Acoustics & Music Theory
why it sounds like that
Theme

Somebody measured it

A surprising amount of this subject rests on a published measurement — of four drummers, of seventy-six speakers, of a room full of tuned pianos. Which somebody, on what material, and how far it generalises is part of the claim.
Key character in Werckmeister III. The twelve major keys in circle-of-fifths order, each with a bar as long as its major third is sharp of a pure 5:4. The bars run from 3.9 to 21.5 cents, so the keys genuinely differ. Pitch and tuning

Keys that had characters, and could be measured

Eighteenth-century writers described E flat major as devotional and F sharp major as harsh, and modern readers file it under synaesthesia. On the instruments those writers used, the difference between the two keys was seventeen cents of third, and that is a quantity.

 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.

 Maqam Rast, Arabic theory. Maqam Rast, Arabic theory: its degrees in cents above the tonic, drawn against the twelve equal steps of a keyboard. the third and the seventh sit halfway between major and minor — by convention, exactly halfway. Source: the quarter-tone convention fixed at the Cairo congress of 1932. Scales and modes

A scale is not a set of pitches

Two ragas can have identical pitch sets and be different ragas. Two national theories of one maqam put its third degree thirty-five cents apart. Both facts are fatal to the idea that a mode is a collection of notes, and both are ordinary in the traditions concerned.

 How much swing there is, against how fast the music goes. The ratio of the long note to the short note of a swung pair, as a function of tempo, derived from the finding that the short note holds a roughly constant hundred milliseconds. The notated readings are horizontal lines, and each is correct at exactly one tempo. Above about three hundred beats a minute the ratio reaches one and the swing has gone. Rhythm and metre

Swing is a ratio, and it is not two to one

Swung quavers are notated as a triplet figure, which is a ratio of two to one. Measurements of what drummers actually play give anything from three and a half to one down to one to one, and which of those it is depends almost entirely on the tempo.

 Where the beats actually fall. Measured timing deviations from a strict grid, in milliseconds, for three published profiles. The right-hand column converts each deviation into the note value it would have to be written as, at three tempi — and because a fixed number of milliseconds is a different fraction of the beat at every tempo, no single notated rhythm describes any of these. Rhythm and metre

The milliseconds that are the groove

A Viennese orchestra plays the second beat of a waltz about fifty milliseconds early, every bar. A jazz soloist sits thirty behind the ride cymbal. Neither deviation can be notated, and not because notation is coarse — because it measures the wrong quantity.

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

The piano is tuned wrong on purpose

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

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

A tuner counts beats, and that is the whole method

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

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

The pitch nobody agreed on, for four hundred years

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

 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.

 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.

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

A memory for the note itself, and it is dated

Absolute pitch is usually described as a rare perceptual gift. It is better described as a memory for a convention — and conventions have dates. Possessors trained on A=440 mis-name Baroque pitch by a semitone, their own labels drift sharp with age, and meanwhile most listeners without it start familiar songs within a semitone of the record.

 The probe-tone profile, major key. How well each of the twelve pitch classes was rated as fitting, after a context establishing the key — Krumhansl and Kessler, 1982. The shading is not part of the measurement: it is the tonic, the rest of the tonic triad, the rest of the scale and the remaining five notes, which are categories this subject had before anybody ran the experiment. The profile separates all four without overlap. Perception and the listener

Counting produced the hierarchy

Ask listeners how well each of the twelve notes fits after a passage in C major and the answers are not a smooth gradient. They fall into four groups with no overlap at all: the tonic, then the rest of the tonic triad, then the rest of the scale, then everything else — categories the subject had names for centuries before anybody ran the experiment.

 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.

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

A hammer is not an impulse

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

 How much bow force is allowed, and where. Schelleng's diagram. The lower bound is the least force that will trigger a slip on every pass of the corner and goes as one over beta squared; the upper bound is the most the string will take before it sticks for more than a period and goes as one over beta. At beta = 0.09 the usable range spans a factor of 9.0; at 0.03, near the bridge, it is 3.0, and at 0.2, over the fingerboard, 20.0. The window closes in proportion to beta, so the difficulty of playing near the bridge is a slope on this picture rather than a matter of opinion. Instruments and their design

How much bow is allowed

Too little force and the corner fails to trigger a slip on every pass; too much and the string sticks for more than a period. Both bounds depend on where the bow is, and they depend on it differently — one as the square of the distance from the bridge and one linearly — so the window between them closes in proportion as the bow approaches the bridge. Sul ponticello is difficult by a power law.

 A bell tuned to 294 Hz, and the note it is heard at. The partials of a well-tuned church bell, as ratios to the prime, with the three that imply the strike note marked. The nominal, twelfth and double octave sit at 2, 3 and 4, which is a harmonic series on 1 — so they imply a fundamental at 294 Hz, an octave below the loudest partial the bell has. The tierce at 1.2 is a MINOR third above the prime, which is why a bell has a minor quality by construction rather than by choice. Instruments and their design

A bell has no fundamental

The note a listener names when a church bell is struck is not any partial the bell has. Its nominal, twelfth and double octave sit at 2, 3 and 4 times the prime, which is a harmonic series on a pitch an octave below the loudest thing in the sound — and that pitch is supplied by the listener. Founders have been tuning it by ear since the fifteenth century.

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

The body is the filter

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

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

The other wolf

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

 Redundancy in bits, and why the number needs a length beside it. Left: the LZ78 cost of each scheme divided by the cost of sending the same symbols flat, against how many bars are sent, with each bar coded as its chord and key. Every scheme is above 1 at a single chorus — the coder loses — and every one falls under it as the piece runs. Right: new dictionary phrases per bar at one chorus, which is the statistic that survives at short lengths. Form and structure

How much of this is new

Repetition can be counted rather than looked at. Feed a piece's bars to a compressor and the bits it needs are a measure of how much of the piece is a repeat of an earlier part of itself. The measurement works, the number is real, and it turns out to be a statement about the description rather than about the music — which is the most useful thing it has to say.

