, and where that maximum sits moves nearly two octaves with the dynamic.

 Who owns clarinet, oboe, voice at every balance. The composite of three players at 392 hertz belongs to whichever of them it is nearest in log-spectral distance, and here that is drawn over the whole plane of balances a conductor could set — the second and third players from 24 decibels below the first to 24 above. voice owns 79 per cent of the square. The three regions meet where all three distances are equal, which is the only balance at which the composite belongs to nobody: it is at -0.3 and -19.1 decibels, inside the square and therefore a balance an ensemble could actually be asked for. A trio has a colour of its own at one point, not over a region. Timbre and acoustics

A section has a loudest member, not a colour

Two players on one note have a balance at which the composite belongs to neither, and that is what blending means. Three should have three such balances and no reason for them to agree — a trio with a rock-paper-scissors ownership would have no strongest member at all. Twenty trios, sixty pairwise comparisons, and not one disagreement: the possibility is real, arbitrary spectra do it once in twenty, and instruments never do.

 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.

 What the chord before takes out of the chord after. Five chords at 1.2 seconds each, with the roughness each one has on its own — its simultaneous masking and the threshold of hearing already applied — and the roughness it actually has once the chord in front of it has raised the threshold. Four of the five are untouched. The fifth, i, two parts, follows the only step in this passage that falls more than fifteen decibels, and it arrives into a hole: it is entirely below threshold for its first 13 milliseconds and takes 240 to get all of itself back. Masking can only remove partials, so it can only lower a roughness — and the chords it can reach are the ones a written dynamic has just made quiet, which are already the smooth ones. Across the passage the dissonance contrast goes from 3021 to 3113: the mask widens it by 3.0 per cent rather than eating it. Form and structure

A soft chord has to fade in

Forward masking sits between the two integration times already in play — two hundred milliseconds against a thirty-seven millisecond roughness window and a two-second loudness release — and it was owed as the term that might eat the dissonance contrast. It does not. It widens it, by three per cent at a chorale's pace and fifty-nine at four chords a second, because it can only ever remove partials and it can only reach the chord a dynamic has already made quiet. What it does instead is stranger: one chord in the passage is entirely inaudible for its first twelve milliseconds and takes a quarter of a second to arrive whole.

 Of 8 beats at A3, 3 can be attended to. Every member of the beat family a 2:1 mistuned by 6.0 cents makes on a real string at A3, placed by how separable its coincidence is from the partials beside it — in auditory filter widths, across — and by how fast it beats, up. The shaded region is the set a listener can receive: wider than one filter, and between 0.4 and 15 fluctuations a second. 4 of 8 members fall inside it, and once rates within a factor of 2 are counted as one modulation channel there are 3. The members that fail do so for two different reasons: the low ones sit under the roughness ceiling but their coincidences are buried in a filter that holds three partials, and the high ones are resolved and far too fast. Intervals and chords

Three beats at most, and only in the middle of the keyboard

A mistuned octave on a real piano makes eight beats at once, and a listener attending to one of them is doing something that has a threshold. Two thresholds, in fact — a rate and a place — and once both are applied the eight become four at A3, one at A1 and one at A5. Every interval a tuner sets goes to zero at both ends of the compass and peaks at eight countable beats in the octave the bearing is laid in.

 The top of a struck note's series falls while the note lasts. The highest partial still above the threshold of hearing, against time, for a note struck at 80 decibels on a fundamental of 130.8 hertz with a 1/n spectrum and a loss rising as the partial number to the power 1. It starts at partial 152 and it is falling from the first millisecond. The horizontal lines are the three tops computed from frequency alone, all of which assume a note that never ends: the note's own top drops past the difference-limen top after 0.03 seconds, past the semitone top after 0.32, and past the resolvable top after 0.64. After that the series is shorter than the ear could have resolved, and what stops it is the clock. Timbre and acoustics

The top that falls while the note lasts

Four earlier essays have asked where the harmonic series stops, and all four answered with a number computed for a tone that never ends. A struck C3 has 152 audible partials at the strike and eight after two-thirds of a second, so the ear's own resolution limit governs the first ten per cent of the note and the decay governs the rest. Playing ten decibels louder buys the ear a further tenth of a second, and doubling its reign would take fifty-eight.

 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.

 Where a section's fluctuation stops being a beat, on 220 hertz. Two rates up the spectrum of a 220 hertz note sung by a section whose voices are spread by 15 cents. The rising line is the beat rate between a typical pair of them, which grows with the partial because a mistuning in cents is a difference in hertz that scales with frequency; it reaches the 15 hertz at which a beat stops being a beat by partial 5.5, at 1217 hertz. The flat line is the amplitude modulation the vibrato imposes through the formants, which is 6.0 hertz at every partial because the vibrato modulates every partial by the same number of cents at the same rate. The two are equal at 487 hertz. Above 1217 hertz the beating has become roughness and the only fluctuation left is the vibrato's — and that frequency is the same one an octave up, where it is partial 2.8 instead. Instruments and their design

The rate that does not rise with the partial

Twelve earlier essays give every vibrato the same six hertz, and the measured spread is 5.5 to 7.5. Putting the two fluctuations a choir contains on one axis shows why the rate matters: the beating between mistuned voices rises with the partial and leaves the range a listener follows as fluctuation at 1,217 hertz, while the vibrato's own modulation is six hertz at every partial. Above that frequency a section fluctuates by vibrato alone — and if every singer had the same rate, it would barely fluctuate at all.

 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.

 A head model a few millimetres out reads every azimuth but the front. The azimuth a listener reports against the azimuth a source is at, for internal head radii from 8.22 to 9.28 centimetres against a true radius of 8.75. The delay a source produces is (r/c)(θ + sin θ) and the listener inverts it with the radius they believe they have, so their answer solves θ̂ + sin θ̂ = (r/r̂)(θ + sin θ). Every curve passes exactly through the origin: on the median plane there is no delay and therefore no error, whatever the head model is. The error grows with azimuth and is largest at the side. An internal head 5.3 millimetres too small runs out of azimuth at 82 degrees: beyond that the world is delivering a delay larger than any its owner's model can produce, and every source out there collapses onto the side. Perception and the listener

Where a wrong head gives itself away

Every claim so far maps a delay to a direction through one fixed geometry, and the listener acquires that map while the geometry grows under them by seventy per cent. So the map can be wrong — and the essay before this one said the error would be largest on the median plane, where the delay curve is steepest. It is exactly zero there. The steepness is in the error and in the threshold and cancels between them, which leaves a listener whose internal head is 1.3 millimetres out with one place to catch it: hard to the side, where nobody localises well.

 A 20-decibel crescendo is 27 phons on a bass note and 20 on a high one. The same change of level, from 60 to 80 decibels, converted to loudness at each register through ISO 226's equal-loudness contours rather than at one kilohertz. The heavy curve gives each note a string spectrum, so its partials are converted in their own bands and summed; the pale one is the fundamental alone. On the spectrum-aware curve the crescendo is worth 27.3 phons at C1 and 20.3 at C7. On the fundamental alone it is 77 at C1, which is not a finding but an artefact: a 60-decibel tone at 33 hertz sits 1.8 decibels above the threshold of hearing and is very nearly nothing. The honest correction is the smaller one, and it is still a difference of 7.1 phons across the compass for a mark written in the same ink. Perception and the listener

A subito piano is four seconds longer in the bass

Every loudness figure with time in it converts level to loudness at one kilohertz, and the equal-loudness contours say that no other frequency works that way. Joining the two sorts the published numbers into those that were about the treble and those that were not. Three move a great deal — a twenty-decibel crescendo is worth 27 phons on a bass note and 20 on a high one, and the seven seconds a subito piano takes becomes eleven and a third. Three do not move at all, and the reason they do not is the same reason in every case.

 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.

 Both notes of a fifth, and the coincidences going out. The highest partial still above the threshold of hearing for each note of a fifth struck at 80 decibels on 130.8 hertz, against time, with a 1/n spectrum and a loss rising as the partial number to the power 1. Both curves come down from off the top of the frame — the lower note starts with 152 partials and the upper with 102, because twenty kilohertz is a ceiling in frequency and not in partial number. The rings are the interval's partial coincidences at the moment they stop existing — successive multiples of one ratio, which go out from the top down: 12:8 at 0.47 s, then 9:6 at 0.64 s, then 6:4 at 1.04 s, then 3:2 at 2.14 s. The lowest, 3:2, is the last, and after it the two notes have no partial in common that either of them can still supply. Timbre and acoustics

A fifth on a piano is not a fifth a second later

Nine essays draw one note in silence, and a listener is given a texture. Put two struck notes an interval apart and consonance comes apart into two quantities that had agreed while nothing moved: the partial coincidence that names the interval outlives the strike in exactly the order common-practice theory ranks its intervals, and the roughness that scores it reorders itself inside a third of a second, with the fifth overtaken by four intervals every treatise calls harsher.

 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.