 How surprising each chord is, in bits. Each step's information content, −log₂ of the probability the root-motion weights used here give it. a perfect cadence totals 6.4 bits over 3 steps; a deceptive cadence totals 7.3 bits over 3 steps; I – IV – V – vi totals 7.3 bits over 3 steps. The single most surprising move drawn is IV to V at 2.7 bits, which is 42 per cent of everything its passage spends. The eight weights are ordinal and stipulated rather than counted, so these are the numbers that ordering implies and not a measurement of any repertoire. Harmony and voice leading

The chord that did not come

A deceptive cadence is described as a surprise, and the explanation offered is that the wrong chord arrived. Measured against the tonal hierarchy already in use, the wrong chord is the second best-fitting triad in the key — and two of its three voices do exactly what they would have done in the right one. The surprise is not statistical. It is one voice, and it is the bass.

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

The room chooses the harmonic rhythm

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

 thirty-two-bar AABA, as a strip of time. thirty-two-bar AABA laid out one cell per bar, coloured by section, with the roman numeral in each bar. the A section's turnaround is the ii-V every variant keeps; the bridge is a chain of applied dominants. At 108 beats a minute in 4/4 the whole of it lasts 71 seconds. Cut into 4 repeat units of 8 bars, 3 pairs of units agree on more than 50 per cent of their bars. 2 of them are not identical, and 2 of those 2 differ in a run of bars ending at the last bar of the unit; the changed bars are marked in orange. Form and structure

Where a repeat is changed

Cut every scheme into its own repeat unit, compare each unit with every other, and ask where a repeat stops agreeing with what it repeats. The answer is that it stops at the end, in every case the corpus contains — and the number of cases the corpus contains depends entirely on where the threshold for "a repeat" is put. Moving it by nothing at all takes the count from two to twenty and the finding with it.

 What each rule costs, in semitones of extra motion. Each prohibition priced on I – ii – iii – IV as the difference between the cheapest realisation that obeys it and the cheapest realisation of all, averaged over every one of the 144 melodies the progression admits inside an octave. The dearest rule costs 1.27 semitones a melody and the cheapest costs nothing, so the bars run from 1.27 to zero; the notes beside them give the share of melodies that pay nothing, or that cannot obey the rule at all. Harmony and voice leading

What the rules cost

The prohibitions of counterpoint are constraints on a minimisation already computed here, so each one has a price in semitones of extra motion. Priced over every melody a progression admits, most of them turn out to be free, the dearest is not the famous one, and two of them cost no motion at all — they cost tunes.

 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.

 Two kinds of systematic timing, which share a word. Each measured profile split into a constant offset from the grid and a pattern that varies by position in the bar. Viennese waltz, second beat is −10.0 ms of offset and 33.4 ms of pattern; jazz soloist against the ride is 28.8 ms of offset and 1.9 ms of pattern; quantised is 0.0 ms of offset and 0.0 ms of pattern. A motor deviation anywhere in the 8 to 20 ms range published for skilled performers leaves 74–95% of Viennese waltz, second beat's variation systematic, 1–5% of jazz soloist against the ride's variation systematic. The two quantities are independent and no single deviation figure distinguishes them. Rhythm and metre

The deviations are not noise

Two published timing profiles, split into the quantities they actually carry. A jazz soloist thirty milliseconds behind the ride is almost pure offset and has no pattern at all. A Viennese second beat is almost pure pattern and has no offset. They are different things, they are reported under one word, and no figure of total deviation tells them apart.

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

Two players and no clock

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

 The price of a tonic. Every note of Dorian is given the same duration except its tonic, which is lengthened; the horizontal axis is the share of the total that goes to it. The key-finder answers with the parent key until 25.0 per cent of the time is spent on the modal tonic, and with D minor above it. At the left-hand edge every note has equal weight, which is the pitch-class set itself — and with every weight identical the correlation is not merely low but undefined, because a flat histogram has no variance to correlate with anything. Perception and the listener

What a tonic costs in seconds

The standard key-finding algorithm cannot be run on a pitch-class set at all — a flat histogram has no variance and the correlation is undefined. Give it durations and it answers with the parent key for all seven modes identically, and it takes between 15.8 and 30.0 per cent of the total time spent on one note before it names that note instead.

 Three answers to how finely a pitch can be heard. Three resolutions across five octaves, on a logarithmic scale of cents. Two notes one after the other are told apart at 4.0 cents at A440 and 8.6 cents three octaves down. Whether a melodic interval is in tune is a judgement an order of magnitude coarser, 25 to 50 cents. And two notes held a fifth apart are heard to beat once every 2 seconds at 1.31 cents, which is finer than either. The horizontal lines are the step sizes of the equal divisions that have been built: 12 at 100.0 cents, 24 at 50.0 cents, 53 at 22.6 cents, 72 at 16.7 cents. Every one of them is coarser than discrimination and finer than melodic judgement. Perception and the listener

Three answers to how finely a pitch can be heard

Two notes one after the other are told apart at about four cents at A440. Whether a melodic interval is in tune is a judgement an order of magnitude coarser. And two notes held together are heard to beat at a third of a cent, because the question is answered by counting rather than by hearing pitch at all. Every equal division ever built sits between the coarsest and the finest.

 Three ways to arrive at the same final tempo. Tempo against position in the closing passage, ending at 35 per cent of the opening tempo, for curvature exponents 1, 2, 3. All three begin and end at the same tempo, so what separates them is the middle: at the halfway point they read 68 per cent for linear in score position, 75 per cent for constant deceleration, 80 per cent for q = 3. The straight line is the one nobody plays. Measured ritardandos fit the decelerating curves, which is the whole of Kronman and Sundberg's argument: a closing gesture has the shape of a body stopping rather than of a dial being turned, and the parameter that varies between performances is the final tempo rather than the shape. Form and structure

An ending is a deceleration

Every performance slows down at the end and the slowing has a shape. Tempo read against score position is the velocity of a body stopping — a square root rather than a straight line — and the three candidate curves agree at both ends by construction, so the whole audible difference is in the middle, where they part by fifteen per cent of the passage's length.

 The boundary operator run over only what has been heard. Foote's checkerboard novelty on thirty-two-bar AABA at a kernel width of 4 bars, computed twice: once with the whole piece available, and once using only the bars heard up to and including each bar. The kernel reaches 4 bars forward, so every cell it needs has been heard 3 bars after its centre — the retrospective curve replotted 3 bars to the right lands on the causal one, and the operator turns out to be causal at a fixed delay rather than blind. The dashed verticals are the encoding's real section boundaries and are not an input. Form and structure

An ending that can be heard coming

Two measurements are both called hearing an ending coming and they point in opposite directions. By the halfway mark of an ordinary form almost nothing new arrives — and the cost of coding each bar has not fallen at all. Neither statistic says anything is about to stop, because no statistic over content can: predicting the next event well is not predicting that there will not be one.