 Every beat in a family is the same depth, and none is near the threshold. The 8 members of the beat family a octave mistuned by 6.0 cents makes at A3 on a real string, each placed at the rate it beats and at the modulation index a listener's filter delivers there. The rising curve is the published detection threshold for amplitude modulation, which is flat at 0.03 below about fifty fluctuations a second and rises above it. The flat dashed line at 0.667 is the depth the pair has in isolation, and it is the same for every member: on a spectrum falling as one over n, the k-th member pairs partials 2k and 1k, whose ratio is 2.00 whatever k is. What the filled points show is the smaller effect that does depend on the member — the partials on either side of the coincidence leak into the same filter, add level without adding fluctuation, and dilute the index from 0.662 to 0.405. Inside the countable rate window the narrowest margin over the threshold is a factor of 16.7, on member 4. The depth criterion removes nothing. Intervals and chords

Every member of a beat family is the same depth

A mistuned octave's beats have been counted on their rate and their place, with the depth recorded as the thing left out and a prediction that the shallow upper members would take the count from four to two. The depth turns out not to fall at all: on any power-law spectrum every member of a family has exactly the modulation index its interval's own ratio gives, at every register, on every wire. The count does fall to two, and the thing that takes it there is the criterion that essay was already using.

 An entrance stops being a loudness event and never stops being a colour one. The same oboe entering on the same note at the same level, against how many players were already sounding. Its contribution to the loudness falls from 15.5 phons to 0.32 — a factor of 48 — and crosses the one-phon difference limen at 5 players already playing. Its contribution to the roughness rises by a factor of 12.3 over the same range, because roughness is a sum over pairs and the entrant makes one new pair with everybody. Both curves are drawn as a share of their own largest value, since a phon and a squared pascal have no exchange rate. The claim is the two directions, not the crossing point of two units. Form and structure

An entrance is a change of colour

Eight essays on orchestration move the assignment and hold the ensemble still, and a score does the opposite: it brings players in and takes them out. Loudness is a sum over parts and roughness is a sum over pairs, so the player who joins adds one term to the first and one to the second for everybody already there. What the entrance is worth in phons falls by a factor of forty-eight across the range an ensemble spans and crosses the difference limen at five players; what it is worth in roughness rises by twelve, and by a further factor of ten for every ten decibels the passage is played at.

 Three clocks receive one entrance, and they do not agree about when. An oboe joining 5 players already sounding, at time zero, with each of the listener's three readings drawn as its own share of the change it eventually makes. The roughness window is 49 milliseconds wide and has half the change at 25; the short-term loudness smoother has half at 15; the long-term one, whose release is the two seconds an earlier essay is about, has half at 90. The two-second release is on the wrong side of the smoother to hide an entrance. Its attack is 99 milliseconds, so an entrance is received promptly and it is a departure that is not. Form and structure

The release is on the wrong side

Whether the loudness model's two-second release makes an entrance inaudible has the answer no, for a reason the question did not anticipate. The smoother is asymmetric — ninety-nine milliseconds going up and two seconds coming down — so a rise is tracked twenty times faster than a fall, and an entrance is received promptly by every one of a listener's three readings. The colour of it arrives first, at twenty-five milliseconds against ninety, and the reading that moves with the ensemble is the one nobody would have picked.

 Which chord of a passage has room for the part that is entering. An oboe entering on one note, tried at each chord of a five-chord passage, scored by how far its own partials sit above the threshold the ensemble already sounding puts over them. The best moment gives it 9.0 decibels of margin and the worst 0.3, a spread of 8.7 — and the best moment is not the quietest chord, which is vi, close below. Room for an entrance is spectral rather than dynamic. A chord with a hole in its written spacing need not have one in its spectrum, because the partials of its bass fill the middle whatever the notes above it do. Form and structure

The chord that has room for an entrance

Three essays have made the ensemble something a score can change and none of them has asked when. The ensemble already sounding puts a masked threshold over whatever register an entering part takes, and that threshold is set by the voicing rather than by the dynamic — so the five chords of one passage differ by 8.7 decibels in how much of an entering oboe survives them, and the quietest chord of the five is the worst place in the passage to bring somebody in. Swept over the entrant's own pitch, the choice of moment is worth as much as the choice of register.

 The channels a fluctuation is analysed into, and how wide they are. A modulation filterbank of quality factor 1, drawn every half octave, with two of the beats a mistuned octave at middle C produces marked at 0.89 and 1.26 a second. A channel of quality one has its half-power points at 0.618 and 1.618 of its centre, so two fluctuations closer than a factor of 1.618 never end up in different channels. The two marked rates differ by a factor of 1.42, which is inside that. They are one fluctuation and not two — which is the question an earlier essay asked and left open, answered by a bank rather than by a convention. Pitch and tuning

One fluctuation or two

Whether two members of a beat family are one thing or two was decided by asking whether their rates differ by a factor of two, and the factor was written down as a stand-in for a modulation filterbank nobody had run. Run, the bank gives 1.618 — the golden section, and not by accident, since a channel of quality one has its half-power points there. The stand-in was conservative rather than optimistic, and the recomputed counts do not change at a single register, because the criterion was never what was binding.

 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 tempered interval moves its difference tone several times further than itself. For every interval inside the octave tuned to twelve equal steps, how far the difference tone f₂ − f₁ lands from where the just interval would put it, in cents, with the interval's own departure from just drawn as the thin bar beside it. minor second −198.0 (the interval −11.7); major second −35.5 (the interval −3.9); minor third −96.0 (the interval −15.6); major third +67.4 (the interval +13.7); fourth +7.8 (the interval +2.0); fifth −5.9 (the interval −2.0); minor sixth −36.7 (the interval −13.7); major sixth +38.8 (the interval +15.6); minor seventh −39.8 (the interval −17.6); major seventh +25.0 (the interval +11.7). The major third's product is +67.4 cents out and the minor third's −96.0, and the largest error is the minor second's, at −198: a product moves p/(p − q) times as far as the interval p:q that made it. Intervals and chords

The third sound magnifies cents, not hertz

Tartini's third sound is said to be a few cents off on a tempered interval. It is sixty-seven cents off on a major third and ninety-six on a minor third, because a difference tone moves p/(p − q) times as many cents as the interval p:q that made it. In hertz it moves exactly as far as the note that moved, and no further — so what the magnifier is worth is the ear's finer resolution at the low frequency where the product lands, which is a factor of two for a long note and nothing at all for a short one.

 Every product of a just interval is a harmonic of the note it implies. Each interval drawn as two harmonics of a fundamental it does not contain — the lower note is harmonic q and the upper harmonic p — with its three combination tones placed on the same numbering: the difference tone at p − q, the cubic product below the pair at 2q − p, and the one above at 2p − q. minor second 16:15: 1, 14, 17; major second 9:8: 1, 7, 10; minor third 6:5: 1, 4, 7; major third 5:4: 1, 3, 6; fourth 4:3: 1, 2, 5; fifth 3:2: 1, 1, 4; minor sixth 8:5: 3, 2, 11; major sixth 5:3: 2, 1, 7; minor seventh 9:5: 4, 1, 13; major seventh 15:8: 7, 1, 22. The shaded column is the fundamental itself. The difference tone sits on it for every interval up to the fifth, the cubic product for the fifth and every interval above except the minor sixth, whose products are its fundamental's octave and twelfth. Intervals and chords

The tone on the root changes hands at the fifth

Every combination tone of a just interval is a harmonic of a fundamental neither note contains, and which harmonic is fixed by the ratio. The difference tone lands on that fundamental for every interval up to the fifth; the cubic product lands on it for the fifth and every interval above except the minor sixth. So the loud product names the root of a narrow interval and the quiet one names the root of a wide one — and a just major seventh's difference tone is a note seven harmonics up that no keyboard has.

 A major triad's combination tones, against its own notes. The three notes of a major triad on C4 in root position, voiced C4–E4–G4, as tall lines, and every combination tone its pairs make, as short ones: difference tones lowest, cubic products taller. In just intonation 2 cubic products land exactly on a note of the chord, and none comes within forty hertz of one. In equal temperament no cubic product lands on a note of the chord, and the nearest miss is 5.63 hertz. Intervals and chords

A major triad's combination tones are its own notes

Play a just major triad of pure tones and two of the ear's cubic products land exactly on its root and its fifth. The reason is a condition rather than a coincidence — a chord's cubic products fall on its own notes when its middle note is the mean of the outer two in hertz — and it holds for the major triad in root position and in the six-four, and for no minor triad in any position or tuning. Equal temperament misses the landing by one number, 5.6 hertz on middle C, which is a beat that belongs to no pair of notes in the chord.