 A wind instrument is a thermometer, and a string is not. Cents from the pitch at 20 degrees, against the temperature of the air inside a wind instrument and of a steel string, computed from the speed of sound as 343.2 metres a second times the square root of absolute temperature, and from a string's tension falling by Young's modulus times the expansion coefficient per degree. The wind slope is 2.95 cents a degree at 20 degrees, so 0.0 cents at 20, 11.7 cents at 24, 23.3 cents at 28, 34.7 cents at 32. The Pythagorean comma is reached at 28.1 degrees — 8.1 degrees of warming, which a wind instrument does from breath alone within a few minutes. The string goes the other way, 36 cents flat at 34 degrees, so the gap between the two sections opens at 5.4 cents a degree. Pitch and tuning

A wind instrument is a thermometer

Pitch goes as the square root of absolute temperature, so a warming clarinet sharpens by about three cents a degree and passes a Pythagorean comma after eight. The strings beside it go flat as they warm. Nobody chose either number, no temperament addresses either of them, and together they are twice the size of the discrepancy this whole field is named after.

 How much of the rule a walk with no rule reproduces. Post-skip reversal in 20,000-note random walks with no melodic knowledge of any kind. An unbounded walk reverses after 50.0 per cent of leaps, which is the chance rate and is the check that the measurement is right. Confining it to 12 semitones raises that to 61.3 per cent. Reaching the 70 per cent that corpus studies report needs a central tendency of 0.95 — an almost deterministic pull back toward the middle at the edges of the range. A wall is not enough; there has to be a spring. Form and structure

The leap that pays itself back

Every melody textbook teaches that a leap should be followed by a step in the opposite direction, and every corpus that has been counted agrees — around seven leaps in ten are answered that way. A random walk with two walls, no memory of the leap and no rule of any kind reverses after 61 per cent of them, and the residue is not a rule either. What is left when the walls are accounted for is a prediction the rule does not make, and it is the prediction that decides between them.

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

The note that is sharp because of where it goes

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

 The 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 this site measured describes only ensembles that hold their pitch still.

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

The series is not a chord

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

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

The note that moved by a minor third

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

 The same steps, counted by the clock. The step distribution of Ode to Joy and Twinkle, twinkle counted two ways: once per interval, which is what every earlier figure did, and once weighted by how long the note it leaves is held. The two disagree because a tune's long notes are not distributed evenly over its interval sizes — in Ode to Joy the 2-semitone step is 55.2 per cent of the moves and 60.0 per cent of the time. Which of the two a claim about melodic motion means has never been stated here, and the answer matters most exactly where a tune slows down, which is at the ends of its phrases. Rhythm and metre

The note that has a length

Every melodic figure so far reads a table where each note is a pair — a pitch and a duration — and throws the second number away. A step between two minims and a step between two quavers have been one event in every histogram it has drawn. Weighting the same statistics by time moves the step distribution by up to seven points, changes forty-four of a hundred and one contour signs, and turns up an off-by-one in the one figure that did use the durations: it took the length of the note arrived at where the time between two onsets is the length of the note left.

 What a contour costs to remember. A melody of n notes over 8 degrees carries 3 bits a note. Its contour carries fewer, and fewer than the number of distinct contours suggests, because the contours are not equally likely: at 6 notes there are 243 of them but the entropy is 6.59 bits, an effective alphabet of 96. Each further note adds 1.28 bits of contour against three of melody, so the shape keeps a stable 37 per cent of what is there however long the tune. Perception and the listener

The part of the tune that is kept

Contour survives transposition, retuning, a change of instrument and a doubling of every interval, and the usual explanation is that it is what a listener retains. That can be counted rather than assumed. A six-note melody over eight degrees carries eighteen bits; its contour carries 6.59 — not the 7.92 the number of distinct shapes suggests, because the shapes are wildly unequal — and the effective alphabet is ninety-six out of two hundred and forty-three. Each further note adds 1.28 bits of shape against three of melody, and at about nine notes a contour is specific enough to pick one tune out of a thousand.

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

The twelfth, and where it comes from

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

 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.

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

A standard is a specification

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

 Where the page ends a phrase, and where the ear does. Twinkle, twinkle with two sets of phrase boundaries on it. The lower curve is a local boundary detector — a peak in how much the interval and the note length change from one to the next, with nothing in it about bar lines or harmony — and the marks above it are where the notation puts the phrase ends. It finds 100 per cent of them and 2 boundaries the page does not have. Where the two agree it is because a long note is sitting at the join; where they disagree the page is marking a grammatical unit and the detector is finding a perceptual one. Form and structure

Where a phrase ends

Run a boundary detector over the three tunes used throughout and it agrees with the notated phrasing on one of them perfectly and on another almost not at all. The reason is which cue each tune uses: Twinkle's phrases all end on a long note, so a duration-weighted detector finds five of five with no false alarms; Ode to Joy's run on in crotchets and its phrasing is in the intervals, where a duration detector finds one of three and a pitch detector finds all three and eight others. No fixed weighting serves both, and the published one is worse on each tune than the single cue that tune uses.

 How much of the writing the prohibitions forbid. I – vi – ii – V – I in C major, written in 3, 4, 5, 6 parts, with every ensemble covering the same total compass. A triad has three pitch classes, so n parts double n − 3 of them and a duet cannot state one at all. Of every ordered pair of complete voicings of V and I, the share with no parallel fifth or octave between any pair of voices: 91.2% at 3, 75.0% at 4, 44.3% at 5, 17.9% at 6. Harmony and voice leading

Why the exercise is in four parts

Eight earlier essays move four voices, and nothing here ever chose four. Cut one choir's compass into three parts instead and the two most famous prohibitions in music cost exactly nothing — the cheapest realisation already obeys them. Cut it into six and they cost more than half a semitone per voice per chord change, because a parallel octave needs a doubled note and six voices sharing three notes can hardly avoid one. Four is the smallest number of parts at which the rules have a price at all, and it is the largest at which the price is small.

 How many modes a set has, and why some have fewer than notes. Every non-empty subset of the twelve — 4095 of them — sorted by size, with how many have a transposition that returns the same set. 75 do, which is 1.8 per cent, and they reduce to 16 distinct step patterns. A set of size k with a symmetry of order s has exactly k/s distinct rotations, so the number of modes is arithmetic rather than musical. Sets of five, seven and eleven notes have none at all, because those sizes share no factor with twelve — which is why every seven-note scale has seven modes before any musical question is asked. Scales and modes

The set with fewer modes than notes

Eight earlier essays rotate one collection, and the seven modes were never a result — they are arithmetic. A set has as many modes as it has notes divided by the order of its own transposition symmetry, so the whole-tone scale has six notes and one mode, the octatonic has eight and two, and the question asked just before — how much has to be heard before the mode is settled — is not merely hard for those collections but undefined. Seventy-five of the twelve-note universe's 4,095 subsets are built this way and they reduce to sixteen step patterns. Five, seven and eleven notes cannot be among them, which is why every seven-note scale has seven modes before anybody plays one.

 What survives a change of encoding: verse and chorus. The same 32 bars of verse and chorus under five encodings, scored on the three things measured here measures. Mean off-diagonal similarity says how alike the piece looks to the arithmetic. Recall and precision are the novelty operator's boundaries against the 3 the section plan has, at a kernel of four bars. The period is the strongest peak of the lag profile, in bars. Under the bag of pitch classes every other figure uses, the piece is 90 per cent self-similar and the operator finds 0 per cent of the boundaries; under how far the root moved it finds 100 per cent. The period is the quantity that does not move. Form and structure

The repeat that is not in the notes

Eight earlier essays compare bars by writing each one as a bag of pitch classes and taking a cosine. Nothing chose that encoding — the first used it and the other seven inherited it. Encode the same six schemes four other ways and one of the three findings survives untouched, one survives with different numbers, and one turns out to have been a statement about the encoding all along: the boundary operator finds none of the section edges in three schemes as a bag of pitch classes and every one of them as tonic, subdominant and dominant.

 note length against the onsets. Every candidate metre's fit to a 16-step pattern with 4 onsets, plotted against how strongly note length is weighted. At a gain of zero the scoring is the onset-only one every earlier model used, and the winner is step 1 and step 4. At a gain of 0.05 the answer becomes step 4. 2 candidates are exactly flat — step 2 and step 3 have no onset on any strong position, so there is no credit for the cue to multiply and no weighting of it can move the line. Form and structure

What the onsets left out

Eight essays induce a metre from a list of ones and zeros, and every failure they recorded was argued about as a failure of the rules. Two of the three are not. Note length is already in that list and the scoring throws it away: put it back and the son clave's two-way tie resolves to the notated downbeat. But the groove with its beat removed cannot be repaired by any cue at any strength, and the reason is arithmetic rather than empirical — the true phase has no onset on any of its strong positions, so there is no credit for a cue to multiply and its line is exactly flat.

 A model that cannot say two keys does not say it is unsure. I – IV – V – I played in C major and in a second major key at the same time, with the second key moved round the circle of fifths. For each separation: the correlation the standard key-finder gives its single best answer, and the correlation reached by the best PAIR of key profiles — a hypothesis the finder does not have. The pair recovers both keys that are sounding at every separation, 7 of 7. The single answer names neither of them at 4 of the 7, and its confidence does not fall when it is wrong: at four steps apart it reports E minor at r = 0.886, against 0.959 for the same progression in one key. Harmony and voice leading

Two keys at once

Every key figure so far assumes one key is sounding, and the standard key-finder has no value it can return that means two. Play one progression in C and the same progression a major third away at the same time, and the model does not report uncertainty: it reports E minor, at a correlation of 0.886, against 0.959 for the same progression in one key. It is as confident as it ever is, and neither key it names is being played. Give it the missing hypothesis — pairs of key profiles rather than single ones — and it recovers both keys at every separation, all seven of seven.

 How many notes an octave can hold, asked twice. The share of trials on which a category is named correctly, against how many equal categories the octave is cut into, for a listener whose internal estimate carries 11 cents of noise — the logistic scale this site's identification figures already use, which is the thirty-cent transition the studies report. At 95 per cent accuracy the ceiling is 6 categories, and seven scores 94.9 per cent — on the line. The other ceiling is resolution: 151 to 356 difference limens fit in an octave depending on register, which is a factor of forty larger. Every system marked below sits between the two, and the marks separate: the number of degrees a mode uses clears the criterion, and the size of the gamut it chooses them from does not. Intervals and chords

How many boxes an octave holds

An identification model of pitch is usually handed twelve categories, and nothing ever asked how many an octave can hold. There are two answers and they are a factor of forty apart. Resolution allows between 151 and 356 — a listener can tell that many pitches apart in a direct comparison. Naming one of them without a comparison is a different faculty and it runs out at six or seven, which is where every mode in every tradition compared here sits. Turkish theory names fifty-three commas to the octave and a makam uses seven of them, and the gap between those two numbers is the whole of the argument.

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

The chord that is not played at once

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

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

Three strings, and the note that comes back

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

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

The hammer is not a point either

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

 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.

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

The note that gets duller as it dies

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

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

An instrument is not one timbre

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

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

Blowing harder is playing sharper

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

 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.

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

Which instrument is underneath

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

 How finely a fifth can be heard, against how fast it goes by. The smallest audible mistuning of a melodic fifth above 440 hertz, against how long each of its two notes lasts. The lower solid line is one note's own limen; the upper one is the interval's, larger because two independent errors add in quadrature. A tenth of a second gives 23.5 cents against the note's own 19.6, the floor for long notes is 5.4, and the syntonic comma is not cleared until each note lasts 111 milliseconds. The dotted line at 1.31 cents is the same interval heard as a simultaneity, where partials 3 and 2 coincide at 1320 hertz and one beat every 2 seconds can be counted there. Intervals and chords

An interval is two errors

Two notes in succession are two pitch estimates, and what a listener judges is their difference — so a melodic interval is heard less finely than either of the notes in it. At a semiquaver the limen is nineteen cents, which lands inside a published range long quoted from the literature and never derived. Sounded together instead of one after the other, the same interval is judged eighteen times more finely.