 A scale in parallel thirds has a line underneath it that nobody plays. A major scale on C4 harmonised in parallel diatonic thirds, with the difference tone f₂ − f₁ of each pair drawn as a third line. In five-limit just intonation that line is C2, A1, C2, F2, G2, F2, G2, C3. In equal temperament it moves to C♯2, A1, B1, F♯2, A♭2, E2, F♯2, C♯3, departing from the just line by +67, +33, −82, +69, +65, −80, −84, +67 cents. Intervals and chords

The bass line under a passage in thirds

A major scale harmonised in parallel thirds gives the ear a difference tone under every pair, and in five-limit just intonation those tones are a diatonic bass line — C, A, C, F, G, F, G, C — made of the scale's own notes. Tempered, the same line moves only by whole tones, a neutral third and a fourth stretched to 650 cents, and wobbles by up to 84 cents from note to note. In sixths the bass is drawn by the other product, because the cubic product of a pair is the difference tone of the same pair inverted.

 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 schedule that hears every entrance best holds the high parts back. Six parts waiting to enter a five-chord passage over four sounding players, each entering once and staying: the schedule under which the least audible entrance is as audible as it can be made. brass on E3 enters at I, open with -1.4 decibels of mean margin over the mask; oboe on E4 enters at vi, close below with -2.6 decibels of mean margin over the mask; clarinet on G4 enters at I, open with -1.4 decibels of mean margin over the mask; voice on C5 enters at IV, close above with 2.1 decibels of mean margin over the mask; violin on G5 enters at V, bracketing with 6.9 decibels of mean margin over the mask; flue pipe on C6 enters at I, hollow with 4.0 decibels of mean margin over the mask. The least audible entrance is at -2.6 decibels and the margins sum to 7.6; of all 15625 schedules 0 have a better least audible entrance and 576 a larger sum. Form and structure

Room is used up by whoever enters first

The chord with the most room for a part entering alone is a fact about that chord. It stops being a fact the moment two parts want it, because each part that comes in raises the mask over everybody after it. Given six parts waiting to enter a five-chord passage, choosing each part's moment the way one part's moment is chosen puts three of them into the same chord and lands in the bottom fifth of all 15,625 schedules. Placing them one at a time does no better. The schedule under which the least audible entrance is heard best is unique, and it brings the low and middle parts in while the texture is thin and holds the three highest back for the last three chords — because a high part keeps its room over a full texture and a middle part does not.

 Level does not dilute the register's roughness, it multiplies it. The mean roughness of the I – vi – IV – V – I arrivals at 4 registers, each relative to the register as written, read three ways. Level-free, the bass is 8.6 times rougher than the treble. With every note at 70 dB it is 8.6 times, the same factor, because one level rescales every pair alike. With each chord played at the level that makes it as loud as the written register's chords — 82.8 dB −2 octaves, 75.5 dB −1 octave, 70.0 dB as written, 66.8 dB +1 octave — the bass is 343 times rougher than the treble, because roughness grows with the square of the pressure and the bass needs more of it to be heard at the same loudness. Harmony and voice leading

A rough arrival is rough because of its spacing

The pair the expectation essays report for every chord — how surprising it was, how rough its voicing is — has no level in it. Putting level back in answers the question it left open, and not the way it was framed. At one written dynamic the arrivals keep their order from 40 to 90 dB at three registers of four, and the bass stays 8.6 times rougher than the treble. Made equally loud, the bass has to be played 12.8 dB harder, and it is 343 times rougher: level does not explain the register's roughness away, it multiplies it.

 The thirds' difference-tone line, note by note, against the dynamic. Each note of the line the difference tone f₂ − f₁ draws under a scale in just thirds, as its level above the higher of the threshold of hearing and the primaries' masking, against the level of the primaries. The product sits 50 dB below the primaries at 60 dB and grows twice as fast as they do. C2 above the limit from 73.5 dB; A1 above the limit from 76 dB; C2 above the limit from 73.5 dB; F2 above the limit from 70 dB; G2 above the limit from 68.5 dB; F2 above the limit from 70 dB; G2 above the limit from 68.5 dB; C3 above the limit from 66 dB. Intervals and chords

A combination-tone bass needs a forte

A scale in just thirds draws a diatonic bass line through its difference tones, and in sixths the cubic product draws one. Given the two published level laws, with their constants swept, the thirds' bass is not heard at all below primaries of about 66 dB and is heard whole only from 71 to 81. The cubic products are a different kind of object: the primaries mask them decibel for decibel as they rise, so no dynamic changes whether they are heard. Most of the thirds' inner line never is, and the sixths' bass needs a forte and a gentle law.

 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.

 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.

 Every product of every pair of partials is a harmonic of one absent note. A 4 : 5 interval on 261.6 and 327.0 hertz, just, each note carrying 6 partials at one over n, with the fundamental at 80 decibels. Every pair of partials makes its own difference tone, and there are 19 distinct frequencies among them. All of them are exact multiples of 65.41 hertz — the note neither instrument is playing — because partial j of a p·f₀ note and partial k of a q·f₀ note differ by (kq − jp)·f₀ whatever j and k are. The heavy line is what each product has to clear — the threshold of hearing at its own frequency, or the masked threshold the two primaries cast there, whichever is higher. 11 of the 19 get through. Intervals and chords

Three harmonics of the bass arrive before the bass

Every product priced until now was between two pure tones, and nothing that plays thirds is pure. Give each note a spectrum and the ear receives every pair of partials — and for a just interval p:q every one of their products is an exact multiple of the same absent fundamental. That crowd lands where the threshold of hearing is tens of decibels cheaper, so it names the bass at 71 decibels where the component at the bass's own frequency needs 74, and at 75 against 85 an octave lower. Tempered, the crowd still forms and names a note seventy cents flat.

 The ghost bass drops when the passage gets louder. The note the whole crowd of products names, as a multiple of the fundamental the interval implies, against how loudly the interval is played. a major third: 2.9999999999999996 times the fundamental below 70 decibels and 1 times above it, a drop of 19 semitones; a minor third: 4 times the fundamental below 62 decibels and 2 times above it, a drop of 12 semitones; a fourth: 2 times the fundamental below 72 decibels and 1 times above it, a drop of 12 semitones. Softly, only the cubic products clear their thresholds, and they are an exact series on (2p − q) times the fundamental with no gaps in it. Loudly, the difference tones fill in the low harmonics, no template on the higher note can explain them, and the fit falls. Nothing about the interval has changed; the listener is simply being given a different subset of the same harmonic series. Intervals and chords

The ghost bass drops a twelfth at a forte

Both crowds arrive at once and every member of both is a multiple of the same absent fundamental, so a listener is never given a choice between them — only a different subset of one harmonic series at every dynamic. Softly, the subset is an exact gapless series on three times the fundamental. Loudly, the difference tones fill in the low harmonics and no template on the higher note survives them. Between 62 and 72 decibels, depending on the interval, the note the crowd names falls by an octave or a twelfth, and the two qualities of third cross at different levels.

 A displaced map is displaced by the same amount everywhere. How far a listener's heard direction is displaced, in units of the smallest angular change they could detect at that azimuth, for four constant offsets added to every interaural delay. Each curve is flat. an offset of 5 microseconds is worth 0.33 just-noticeable steps at every azimuth; an offset of 10 microseconds is worth 0.67 just-noticeable steps at every azimuth, and past 88° hands the listener a delay their own head cannot produce; an offset of 20 microseconds is worth 1.33 just-noticeable steps at every azimuth, and past 86° hands the listener a delay their own head cannot produce; an offset of 40 microseconds is worth 2.67 just-noticeable steps at every azimuth, and past 82° hands the listener a delay their own head cannot produce. The reason is exact: differentiating Woodworth's curve gives a slope proportional to (1 + cos θ), so the angular displacement a fixed offset produces carries a factor of 1/(1 + cos θ) — and so does the smallest detectable angle, so the ratio has no azimuth in it. That is the opposite of a wrong head radius, whose displacement is zero on the median plane and grows toward the side. Perception and the listener

The error that moves straight ahead

The essay before this one found that a listener whose internal head is the wrong size makes no error at all on the median plane, and has to look hard to the side to catch it. Every head drawn here has its ears at equal radii, which makes the delay curve odd and every error a factor — and a factor cannot move a zero. Real heads are not symmetric. A constant offset of twenty microseconds displaces a listener's straight ahead by two and a quarter degrees, and it displaces every other direction by the same number of just-noticeable steps, exactly.

 Counted in beats, the abandoned chord comes back sooner and by steps. The piece in standard tuning with its G major, open chord given more and more of the piece, searched twice: once minimising the mean departure from just in cents, once minimising the mean beat rate. The vertical axis is how far that chord sits from just at each optimum. Counted in cents it stays at 13.29 cents until it holds exactly 44 beats — the same as the rest of the piece — and then drops to zero in one step. Counted in beats it drops at 30.4 beats to 4.96, then at 47.4 beats to 3.91, then at 73.1 beats to 0.00 cents. At the piece's own 8 beats the two counts agree exactly: the corner does not move, and what the beats change is where the steps are. Pitch and tuning

Counting beats moves the price of a chord, not the tuning

The search that found a guitar piece's best tuning counted every cent of error alike, and the obvious objection is that a cent of a third beats faster than a cent of an octave. Counted in beats instead, every piece gets exactly the same tuning back. What changes is how much of the piece the abandoned chord has to hold before it is rescued — thirty beats instead of forty-four, arriving in three steps instead of one.