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

The pair tuned apart on purpose

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

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

The hammer that is heavier than its string

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

 How fast two keys can alternate before the finder stops following. The share of bars a moving key-finder names correctly, once its reading is shifted back by its own lag, against how many bars each key holds for. One line per window. Below a block of three bars the second key is never named at all — 2 of the sweep's readings report a single key for the whole passage — and above about twice the window the tracking is over ninety per cent. The lag itself is about half the window: 0 bars at a window of 3, 0 bars at a window of 4, 3 bars at a window of 8. Harmony and voice leading

The alternation a key-finder cannot follow

Two keys sounding together are not in the key-finder's vocabulary, and an earlier essay ended by pointing at the other case and saying what was missing: a passage whose alternation rate can be varied while everything else is held still. Built, it gives a rule with three numbers in it — the second key is never named below a block of three bars, the tracking clears ninety per cent above twice the window, and the reading is late by half the window throughout.

 What the joint search changes, and what it never changes. Over 552 constructed passages of eight slots with rests, how often the joint reading differs from the pipeline's. The chord differs in 29 per cent and the barline in 31, with both differing in 20. The key differs in 0 per cent — never — because the key is read from a pitch-class histogram, which does not know where the bar starts or which notes are chord tones. Two of the three decisions are entangled and the third is not. Harmony and voice leading

Three decisions that constrain each other

Every model here decides one thing at a time — the key from the pitch classes, the metre from the onsets, the chords from the metre — and an earlier essay ended by saying a listener does all three at once. Resolving them jointly costs a hundred and fifty-seven times the search and changes the reading of two passages in five. It never once changes the key, and the reason it cannot is the reason the whole account is built the way it is.

 5 onsets in 16, by evenness against locatability. Every one of the 273 rotation classes of 5 onsets in 16 steps, placed by how uneven its gaps are against how much of the cycle has to be heard on average before its position is known. The two objectives are opposed — the perfectly even pattern is the hardest to locate and the most clustered ones are the easiest — so there is no best pattern, only a frontier, and 22 of the 273 are on it. son clave and the bossa-nova pattern are among them. The named timelines are marked. Rhythm and metre

What the clave buys with its unevenness

The essay before this one found that the two best-known timelines in the world are not Euclidean at any rotation, and asked whether they maximise something else. They do: how quickly a fragment of the cycle says where in the cycle it is. The son clave locates itself in nine of its sixteen steps where the even pattern needs fifteen — and of two hundred and seventy-three patterns, twenty-two are on the frontier between the two objectives and the son clave is one of them.

 A duration category has a tempo range of its own. Each simple ratio's short note is the beat divided by one more than the ratio, so at a high enough tempo it falls under the fastest interval that can be a beat at all — 100 milliseconds. Each bar here runs from the slowest tempo at which the ratio's long note still belongs to a beat to the fastest at which its short note is still a note: 1:1 ends at 300 bpm, 2:1 ends at 200 bpm, 3:1 ends at 150 bpm, 4:1 ends at 120 bpm. The line is this site's swing curve, and where it crosses a category boundary the category it is leaving has already ceased to exist. Rhythm and metre

The short note is sitting on the floor

Swing is modelled here as a short note of constant absolute length, which was measured from drummers and left as a fitted parameter. The tempo window's fast edge — the shortest interval a series of events can be a beat at — was measured from listeners tapping. Both are a hundred milliseconds, and if that is not a coincidence then the swing ratio has no free parameter in it at all: the short note is not held constant, it is resting on the floor.

 The boundaries that survive each amount of smoothing. The local boundary strengths of Twinkle, twinkle read at every scale: the curve is smoothed with a Gaussian of the width on the horizontal axis and the peaks that survive are counted. Small scales give 11 boundaries and large ones give one, and the notation marks 5. The level with that many falls at a width of 2, where the model finds 100 per cent of the notated boundaries and 100 per cent of what it finds is notated — a comparison with no threshold in it, which is what the scale parameter buys. Form and structure

A boundary at a stated level

A boundary detector run over three tunes agreed with the notation on one and barely at all on another, and left two things owing: a version with a scale parameter, and a version run on performance timings. Both are paid here, and they pay differently — the scale removes a free parameter from the comparison and does not rescue the hard case, while two per cent of rubato does.

 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.

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

The players who have to be early

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

 The distribution that can be measured is not the one the player has. A player aiming at a short note of 100 milliseconds with a standard deviation of 15, against a floor at 100. The pale curve is what the player is doing and the heavy one is what can be recorded, because the 50 per cent of the parent below the floor arrives at the floor instead. The measured mean is 112.0 milliseconds rather than 100 and the measured standard deviation is 9.0 rather than 15. At 160 beats per minute that turns an intended swing ratio of 2.75 into a measured 2.35. Rhythm and metre

A quantity resting against a wall

The short note of a swung pair was found sitting exactly on the fast edge of the tempo window. If that edge is a floor rather than a fitted number, then every swing statistic computed until now was computed on a censored sample — and a censored sample has a mean that is 0.80 standard deviations too high, a spread that is 40 per cent too low, and a correction gain that can come out twice what the players actually have.

 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.

 How much correlation it would take to matter. The limen of a 7-semitone interval at a note length of 0.25 seconds, against the correlation between the two notes' errors. The independent model at the left gives 9.44 cents. Halving that needs a correlation of 0.75; a fifth off it needs 0.31. The curve is √(1 − ρ) and nothing else, so the correlation required for a stated improvement is arithmetic — which turns the question from “does a key help?” into “by how much, and here is the number it must reach”. Intervals and chords

How much an anchor would have to be worth

Two pitch errors added in quadrature assume an independence nobody measured — a listener inside a key hears a note as a scale degree, and a shared reference is exactly a correlated error. Turning the dial is not evidence. What is evidence is that the dial is not free: a shared error cancels out of a difference completely, so a listener's single-note limen and their interval limen give the two components with nothing left over, and halving the interval limen needs a correlation of exactly 0.75.

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

The same note is a different width

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

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

The pulse that was assumed

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

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

What the climb was a search for

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

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

The hand goes in, and the note jumps

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

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

Two dials the player turns together

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

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

The corner does not come back a corner

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

 How far a wall moves the mean, in spreads. The bias a floor puts into an observed mean, in units of the parent's own spread, against how far the parent sits above the floor. With the mean exactly on the wall the truncated bias is 0.797 spreads and the censored bias is 0.399 — half of it, exactly, because half the mass sits at one point and the other half is an upper half-normal. The earlier essay used the upper figure for a process that produces the lower one, so every bias it quoted is twice what a floor on execution actually causes. Rhythm and metre

The shape a wall leaves behind

A floor on execution biases every statistic computed on swing timing, and the bias was priced with Pearson's truncated-normal formulas. Those are the formulas for a sample with everything below the wall thrown away. A player who cannot execute a short gap does not throw the attempt away — it comes out at the floor. That is a censored sample, its bias is exactly half, and its skew is two thirds larger.