 A louder final chord stands higher and still stands for a fraction of a second. How far a final chord stands above the listener's running impression at the instant it is released, against how long it lasts. The chord is a struck six-note tonic; "a step louder" is the hammer velocity doubled, which raises its loudness 7.12 phons above the tutti's. At the tutti's level, after 1.2 s of silence, the impression is 8.09 dB below the chord and the chord stands highest, 5.14 dB, at 54 ms; a step louder, straight out of the tutti, the impression is 7.12 dB below the chord and the chord stands highest, 4.51 dB, at 36 ms; a step louder, after 1.2 s of silence, the impression is 15.21 dB below the chord and the chord stands highest, 9.46 dB, at 38 ms. Every curve returns to zero by half a second: the impression climbs to whatever level the chord is played at, so a louder mark is not a stand that lasts but a deeper fall to climb out of, and a silence and a mark add as depths. Form and structure

A louder final chord is a deeper silence and a brighter sound

A final chord marked a step louder than the passage was supposed to stand above a listener's running impression for as long as it sounded, since the impression can climb no higher than the chord. It climbs exactly that high, and the stand closes in half a second as it always did. What a louder mark actually buys is depth — about seven phons, the same depth a second of silence buys — and a spectrum whose balance point sits most of a whole tone higher, which, unlike the stand, lasts for the whole chord.

 With 4 of six required at the end, the best schedule hands parts over. Six parts entering, leaving and re-entering a five-chord passage over four sounding players, in the walk through all 64 sets of sounding parts that makes the least audible entrance as audible as possible, with no memory of the chord before, at least 4 of the six sounding at the last chord. I, open: clarinet on G4 enters at 0.6 dB; vi, close below: oboe on E4 enters at 0.3 dB, and clarinet leaves; IV, close above: brass on E3 enters at -0.7 dB, voice on C5 enters at 2.2 dB, and oboe leaves; V, bracketing: violin on G5 enters at 7.2 dB; I, hollow: flue pipe on C6 enters at 4.0 dB. The least audible entrance is -0.72 dB against -2.65 for the best schedule in which nobody leaves; the walk has 2 exits and 6 entrances. Form and structure

An exit is worth nothing until the tutti is given up

Six parts entering a five-chord passage have a best schedule when each enters once and stays, and letting parts leave and come back was supposed to improve it. Searched over every set of sounding parts at every chord, it improves it by exactly nothing, with or without the chord before still masking — as long as all six must be playing at the end. Let one part be missing from the final chord and the weakest entrance gains 1.4 decibels; let two be missing and it gains 1.9, by a relay in which the parts with least room come in, are heard for one chord, and give way.

 Put back beside its notes, the crowd names the bass at every dynamic. A just major third on complex tones, drawn on one axis of harmonic numbers of the fundamental its ratio implies: the partials of the two played notes, and the products of those partials that clear threshold, at 55 and 80 dB. At 55 dB the products alone name 3 times the fundamental with 0 empty slots; the products and the notes together name 1 times it with 4 empty slots, and the notes alone name it with 9. At 80 dB the products alone name 1 times the fundamental with 0 empty slots; the products and the notes together name 1 times it with 0 empty slots, and the notes alone name it with 9. The soft reading on a higher note exists only when the loud notes are set aside. Taken together, the products do not decide which fundamental is named; they decide how many holes its template has. Intervals and chords

The played notes already name the ghost bass

The products of a just third's partials, fitted on their own, name a note a twelfth above the bass when the interval is soft and drop to the bass when it is loud. Put the two played notes back beside them and the drop disappears: the notes and their products name the bass at every dynamic, because the notes' own partials are harmonics of it already. What the dynamic changes is not which note is implied but how complete its harmonic series is — nine holes from the notes alone, four when soft, none when loud.

 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.

 Counted over what arrives, the balanced bass is not the roughest register. The mean roughness of the I – vi – IV – V – I arrivals with each chord played as loud as the written register's, relative to the written register, counted over every partial and over the partials that stand above what the rest of the chord masks. Every partial: 70 −2 octaves, 7.48 −1 octave, 1.00 as written, 0.20 +1 octave. Delivered partials only: 6e-9 −2 octaves, 2.83 −1 octave, 1.00 as written, 0.19 +1 octave. Over every partial the lowest register is 343 times rougher than the highest; over what arrives it is the smoothest of the four, and the roughest is −1 octave, 2.8 times the written register. Perception and the listener

A bass chord low enough to balance has already hidden its tenor

Played as loud as the written register, a progression two octaves down is 343 times rougher than the same progression an octave up — if every partial on the page is counted. Count only the partials that stand above what the rest of the chord masks and that register is the smoothest of the four, with nothing left that beats. The balance is not what does it: the extra thirteen decibels move no voice by more than two partials. The register had already buried the tenor at the written dynamic.

 A tempered major third and its own ghost are not one sound. The multiple of the implied fundamental that each reading names, against how far the upper note is tempered from the just major third, at 55 and 80 dB. At 55 dB the notes and products together name the fundamental from −9.0 cents to +8.5 cents and no template at all outside that; the products alone name 3 times it and go on naming very nearly that across the whole range. At 80 dB the notes and products together name the fundamental from −2.0 cents to +2.0 cents and no template at all outside that; the products alone name 1 times it and go on naming very nearly that across the whole range. Equal temperament's major third is +13.7 cents, which is outside the window at both dynamics. The notes alone name the fundamental throughout, because tempering moves every partial of the upper note by the same amount and a template can follow it. Intervals and chords

A tempered third cannot be one sound with its own ghost

The essay before this one fitted one harmonic template to a just third's played partials and the products of those partials together, and it worked at every dynamic. Tempered, it does not work at all — and not by naming a different note. There is no template. The lower note's partials sit exactly on their harmonics, the upper note's are every one fourteen cents out, and the products are twenty-three cents out the other way, a span of thirty-seven cents across a tolerance of thirty. The window is nine cents wide when the third is soft and two when it is loud, so equal temperament's fourteen falls outside it — while the fifth's two cents fall inside. The consonances a temperament leaves nearly alone are exactly the ones that stay single sounds.

 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.

 A rest can balance a louder stamp only if the stamp is shorter than a ninth of a second. The level a final chord a step louder than the tutti, 7.12 phons, leaves the running impression at, in phons over the tutti, against the length of the rest before it, for stamps of 50 ms, 70 ms, 100 ms, 120 ms, 200 ms. 50 ms: 0 s +2.21, 0.25 s +0.16, 0.5 s -0.92, 0.75 s -1.91, 1 s -2.83, 1.5 s -4.46, 2 s -5.84, 2.5 s -7.00, 3 s -7.95, 4 s -9.38, 5 s -10.30, 6 s -10.89, balanced by a rest of 0.29 s; 70 ms: 0 s +3.16, 0.25 s +1.47, 0.5 s +0.67, 0.75 s -0.05, 1 s -0.71, 1.5 s -1.84, 2 s -2.76, 2.5 s -3.51, 3 s -4.12, 4 s -5.00, 5 s -5.55, 6 s -5.89, balanced by a rest of 0.73 s; 100 ms: 0 s +4.27, 0.25 s +3.07, 0.5 s +2.55, 0.75 s +2.08, 1 s +1.67, 1.5 s +0.97, 2 s +0.42, 2.5 s -0.02, 3 s -0.37, 4 s -0.87, 5 s -1.17, 6 s -1.35, balanced by a rest of 2.47 s; 120 ms: 0 s +4.83, 0.25 s +3.88, 0.5 s +3.48, 0.75 s +3.12, 1 s +2.81, 1.5 s +2.29, 2 s +1.87, 2.5 s +1.55, 3 s +1.30, 4 s +0.94, 5 s +0.72, 6 s +0.59, and no rest brings it to the tutti; 200 ms: 0 s +6.14, 0.25 s +5.76, 0.5 s +5.60, 0.75 s +5.47, 1 s +5.35, 1.5 s +5.16, 2 s +5.01, 2.5 s +4.89, 3 s +4.80, 4 s +4.68, 5 s +4.60, 6 s +4.56, and no rest brings it to the tutti. Past a stamp of 115 ms no rest is long enough: the stamp lifts even a fully fallen impression past the tutti. Form and structure

No rest balances a louder chord held past a ninth of a second

A rest before a final chord leaves the passage quieter than the tutti and a louder mark on the chord leaves it louder, so the two together can leave it exactly where it was, while the chord stands out by both at once. They can — but only for a stamp. A chord a step louder and longer than 115 milliseconds lifts even a listener's fully fallen impression past the tutti, so no rest of any length balances it; two steps louder, the limit is 67 milliseconds, and three, 45. The balanced ending exists, and it is decided by the release, not by the rest.

 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.