 What the notes in between do to the anchor the interval is measured against. How finely a 7-semitone interval can be judged when its two notes are separated by other notes rather than by silence, under the two published accounts. Confirming material restates the key and refreshes the shared reference, so the correlation climbs from 0.5 toward a ceiling and the limen falls to 4.81 cents. Overwriting material competes for the same memory, so the correlation decays to 0.04 and the limen rises to 9.28. By 8 notes the two accounts differ by 4.5 cents, which is 47 per cent of the limen with no anchor at all — and no experiment here distinguishes them. Intervals and chords

The notes in between

Every figure until now is about two notes with nothing between them, and a melody is notes with other notes between them. Two published accounts of what the intervening material does predict opposite signs — one says the key is restated and the shared reference is refreshed, the other says each note competes for the same memory and it decays. By eight notes they differ by four and a half cents, which is nearly half the limen the interval would have with no anchor at all.

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

It was never the strings that stopped the climb

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

 What the engraver used here gives a note, measured off the page. The horizontal distance VexFlow allots each duration, read back off a formatted system rather than quoted from a manual. It is not a power of the duration: it is a constant of 58 points plus 15 points a crotchet, and the constant is 93 per cent of the width the shortest note here gets. A note four times as long as another is about 2 times as wide, not four. The floor is the notehead, its stem, its accidental and the space a reader needs to see them as separate events — which is a claim about legibility and not about time at all. Scales and modes

The axis that is not a time axis

Eight earlier essays have measured the staff's vertical axis to a position. Its horizontal one has never been asked about, and the answer is that it is proportional to nothing: measured off the typesetter used here, a note gets 58 points before its duration is considered at all and 15 points a crotchet after — so the constant is 93 per cent of what the shortest note gets, and a note four times as long is not four times as wide.

 Every resolution claim here, against the number it rests on. How many equal steps of the octave can be named at 95 per cent accuracy, against the internal noise the model gives a listener. The laboratory value this collection quotes everywhere is 11 cents, which gives 6 nameable categories — the "about seven per octave" every claim here has been repeating. The laboratory measures it on isolated intervals and music never presents one, so the effective value inside a piece is smaller by an amount nobody has measured: at 4 cents it is 18, which is the chromatic scale, and the conclusion changes from "the ear has fewer boxes than the notation" to "it has exactly as many". Nothing here measures it. This is what it is worth if it moves. Scales and modes

The number every claim here has been quoting

Sigma is the internal noise a listener's pitch judgements carry, it is measured in a laboratory on isolated intervals, and music never presents an isolated interval. Every resolution claim here rests on the laboratory value. Sweep it and the headline finding moves: at eleven cents the ear has about six nameable categories per octave, and at six it has twelve — which is the chromatic scale, and turns 'the ear has fewer boxes than the notation' into 'it has exactly as many'.

 The one number the ordered key-finder was tuned on. For each rate of alternation between two keys, the cost of changing key at which the model stops hearing two keys and starts hearing borrowed chords in one. The threshold rises with the period — 0.95 at 1 bar, 0.95 at 2 bars, 0.95 at 4 bars, 2.00 at 8 bars, 3.50 at 16 bars — so the parameter and the rate trade off against each other exactly. The value tuned earlier, 2.2, sits above every threshold on this axis, which means its verdict about fast alternation was a consequence of the tuning rather than a finding about the music. Filled means the model names two keys; hollow means it names one and calls the rest borrowings. Scales and modes

A modulation and a borrowing are one number apart

The key-finder that keeps the order has one tuned parameter, and said so. Sweep it and the parameter turns out to be the whole verdict: below a threshold the model hears two keys alternating, above it one key with borrowed chords. The threshold rises with how slowly the keys alternate — and the value it chose sits above every threshold in range, so its finding about fast alternation was a consequence of the tuning.

 Two kinds of evidence, each measured against its own chance. Every key as a point: across, how many standard deviations its cadence count is above what the same bars resampled would give; up, the same for its pitch-class profile correlation. The winner on cadences is key C at 6.2 standard deviations and on profile C at 1.1, and they agree. Standard deviations above chance are the same unit whatever produced them, which is the commensuration this figure is for — and it costs something: it assumes the two chances are equally interesting, which is a weighting in disguise. The obvious null does not work at all for one of the two: shuffling the bars leaves a pitch-class histogram exactly as it was, so its spread is zero and every key scores nothing against it. Harmony and voice leading

A count and a correlation

Cadence evidence is a count of ordered pairs and profile evidence is a correlation with a template, and the cadence essay refused to total them because they are not in the same units. Score each against its own chance and they are — standard deviations above chance are the same unit whatever produced them. Then the trouble moves: the obvious null does nothing at all to one of the two, because shuffling the bars leaves a pitch-class histogram exactly as it was.

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

Who plays what and how loud is one question

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

 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.

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

The cutoff a maker can actually measure

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

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

Four terms, and only one of them binds

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

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

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

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

 The same interval, started on each of the twelve. An interval of 7 semitones started on each pitch class of a major key, against how strongly the key specifies its two notes — the mean of the probe-tone profile at each. The interval account says the listener encodes a distance, so the key cannot enter and the prediction is a horizontal line at 5.4 cents. The degree account says the listener refers each note to the key, so its precision on a note falls as the key's specification of that note weakens; scaled to agree at the most stable start, it rises from 5.4 cents on C to 8.5 on E♭. Every earlier figure measures a quantity the second account says is not being formed at all. Intervals and chords

The quantity a rival account says is not there

Two earlier essays measure how much two notes' errors are correlated through a shared anchor, and price what that correlation would be worth. There is a rival account in which a listener refers each note to a key and never forms the distance at all — under which the correlation is not small, it is a description of something that is not happening. The two accounts agree on almost everything and disagree on one manipulation, and the manipulation costs an afternoon.