 An octave down, the ghost bass needs notes twice as long before it is there at all. The share of each note of a C major scale in just thirds that has its difference-tone bass, against how long each note lasts, for the passage starting on C3, C4 and C5, with the bass heard only after 4 of its own cycles. Thirds from C3: 40 ms 0%, 60 ms 0%, 80 ms 3%, 100 ms 12%, 125 ms 22%, 150 ms 33%, 200 ms 50%, 250 ms 60%, 300 ms 67%, 400 ms 75%, 500 ms 80%, 750 ms 87%, 1000 ms 90%; no note has its bass below 147 ms. Thirds from C4: 40 ms 3%, 60 ms 20%, 80 ms 38%, 100 ms 50%, 125 ms 60%, 150 ms 67%, 200 ms 75%, 250 ms 80%, 300 ms 83%, 400 ms 88%, 500 ms 90%, 750 ms 93%, 1000 ms 95%; no note has its bass below 73 ms. Thirds from C5: 40 ms 38%, 60 ms 58%, 80 ms 69%, 100 ms 75%, 125 ms 80%, 150 ms 83%, 200 ms 88%, 250 ms 90%, 300 ms 92%, 400 ms 94%, 500 ms 95%, 750 ms 97%, 1000 ms 98%; no note has its bass below 37 ms. Every step down an octave halves the bass and doubles the note length it needs. Intervals and chords

A ghost bass an octave lower needs notes twice as long

A scale played in just thirds has a bass underneath it that nobody plays — the difference tones of its thirds spell a diatonic line. Every essay on that bass has held its intervals still. A moving passage gives the bass a new pitch every note, and a low pitch is not there until it has run a few of its own cycles. With four cycles, a passage in thirds from C4 keeps its bass for 83 per cent of each note at three notes a second and loses it completely below 73 milliseconds a note. An octave down, every number doubles: the passage from C3 has no bass at all in its lowest note at semiquavers, and it is the scale's one Pythagorean third that sets the limit.

 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 listener whose ears differ by a decibel or two has a straight ahead that moves with the dynamic. Where the median plane sits for a listener whose two ears' thresholds differ, against the level of a 500 Hz tone, if the whole of the latency difference the published brainstem latency law gives reaches the comparator that reads interaural delay. 1 dB worse: 19.5° at 30 dB (172 µs), 12.0° at 60 dB (106 µs), 7.4° at 90 dB (65 µs); 2 dB worse: 40.6° at 30 dB (347 µs), 24.4° at 60 dB (214 µs), 14.9° at 90 dB (132 µs); 3 dB worse: 65.6° at 30 dB (524 µs), 37.7° at 60 dB (323 µs), 22.7° at 90 dB (200 µs). Every curve halves its offset every 43 dB and none reaches zero at any level a listener hears; the dashed line is one detectable step on the median plane, 1.68°. Perception and the listener

A quiet bass is further off-centre than a loud violin

An ear whose threshold is a decibel worse than the other is a little later to report every sound, and the published brainstem latency law says how much later: a gap that halves every 43 decibels and grows toward the bass. Taken at full size, a one-decibel asymmetry would put straight ahead twelve degrees round at a moderate level — seven times what the ear's own two thresholds say a decibel is worth, so most of that latency cannot be reaching the comparator that reads direction. What survives any scaling is the shape. The displacement is not one number but a curve in level and in pitch, and it deforms an orchestra rather than moving it.

 The notes name their ghost bass long before the bass has run a single cycle's worth of its count. A just major third, C4 against E4, with every component — each note's first six partials, the difference tone and the cubic tone — entering a harmonic fit once it has run 4 of its own cycles. at 2.0 ms, 1 component: nothing fitted; at 2.4 ms, 2 components: nothing fitted; at 2.5 ms, 3 components: names 103.3 Hz; at 3.1 ms, 4 components: names 82.0 Hz; at 3.8 ms, 5 components: names 86.3 Hz; at 4.1 ms, 6 components: names 81.8 Hz; at 5.1 ms, 6 components: names 65.4 Hz, the ghost bass; at 6.1 ms, 6 components: names 65.4 Hz, the ghost bass; at 7.6 ms, 6 components: names 65.4 Hz, the ghost bass; at 12.2 ms, 6 components: names 65.4 Hz, the ghost bass; at 15.3 ms, 6 components: names 65.4 Hz, the ghost bass; at 20.4 ms, 6 components: names 65.4 Hz, the ghost bass; at 61.2 ms, 6 components: names 65.4 Hz, the ghost bass. The fit names the 65.4 Hz ghost from 5.1 ms; the bass's own 4 cycles take 61.2 ms. Intervals and chords

The notes name the ghost bass before it has a cycle

A ghost bass that had to wait for four of its own cycles needed 61 milliseconds under a third from C4, and so vanished from fast passages. A pitch mechanism fitting the whole object — both notes' partials, which run their cycles many times faster, and the products — names the same bass in five. Seven of the eight thirds of a just scale have their bass for most of every note at any tempo a player reaches. The eighth, the scale's Pythagorean D–F, never has its own: the object hears it as a 6 : 5 and supplies the bass a D a comma lower would give, a semitone above the difference tone, until the difference tone arrives and breaks the fit.

 Quietly, or quickly, the scale's Pythagorean third is heard with the bass of the just third a comma away. The third D4–F4 of a just C major scale, a Pythagorean 32 : 27, fitted as a whole object with every component that has run 4 of its cycles and every product only if it is audible at the level played: what the fit names at the end of a note of each length, at each dynamic. 40 dB: 20 ms 58.4 Hz, 40 ms 58.4 Hz, 60 ms 58.4 Hz, 80 ms 58.4 Hz, 100 ms 58.4 Hz, 150 ms 58.4 Hz, 200 ms 58.4 Hz, 300 ms 58.4 Hz, 500 ms 58.4 Hz; 50 dB: 20 ms 58.4 Hz, 40 ms 58.4 Hz, 60 ms 58.4 Hz, 80 ms 58.4 Hz, 100 ms 58.4 Hz, 150 ms 58.4 Hz, 200 ms 58.4 Hz, 300 ms 58.4 Hz, 500 ms 58.4 Hz; 60 dB: 20 ms 58.4 Hz, 40 ms 58.4 Hz, 60 ms 58.4 Hz, 80 ms 58.4 Hz, 100 ms 58.4 Hz, 150 ms 58.4 Hz, 200 ms 58.4 Hz, 300 ms 58.4 Hz, 500 ms 58.4 Hz; 66 dB: 20 ms 58.4 Hz, 40 ms 58.4 Hz, 60 ms 58.4 Hz, 80 ms 58.4 Hz, 100 ms 58.4 Hz, 150 ms 58.4 Hz, 200 ms 58.4 Hz, 300 ms 58.4 Hz, 500 ms 58.4 Hz; 70 dB: 20 ms 58.4 Hz, 40 ms 58.4 Hz, 60 ms 58.4 Hz, 80 ms 58.4 Hz, 100 ms 58.4 Hz, 150 ms 58.4 Hz, 200 ms 58.4 Hz, 300 ms 58.4 Hz, 500 ms 58.4 Hz; 74 dB: 20 ms 58.4 Hz, 40 ms 58.4 Hz, 60 ms 58.4 Hz, 80 ms 58.4 Hz, 100 ms 58.4 Hz, 150 ms 58.4 Hz, 200 ms 58.4 Hz, 300 ms 58.4 Hz, 500 ms 58.4 Hz; 76 dB: 20 ms 58.4 Hz, 40 ms 58.4 Hz, 60 ms 58.4 Hz, 80 ms 27.1 Hz, 100 ms 27.1 Hz, 150 ms 27.1 Hz, 200 ms 27.1 Hz, 300 ms 27.1 Hz, 500 ms 27.1 Hz; 80 dB: 20 ms 58.4 Hz, 40 ms 58.4 Hz, 60 ms 58.4 Hz, 80 ms 27.1 Hz, 100 ms 27.1 Hz, 150 ms 27.1 Hz, 200 ms 27.1 Hz, 300 ms 27.1 Hz, 500 ms 27.1 Hz; 85 dB: 20 ms 58.4 Hz, 40 ms 58.4 Hz, 60 ms 58.4 Hz, 80 ms 27.1 Hz, 100 ms 27.1 Hz, 150 ms 27.1 Hz, 200 ms 27.1 Hz, 300 ms 27.1 Hz, 500 ms 27.1 Hz; 90 dB: 20 ms 58.4 Hz, 40 ms 58.4 Hz, 60 ms 58.4 Hz, 80 ms 27.1 Hz, 100 ms 27.1 Hz, 150 ms 27.1 Hz, 200 ms 27.1 Hz, 300 ms 27.1 Hz, 500 ms 27.1 Hz. 58.1 Hz is the bass of D–F with D a syntonic comma lower, a just 6 : 5; the Pythagorean third's own difference tone is 54.5 Hz and is audible from 76 dB, and once it has run its cycles the fit breaks. Intervals and chords

The dynamic chooses which D the scale has

A just C major scale has one third that is not 5 : 4 or 6 : 5 — D–F, a Pythagorean 32 : 27 — and a pitch mechanism fitting the notes' partials hears it as the just minor third a comma away, with that third's bass, B♭1, beneath it. Only the Pythagorean third's own difference tone can contradict that, and it is audible only from 76 dB. So below a forte the ghost line under a scale in just thirds is C, B♭, C, F, G, F, G, C for the whole of every note, and the syntonic comma is resolved by the listener rather than the players; above it, and only in notes longer than 73 milliseconds, the B♭ is taken away.

 Made to stay 2 chords, the parts change over in groups rather than one by one. Six parts entering, leaving and re-entering a five-chord passage over four sounding players, in the walk through all 64 sets of sounding parts that makes the least audible entrance as audible as possible, with no memory of the chord before, at least 2 of the six sounding at the last chord, and every part that enters staying at least 2 chords before it may leave. I, open: clarinet on G4 enters at 0.6 dB; vi, close below: nobody enters; IV, close above: brass on E3 enters at -0.7 dB, voice on C5 enters at 2.2 dB, and clarinet leaves; V, bracketing: violin on G5 enters at 7.2 dB; I, hollow: oboe on E4 enters at -1.1 dB, flue pipe on C6 enters at 4.7 dB, and brass and voice leave. The least audible entrance is -1.07 dB against -2.65 for the best schedule in which nobody leaves; the walk has 3 exits and 6 entrances. Form and structure

A part that must stay changes over in shifts

The best way to keep every entrance audible, when the passage need not end with everybody in, was a relay: a part enters, is heard for one chord, and gives way. Make every part stay at least two chords once it is in and the relay does not collapse into overlaps, as predicted. It changes over in shifts: two parts enter together, one chord passes in which nobody enters, and both leave on the chord their successors arrive. Half of what exits were worth survives a two-chord stay and nearly two fifths a three-chord one — and with the chord before still masking, the stay costs less, not more, because it was the one-chord handover that memory had already made leaky.

 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.

 Scored as one arrival, a pair enters as a loud part carrying one that cannot be heard. Six parts entering, leaving and re-entering a five-chord passage over four sounding players, in the walk through all 64 sets of sounding parts that makes the least audible entrance as audible as possible, with no memory of the chord before, at least 2 of the six sounding at the last chord. I, open: clarinet on G4 enters at -10.9 dB, violin on G5 enters at 19.6 dB; vi, close below: oboe on E4 enters at -1.5 dB, voice on C5 enters at 14.5 dB, and clarinet and violin leave; IV, close above: nobody enters, and oboe and voice leave; V, bracketing: brass on E3 enters at -0.0 dB, flue pipe on C6 enters at 25.0 dB; I, hollow: nobody enters. Scored with the parts entering on a chord taken together, at most 2 at a time, as one arrival against what was already sounding: the weakest arrival is 10.93 dB, and the most buried single entrant, measured as itself, is -10.94 dB. Form and structure

A pair entering as one hides the part that cannot be heard

Scored as two entrances, a pair of parts coming in together is charged for masking itself; scored as one arrival — a new colour against what was already sounding — it is not, and the best schedules change completely. Allowed two at a time, the weakest arrival rises from 0.6 dB above the mask to 10.9, and every pair the search chooses is a loud part carrying a quiet one: a violin 19.6 dB clear beside a clarinet 10.9 under. Allowed any number, the six parts enter at once. The staggered build-up of every earlier schedule exists only because each part was meant to be heard as itself.

 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.

 A slow attack is heard where a much quieter note would be. How far a listener whose ears differ by 2 dB hears a 98 Hz note displaced toward the better ear, in detectable steps of 15 µs, against its dynamic, if 14 per cent of the brainstem latency law's difference reaches the delay comparator and the note's position is set by its onset, weighted over 3 ms from the moment it becomes audible. The dashed line is the note treated as a steady tone. struck (2 ms rise): 40 dB 5.73, 50 dB 5.00, 60 dB 4.32, 70 dB 3.70, 80 dB 3.16, 90 dB 2.69. plucked (5 ms rise): 40 dB 5.95, 50 dB 5.32, 60 dB 4.66, 70 dB 4.02, 80 dB 3.45, 90 dB 2.94. tongued (25 ms rise): 40 dB 6.46, 50 dB 6.07, 60 dB 5.52, 70 dB 4.87, 80 dB 4.23, 90 dB 3.63. bowed (100 ms rise): 40 dB 6.69, 50 dB 6.51, 60 dB 6.16, 70 dB 5.64, 80 dB 5.01, 90 dB 4.36. a soft bowed entrance (250 ms rise): 40 dB 6.76, 50 dB 6.67, 60 dB 6.46, 70 dB 6.07, 80 dB 5.52, 90 dB 4.87. Steady: 5.52, 4.70, 4.00, 3.40, 2.90, 2.47. At 90 dB a 250 ms entrance is displaced 1.98 times as far as the steady tone. Perception and the listener

A bowed note is placed as if it were quieter

An ear a decibel or two worse than the other is later to report every sound, by an amount that grows as the sound gets quieter, and a listener takes a note's direction from its onset. A plucked or struck note reaches its dynamic before the onset is over and is placed almost where a steady tone would be. A bowed note is still near the threshold of hearing while its onset is being read, so it is placed as though it were 27 decibels quieter than it is. The prediction was that slow attacks would soften the displacement; they sharpen it, and an asymmetric listener's orchestra is deformed by its articulation as much as by its register.

 Three chords share a ceiling; only the tempered one beats. Five four-note chords over C3, each at the spacing that keeps its partial coincidences longest: how much slow beating its partials make — pairs from different notes within 15 Hz of each other, weighted by the quieter of the two and integrated over three seconds — and how rough it is, by the Plomp–Levelt kernel integrated the same way. diminished seventh, tempered: lasts 0.86 s, beating 1.00 of the diminished seventh's, 125 beating pairs at the strike, roughness 1.00 of the diminished seventh's. dominant seventh, tempered: lasts 0.66 s, beating 1.12 of the diminished seventh's, 131 beating pairs at the strike, roughness 1.06 of the diminished seventh's. half-diminished seventh, tempered: lasts 0.69 s, beating 1.25 of the diminished seventh's, 115 beating pairs at the strike, roughness 0.60 of the diminished seventh's. dominant seventh, 4 : 5 : 6 : 7: lasts 0.87 s, beating 0.00 of the diminished seventh's, 0 beating pairs at the strike, roughness 0.39 of the diminished seventh's. half-diminished seventh, its mirror: lasts 0.87 s, beating 0.00 of the diminished seventh's, 0 beating pairs at the strike, roughness 0.94 of the diminished seventh's. The two chords tuned to a series have no partial pair closer than 21.8 Hz without coinciding exactly. Intervals and chords

A tuned seventh chord does not beat; it buzzes

Three four-note chords share a ceiling of 0.87 seconds on how long their partial coincidences last, and nothing in that measure could tell them apart. Their beating can. A chord tuned exactly to one harmonic series is one periodic sound, so no two of its partials can differ by less than the series' fundamental: the tuned dominant seventh has no slow beating at all, where the tempered diminished seventh has 125 beating pairs at the strike. Its roughness is another matter. Tuning removes the beats and changes the roughness by a few per cent, in either direction, and a seventh on its way to 7 : 4 beats more, not less, until it arrives.

 Every decibel taken off the passenger comes off the arrival. The best walk with parts entering together scored as one arrival, at most two at a time, against a floor under each member: no entrant, heard as itself, more than the floor below its mask. Solid lines are the weakest arrival, dashed the most buried member, for two parts required at the end and for all six. 2 at the end: floor 0: arrival 0.59, most buried 0.59; floor 1: arrival 0.59, most buried -0.86; floor 2: arrival 0.59, most buried -1.57; floor 3: arrival 0.59, most buried -2.87; floor 4: arrival 0.59, most buried -3.47; floor 5: arrival 0.59, most buried -3.47; floor 6: arrival 4.50, most buried -5.42; floor 8: arrival 4.50, most buried -6.16; floor 10: arrival 10.93, most buried -9.90; floor 12: arrival 10.93, most buried -10.94; every part heard as itself, the earlier walks' score, 0.59. 6 at the end: floor 0: no walk; floor 1: no walk; floor 2: no walk; floor 3: arrival -2.65, most buried -2.65; floor 4: arrival 0.50, most buried -3.27; floor 5: arrival 0.59, most buried -4.24; floor 6: arrival 4.50, most buried -5.42; floor 8: arrival 4.50, most buried -6.13; floor 10: arrival 4.50, most buried -6.13; floor 12: arrival 4.50, most buried -6.13; every part heard as itself, the earlier walks' score, -2.65. Form and structure

A floor under each member is paid for in steps

Score parts entering together as one arrival and the best pairs are a loud part carrying one that cannot be heard. Put a floor under each member — no part more than so many decibels under its own mask — and the prediction was a schedule between the two, neither staggered nor buried. There is none. The arrival's margin climbs in three steps as the floor is loosened, each step bought with a passenger buried about as deep as the step is high, and below a floor of five decibels nothing beats bringing the parts in one at a time, because the clarinet can clear its mask only alone.