 The tempo turns, and almost nothing moves. The earlier arrival reading — what is sounding at the final chord over what the listener has been hearing — swept over bar lengths from 0.5 to 5 seconds, which is 480 down to 48 beats a minute, at 3 closing lengths. Every curve is nearly flat. Across a tenfold change of tempo one gesture's reading moves by a factor of 1.201 and the other's by 1.098, while the gap between the two gestures — which is what that essay was measuring — is 1.228. The expectation was that the tempo would decide the answer, on the grounds that a two-second bar against a two-second release is a comparable pair. The premise is wrong in a way the sweep makes obvious: the thing being compared with the release is not a bar, it is the WHOLE ENDING, which is 2 to 8 bars long and is therefore far longer than the release at every tempo anybody plays. The running impression has caught up with the closing texture before the final chord arrives, at 0.5 seconds a bar and at 5, and what is left is the last bar's own jump. Form and structure

The parameter that did not decide the answer

An earlier essay on closure ended by naming the tempo as the thing every number in it was resting on, and said it was the kind of parameter that had caused trouble before by turning out to decide the answer. Turned across a tenfold range at a closing gesture of fixed length it moves the reading by four per cent, against a twenty-three per cent gap between the gestures it is distinguishing. The parameter beside it in the same figure — how many bars the gesture occupies — moves it by twenty, and nobody had named that one at all.

 How far the detector looks, note by note. The number of notes that fit inside a 3.5-second present at each point of the tune, once the performance has lengthened its phrase-final notes by 30 per cent. It runs from 5 to 10 notes against a constant 7 for the unperformed version, and it dips exactly where a boundary is, because a boundary is where the performance slows. Reading the boundary-strength curve with that width at every point instead of one width everywhere gives an agreement of 0.55 with the notated phrasing, against 0.36 for the fixed width the present dictates and 0.71 for a fixed width fitted to this tune. The dips are marked, and the notated boundaries are the vertical lines: the detector narrows itself at the places it is supposed to find, which is the circularity this figure has to be honest about — the lengthening was put there by the notation. Form and structure

A detector whose resolution the performance sets

The boundary detector lost its free parameter when the psychological present became a number of notes at a stated tempo, and what that held still was named at the time: a performance slows into a phrase end, so the number of notes inside the present is not the same everywhere in a tune — it falls exactly where a boundary is. Making the width follow the performance recovers half of what removing the parameter cost, and honestly leaves the other half.

 The surprise of each chord, against the uncertainty it arrived into. The information content of each step — minus the log of its probability under a distribution that multiplies the root-motion weight by how well the destination triad's notes fit the key — with the entropy of the moment before it drawn behind. I – IV – V – I: I→IV 1.71 bits, IV→V 2.80 bits, V→I 1.60 bits, against a mean uncertainty of 2.63; I – IV – V – vi: I→IV 1.71 bits, IV→V 2.80 bits, V→vi 2.61 bits, against a mean uncertainty of 2.63. A surprise larger than the entropy it arrived into is an outcome the model was not expecting even given how uncertain it was; one below it is an outcome the model had already mostly bet on. An earlier essay produced the first of those numbers and had no way to produce the second, because a set of preferences is not a distribution and only a distribution has an entropy. Harmony and voice leading

A chord, given a key and a predecessor

A chord's improbability has been priced from its root motion alone, which left one multiplication unmade: a chord is also improbable because its notes do not fit the key, and that number has been available since the probe-tone profile. Multiplied and renormalised, the two give a conditional distribution — and a distribution has an entropy, which is the quantity a surprise has to be read against and which a list of preferences cannot supply.

 The passage built so the two statistics disagree. A body of chords diatonic to C major, of growing length, ended by a ii–V–I in G. The bag of notes says one key and the ordered pair says the other, which is the case the earlier figures never contained. The line is the cadence evidence for G, and under the axis is what each reading actually names at each length. The cadence reading holds G up to a body of 6 bars and is overturned at 8, so one explicit cadence is worth about that many bars of profile evidence — and it is overturned by the body's OWN incidental root motions rather than by the histogram at all. That is the finding the earlier essay could not have: on real material the two statistics cannot be varied independently, because lengthening the profile evidence adds cadence evidence too, for a third key. Harmony and voice leading

The passage built to make them disagree

A cadence count and a key-profile correlation have been put on one scale and run on a passage where the two agree, which tells nobody anything. Building one where they disagree — the notes of one key and the cadences of another — measures the exchange rate at about six bars per cadence, and finds something the agreement case concealed: the two statistics cannot be varied independently, because lengthening the profile evidence adds cadence evidence too, for a third key.

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

Which notes have to be played early

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

 How sure the reading is, bar by bar. Every earlier essay reports one best reading. A dynamic program that finds a best path has, by construction, the best score into every state at every bar — so the gap between the best reading and the best reading in any other key is already computed and has never been printed. Here it is, in bits, for the thirty-two-bar AABA. The mean margin is 1.14 bits and 13 of 32 bars are inside one bit of a rival reading, which is where a listener would be genuinely undecided. The reading itself names C, E, B, D, G; the margin says what that naming is worth, and at the weakest bar — bar 31, C over F — it is worth 0.07. Scales and modes

The margin the dynamic program already had

Nine earlier essays produce a single best reading, and the passages worth arguing about are the ones where two readings are nearly equally good. What is needed for that has been inside the model from early on: a dynamic program that finds a best path has, by construction, the best score into every state at every bar — so the gap between the best reading and the best reading in any other key is already computed, and printing it turns every analysis here into a measurement of ambiguity.

 How much music each notation fits on a page. The two axes multiplied. Vertically, a system is as tall as the staves the range needs; horizontally, a system holds as many notes as fit once the shortest is wide enough to read. the staff, seven to the octave: 3 staves to the system, 4 systems and 38 notes to a system, 152 notes to the page — 46 seconds at 100 beats a minute, so a page turn every 46 seconds; a chromatic staff, twelve to the octave: 4 staves to the system, 3 systems and 38 notes to a system, 114 notes to the page — 34 seconds at 100 beats a minute, so a page turn every 34 seconds; a whole-tone staff, six to the octave: 2 staves to the system, 7 systems and 38 notes to a system, 266 notes to the page — 80 seconds at 100 beats a minute, so a page turn every 80 seconds. a whole-tone staff, six to the octave holds 2.33 times what a chromatic staff, twelve to the octave does, which is a difference of 1.00 page turns a minute — and a page turn is a thing a player with two hands occupied cannot do. Scales and modes

How much music a page holds

Nine earlier essays have measured notations, and every one of them is a page — a two-dimensional object read in a fixed order by a reader who has to turn it. One measured the vertical axis and another the horizontal, and multiplying them gives the one design constraint on notation that is not about legibility at all: a chromatic staff turns pages a third more often than an ordinary one, and a proportional spacing rule turns them nearly twice as often as a columnar one.