 In a hall the balanced ending is a louder chord, and the hall sets how much louder. The gain over the tutti, in phons, that a final chord held after no rest needs for the listener's running impression to finish exactly at the tutti's weight when the chord's tail has died into the silence, against the hall's reverberation time, for chords held half a second, one second and two. 0.5 s: 0.2 s 0.7, 0.3 s 1.2, 0.45 s 1.9, 0.6 s 2.7, 0.8 s 3.9, 1 s 5.0, 1.25 s 6.6, 1.5 s 8.2, 1.75 s 9.9, 2 s 11.7, 2.25 s 13.7, 2.5 s 15.8, 3 s 20.4. 1 s: 0.2 s 0.7, 0.3 s 1.1, 0.45 s 1.9, 0.6 s 2.7, 0.8 s 3.8, 1 s 5.0, 1.25 s 6.5, 1.5 s 8.1, 1.75 s 9.9, 2 s 11.7, 2.25 s 13.6, 2.5 s 15.7, 3 s 20.4. 2 s: 0.2 s 0.7, 0.3 s 1.1, 0.45 s 1.9, 0.6 s 2.7, 0.8 s 3.8, 1 s 5.0, 1.25 s 6.5, 1.5 s 8.1, 1.75 s 9.9, 2 s 11.7, 2.25 s 13.6, 2.5 s 15.7, 3 s 20.4. The dashed lines are one dynamic step, 7.12 phons, and two; a one-second chord a step louder balances in a hall of about 1.34 s and two steps louder in one of about 2.32 s. Form and structure

In a hall the balanced ending is a louder chord

In a dry room a rest before a final stamp a step louder can leave a passage finishing exactly at its tutti's weight, but only for a stamp shorter than a ninth of a second. The prediction was that a hall would make it impossible: its tail fills the rest and lengthens the stamp. That is right for a small room and wrong for a hall. Read when the chord's tail has died, a live hall pulls every ending below its tutti, and the ending that finishes at its own weight there is a held chord marked louder than the passage — five phons louder at a second of reverberation, a whole step at about 1.3 seconds, two at 2.3.

 The tempered seventh chord repeats where the tuned one does, only less exactly. The autocorrelation of the sound reaching the ear, for a dominant seventh over C3 at its longest-lived spacing — C3, then E4, G4 and B♭4 — tuned 2 : 5 : 6 : 7 and tempered, struck at 80 dB, partials above 5 kHz removed, across every lag that corresponds to a pitch above 30 Hz. The tuned chord's highest peak is 1.000 at 15.29 ms, 65.41 Hz, C2, the series' fundamental. The tempered chord's highest is 0.824 at 15.21 ms, 65.75 Hz, C2 +9¢. At the bass note's own period, 7.64 ms, the two reach only 0.24 and 0.25: neither chord repeats strongly at the pitch of its lowest note. Intervals and chords

A tempered seventh chord already has the tuned one's low note

A dominant seventh tuned 4 : 5 : 6 : 7 is one periodic sound, and its period is a note below the chord that nobody plays. The question left open was whether tuning adds that note. Run through the temporal account of residue pitch, the tempered chord turns out to repeat at the same place, a few cents sharp, at 0.82 of the tuned chord's exactness at the strike and 0.96 a second later. Tuning does not create the low note. It makes it exact, and it gets there by a route on which every step helps, which is not true of the beating.

 In the bass a bowed step keeps the entrance's displacement; in the treble every note is placed afresh. A listener whose ears differ by 2 dB, with 14 per cent of the brainstem latency difference reaching the delay comparator: how far the second note of a bowed legato pair at 60 dB is displaced, in detectable steps of 15 µs, against the interval it moves by, in three registers. A note with no onset of its own keeps the first note's position, drawn at the entrance's displacement; the dotted lines are each register's entrance from silence and the dashed its steady tone. G2, a double bass (entrance 6.16, steady 4.00): -12 6.19, -7 5.02, -4 4.35, -2 4.06, -1 no onset, +1 no onset, +2 no onset, +4 4.08, +7 4.26, +12 4.53. A4, a violin (entrance 3.73, steady 2.15): -12 3.72, -7 3.56, -4 2.82, -2 2.35, -1 2.19, +1 2.15, +2 2.22, +4 2.39, +7 2.67, +12 3.07. A6, a violin's top (entrance 2.13, steady 1.21): -12 2.13, -7 2.13, -4 1.77, -2 1.40, -1 1.26, +1 1.22, +2 1.29, +4 1.43, +7 1.66, +12 1.93. At the dial's up 2 semitones: G2 no onset, keeps 6.16; A4 2.22; A6 1.29. Perception and the listener

A bowed bass line is placed by its entrance; a violin line by every note

A listener whose ears differ slightly places a note by its onset, and a bowed entrance is placed as though it were far quieter than it is. A legato line's later notes do not start from silence. Whether one of them has an onset at all depends on how much of the note before is already in its own auditory filter — and in the bass, where the filters are wide and their lower skirts shallow, a step of a tone or less has none, so the note keeps the entrance's displacement. In the treble nearly every note has its own onset and is placed near its steady position, except where the line leaps down.

 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.

 The tuned close chord flutters at one rate; the tempered one at several. The dominant seventh in close position over C2, tuned 4 : 5 : 6 : 7 and tempered: the modulation spectrum of the envelope in the auditory filter centred at 93 Hz, where the tuned chord's fluctuation stands furthest above the detection threshold, as a modulation index at each rate up to 40 Hz. The dashed curve is the published threshold for detecting amplitude modulation. Tuned: strongest at 16.4 Hz, the series fundamental 16.35 Hz, index 1.33, 42 times the threshold, 100 per cent of the modulation at that rate and its multiples. Tempered: strongest at 15.4 Hz, index 0.34, 45 per cent at one rate and its multiples. Intervals and chords

A tuned close seventh chord low on the keyboard flutters at one rate

A dominant seventh tuned 4 : 5 : 6 : 7 in close position repeats exactly at a frequency two octaves below its bass, and below B2 that frequency is too slow to be a pitch. The question left was whether the repeat is heard as a flutter. In every auditory filter the chord reaches, its envelope pulses at that one rate, forty times over the threshold for detecting a fluctuation in the filter where it is strongest. The tempered chord fluctuates too, but a third as deeply and at rates that change from filter to filter. Tuning adds something here rather than sharpening it, and it adds it only within two cents of 7 : 4.

 Read in its partials' filters, every step of a bowed bass line is an onset. The second note of a bowed legato pair on the double bass's G2 at 60 dB, for a listener whose ears differ by 2 dB: its displacement in detectable steps against the interval, placed by the filter at its fundamental alone (dashed, with pale dots where that filter sees no onset and the note keeps the entrance's position) and by the first of its eight partials' filters to see an onset (solid). By partials: -12 6.19 in partial 1, -7 5.02 in partial 1, -4 3.68 in partial 2, -2 3.12 in partial 3, -1 2.76 in partial 4, +1 2.61 in partial 5, +2 3.10 in partial 3, +4 3.56 in partial 2, +7 3.15 in partial 3, +12 3.51 in partial 2. The entrance, placed the same way, is 4.49 steps by its partial 4, against 6.16 by its fundamental. At the dial's +2: 3.10 steps, first in partial 3. Perception and the listener

A bowed bass line has an onset at every step, in its upper partials

A legato note's onset was read in the auditory filter at its fundamental, and in the bass a step of a tone or less left that filter nearly as full as before: no onset, and the note kept its entrance's displacement. A bowed note is not a fundamental. Read in every filter its partials reach, every step of a bowed bass line has an onset, first seen in the third to fifth partial, where a tone moves a partial right out of its filter. The line is re-placed at every note after all — and nearer the centre, because a partial's filter is at a frequency where the ears' latency difference is smaller. The bowed bass entrance itself is displaced a quarter less than its fundamental said.

 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.

 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.

 One rate in every filter, and whether the filters keep time. The envelopes of four auditory filters — centred at 93, 170, 385, 822 hertz — over the first quarter-second of the dominant seventh in close position over C2, tuned 4 : 5 : 6 : 7. On the left every partial of every note starts together, as in a chord struck at one instant: all four filters pulse once every 61.2 milliseconds and their pulses line up. On the right each note is given its own starting phase, as four voices or four organ pipes have: every filter still pulses at the same rate, but each at its own moment in the period, and the lowest and highest filters' envelopes correlate at -0.34 rather than 0.56. Timbre and acoustics

A tuned chord's flutter binds it only if its notes start together

A close seventh chord tuned 4 : 5 : 6 : 7 low on the keyboard fluctuates at one rate in every auditory filter, and one rate everywhere looked like the ear's evidence that a sound is one thing. It is only half of that evidence. The filters also have to rise and fall at the same moments, and whether they do depends on when each note starts — which tuning does not decide. Spread the notes' onsets by two milliseconds and the tuned chord's filters move together no more than a tempered chord's.

 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.

 A vibrato hides a section's beats above about four hundred hertz. Two voices of a section whose tuning is spread by 15 cents, singing one note: how many nulls of their beating a listener can follow each second — counted only while the beat is at or below fifteen a second — at the frequency of the partial compared, sung straight and with a 71-cent vibrato at about six hertz. 220 Hz 2.2 straight, 5.3 with vibrato; 330 Hz 3.3 straight, 4.4 with vibrato; 440 Hz 4.2 straight, 3.9 with vibrato; 550 Hz 5.0 straight, 3.3 with vibrato; 660 Hz 5.1 straight, 2.8 with vibrato; 880 Hz 5.5 straight, 2.4 with vibrato; 1100 Hz 4.8 straight, 2.0 with vibrato; 1320 Hz 4.6 straight, 1.5 with vibrato; 1760 Hz 3.9 straight, 1.2 with vibrato; 2200 Hz 3.5 straight, 1.1 with vibrato; 2640 Hz 3.2 straight, 0.9 with vibrato; 3200 Hz 2.3 straight, 0.7 with vibrato; 3520 Hz 2.0 straight, 0.7 with vibrato. Below 413 hertz the vibrato adds followable beats, because it sweeps a slow difference faster. Above it the vibrato removes them, because it carries the pair's difference past fifteen hertz most of the time; at 3,200 hertz, the singer's formant, a straight section gives 3.4 times as many. Instruments and their design

A vibrato hides a choir's tuning above four hundred hertz

Thirteen essays drew a vibrato that had always been running; a real one grows out of a straight tone over the first quarter-second of a note. Counting the beats a listener can follow between two voices of a section shows what that front edge is worth. A vibrato takes most of a section's countable beats away above about 400 hertz — seven in ten of them at the singer's formant — and adds some below. The straight start gives them back only while the vibrato is still growing, and in the band where a voice carries that is less than one beat.

 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 section's vibratos must lock almost completely before its tuning is exposed again. Two voices of a section spread by 15 cents, each with a 71-cent vibrato near six hertz, their vibratos pulled together so that their differences of rate and phase shrink to a stated share: followable beats a second at three partial frequencies, against how far they are locked, with the same voices sung straight as the dashed line beside each. 660 Hz (straight 5.14): 0 2.77, 0.25 2.90, 0.5 3.23, 0.7 3.85, 0.8 4.45, 0.85 4.98, 0.9 5.02, 0.95 5.31, 0.98 5.17, 1 5.15; 1217 Hz (straight 4.66): 0 1.60, 0.25 1.80, 0.5 2.04, 0.7 2.58, 0.8 2.88, 0.85 3.29, 0.9 3.88, 0.95 4.41, 0.98 4.67, 1 4.66; 3200 Hz (straight 2.33): 0 0.68, 0.25 0.70, 0.5 0.80, 0.7 1.13, 0.8 1.10, 0.85 1.45, 0.9 1.70, 0.95 1.98, 0.98 2.12, 1 2.23. At the singer's formant, half-locked vibratos expose 34 per cent of what a straight section does, locked to nine-tenths 73 per cent, locked to ninety-five 85 per cent; only at about ninety-eight per cent does it come within a tenth of a straight section. Instruments and their design

A choir's vibratos must lock to two milliseconds

A vibrato hides a section's mistuning from a listener counting beats, and the obvious way it could stop hiding it is for the singers' vibratos to move together. They would have to move very nearly together. Half-entrained vibratos leave a section as hidden at the singer's formant as independent ones; locked to nine-tenths they expose three-quarters of what a straight section does; only past nineteen parts in twenty does the section's tuning show as if they sang straight. At six cycles a second that is two vibratos peaking within two milliseconds of each other — and the better a section is tuned, the tighter the lock has to be.

 A tuned chord struck in step drifts out of step as soon as it is mistuned. The tuned close seventh over C2, every note struck at the same instant and held, each note detuned by a draw with the stated spread in cents: the mean correlation between filters' envelopes, in half-second windows every quarter-second through eight seconds, averaged over eight draws. 0 cents: 0.61, 0.59, 0.59, 0.59, 0.59, 0.59, 0.59, 0.59; 0.1 cents: 0.61, 0.56, 0.50, 0.41, 0.36, 0.32, 0.28, 0.25 (halfway down by 4 s); 0.25 cents: 0.60, 0.43, 0.30, 0.23, 0.20, 0.18, 0.18, 0.17 (halfway down by 1.75 s); 0.5 cents: 0.57, 0.29, 0.19, 0.17, 0.17, 0.16, 0.18, 0.19 (halfway down by 1 s); 1 cent: 0.50, 0.18, 0.17, 0.18, 0.14, 0.18, 0.14, 0.17 (halfway down by 0.5 s), at whole seconds. The tempered chord's level is 0.16. Exactly in tune the chord stays in step; a tenth of a cent takes four seconds to lose half of it, a cent half a second. Timbre and acoustics

A tuned chord stays in step for half a cent-second

A close seventh chord tuned 4 : 5 : 6 : 7 low on the keyboard fluctuates in every auditory filter at once, but only if its notes start within a millisecond of each other — and a tuned chord, it was said, keeps its phases for as long as it sounds. It keeps them only as long as it stays exactly tuned. Mistune its notes by a fraction of a cent and they slide against each other, and the chord drifts out of step in a time inversely proportional to the mistuning: about half a cent-second. A tenth of a cent holds it for four seconds, a cent for half a second. A chord that starts out of step never drifts into it.

 A vibrato short of perfect lock takes an exactly tuned chord out of step. The tuned close seventh over C2, exactly in tune and every note struck at the same instant, held for eight seconds: the mean correlation between filters' envelopes, in half-second windows, averaged over eight draws. Straight: 0.61, 0.59, 0.59, 0.59, 0.59, 0.59, 0.59, 0.59. With a 71-cent vibrato on every note, fully locked: 0.59, 0.57, 0.57, 0.57, 0.57, 0.57, 0.57, 0.57. Locked 98 per cent: 0.59, 0.55, 0.52, 0.47, 0.43, 0.39, 0.37, 0.33, halfway to the tempered chord's 0.16 by 6 s, at whole seconds. A straight tuned chord stays in step for as long as it is held; a vibrato can only keep it there by being locked completely. Timbre and acoustics

A shared vibrato cannot hold a tuned chord together

A tuned close seventh chord binds its auditory filters together only while its notes stay within a third of a millisecond of each other, and mistuned by a fraction of a cent it drifts out of step in half a cent-second. A vibrato shared by every voice looked like a way to hold it: if all four notes swing together, they slide together. They do not stop sliding. A perfectly locked vibrato leaves the drift exactly as fast as before, because it changes every note's frequency by the same factor and leaves their differences alone; a vibrato locked ninety-eight per cent takes an exactly tuned chord out of step in six seconds, and one locked ninety per cent in one.

 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.

 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.

 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.

 A hall fills the gaps a thinning ending leaves only slowly, so the written diminuendo reaches the seats nearly whole. Short-term loudness through the closing bars of a thirty-two-bar scheme of six parts, computed from each bar's part count on the excitation pattern, with a hall of 2 seconds' reverberation under it, its reverberant field in power six decibels over the direct sound in the steady state; the dashed line is the thinning ending with no room. thins to one: arriving -3.83 phons against the running impression; full final chord: arriving +0.13 phons against the running impression; unchanged: arriving +0.00 phons against the running impression; thins then full: arriving +0.43 phons against the running impression. With no room the thinning arrives at -4.46. Perception and the listener

A hall leaves a thinning ending most of its diminuendo

A written diminuendo made by dropping parts arrives about four and a half phons under the listener's impression in a dry room. A hall's tail sounds under every bar and holds the level up as parts leave, and the obvious worry is that it fills the gaps a thinning leaves, as it fills a rest. It barely does. A concert hall leaves the thinning 86 to 100 per cent of its contrast and a stone church about three quarters, where the same rooms leave a two-second general pause three quarters and a half. A thinning steps down a decibel or two a bar, and a hall's tail falls nine decibels in the time a listener takes to register the arrival; only a cathedral is slow enough to blunt the last step, and it takes half.

 Three orders of leaving, one passage: the part left sounding sets the drop, and the room lifts it behind them all. Short-term loudness on the excitation pattern of six string parts, G3 C4 E4 G4 C5 E5 at 62 dB each, dropping one a bar in three orders, with a hall of 2 s in the middle bands, longer in the bass and shorter in the treble. top down: -9.39 phons from the steady texture to the last bar; bass up: -7.40 phons from the steady texture to the last bar; inner first: -8.05 phons from the steady texture to the last bar. Perception and the listener

A hall forgets almost everything but the last part to leave

A hall's bass outlasts its treble, so a thinning ending ought to care which parts it drops first: top down should leave the tail's darkness to fill the gap, and bass up should remove what the tail holds longest. Searched over all 720 orders, it barely does. Give the tail a spectrum and the drop moves by under half a phon, in the direction that helps the order expected to suffer. What sets the ending is two notes: the one left sounding, and the one that went just before it. Everything earlier has gone.

 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.

All themes