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

A note takes a number of periods to speak

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

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

The higher note speaks sooner and takes longer

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

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

A family resemblance in the heights

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

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

The crossing belongs to the felt

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

 A woodwind cannot be pulled to a new standard. Every earlier figure computes strings, because a string's tension and gauge give a closed-form scaling law. A wind instrument does not have one, and this is why. To move from 440 to 415 hertz the player pulls out 11.7 millimetres at the joint, which lengthens the sounding tube of every fingering by the same absolute amount — so the interval each note drops is a fixed length against a shrinking one, exactly the shape of an end correction. The tuning note lands where it should and nothing else does: D3 is 67 cents sharp of where it belongs and G5 is 75 flat, a spread of 142 cents across the compass. A rebuilt instrument has no such problem — scale every length by one ratio and every mode moves by the same interval, and the tone-hole lattice cutoff moves with it, from 1766 hertz to 1666, which is 101 cents and therefore the same instrument transposed. That is the difference between an afternoon and a year. Pitch and tuning

A woodwind cannot be pulled to a new standard

Every earlier essay computes strings, because a string has a closed-form scaling law. A wind instrument does not. Pulling out at the joint lengthens every fingering's sounding tube by the same number of millimetres, which is a fixed length against a shrinking one — so the tuning note lands and the compass spreads by 142 cents. The strings could be regauged in an afternoon; the winds had to be rebuilt.

 How far out of tune an interval has to be before its beat is usable. An earlier essay produced a modulation index for every interval on a stated timbre; whether a fluctuation of that index at that rate can be detected is a published function of both. Running one against the other turns a table of decibels into a window in cents. The bar is the mistuning over which the beat is both deep enough to notice and at a rate a tuner can use — not so slow that a beat takes half a minute to complete, not so fast that it has stopped being a beat. the major third gives the widest window, 1.3 to 60.0 cents, and the minor sixth the narrowest, 0.8 to 38.9. The ordering is the opposite of the dip's: the octave has the shallowest dip on a string spectrum and the widest usable window, because its coincidence sits at a low partial and a given mistuning therefore produces a slower beat. Depth and rate pull opposite ways, and it is the rate that decides. Intervals and chords

The beat a tuner can actually use

There is a modulation index for every interval on every timbre, and the published threshold for detecting a fluctuation is a function of exactly that and its rate. Running one against the other turns a table of decibels into a window in cents — and reverses the ordering, because the interval with the shallowest dip has the widest window.

 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.

 Expectation as a curve, and what a change costs where it lands. Every quantity so far is attached to a chord change: a list of surprises, one per event. A listener's expectation is continuous — it sharpens through a bar and collapses when the change arrives — and the two ingredients for it are already here, the harmonic rhythm and the metrical beat weights. The curve is the hazard: given that the chord has not changed yet, the chance that it changes on this beat. It runs from 0.043 on the weakest beat to 0.290 on the downbeat, a ratio of 6.7, against 0.125 if every beat were alike. The marked beats are where the changes actually arrive, and their timing bill is 3.6 bits against 6.0 for a listener with no metre — so these changes are 1.7 times cheaper to expect than a metreless listener would find them. That term is new: an earlier essay prices which chord arrived and this prices when, and a listener meets the sum. Harmony and voice leading

Expectation is a curve, not a list

Every quantity so far is attached to a chord change: a list of surprises, one per event. A listener's expectation is continuous, sharpening through a bar and collapsing when the change arrives — and the two ingredients for it were already here, in two other accounts. What comes out is a second surprise, for when a chord arrives rather than for which one it is.

 A ritardando does not spend the diminuendo. The arrival reading under a deceleration into the ending, from no ritardando at all to a final tempo 30 per cent of the starting one — which stretches the closing bars from 12.0 seconds to 20.2. The expectation was that it would matter: a ritardando lengthens exactly the bars the gesture is happening in, so a diminuendo that would have been absorbed at a steady tempo gets more of the smoother's own time to be absorbed in. It moves the reading by 0.00 per cent. Every line here is flat to within the thickness of the line, which is the second time a tempo parameter has been swept here and found to do nothing. Form and structure

The reading was a step response

Sweeping the tempo found it did not decide the answer. This one sweeps the deceleration across a factor of three and finds a null to five figures, and then sweeps the length of the closing gesture across a factor of forty-eight and finds it moves the reading by eight per cent — but not as a function of seconds. Sorted by seconds the twelve runs scatter; sorted by how many bars the instruction covers they fall into three tight groups. One sentence explains the null and the not-null together.

 An ensemble finding an asynchrony nobody told it about. An earlier essay produced a map of required leads — which notes of a scoring have to be played early, and by how much — and nothing tells the players those numbers, because they are a property of the instruments' attacks rather than of the music. So an ensemble has to find them, and the mechanism is already here: each player hears sounds rather than onsets and moves their next onset toward the mean of the others'. The spread of arrival times starts at 20 milliseconds and settles at 4, crossing 5 milliseconds after 5 beats — about 1.3 bars of four. The leads it converges on match that map to within 0.2 milliseconds, which is what makes this a convergence rather than a coincidence: the fixed point of players listening to each other is every player leading by their own attack. Rhythm and metre

How many bars an ensemble needs

The map of required leads is something nobody tells the players, because the leads are a property of the instruments' attacks. So an ensemble has to find them, and the mechanism is the one the microtiming essays describe: each player hears sounds rather than onsets and moves toward the others. It converges on the map to within a fifth of a millisecond, in five beats, and there is a best correction gain.

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

Three exciters and three wires

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

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

A note starts twice

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

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

A beat is never one beat

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

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

The passage that separates two players

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

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

The mouth that decides nothing

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

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

The partials the tube makes itself

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

 Five rulers on 5 onsets in 16, and what each frontier keeps. The same 273 rotation classes and the same locating cost, with the frontier recomputed against five different definitions of unevenness. The frontier's size varies from 7 to 22 of 273, so how exclusive membership is depends on the ruler. Of the named timelines of this size, son clave is on 4 of the five, rumba clave is on 0 of the five, the bossa-nova pattern is on 4 of the five. The four rulers that measure how large a pattern's departure from even is rank the whole census together, at no worse than 0.89 between any two of them; the one that counts how many distinct gap lengths a pattern uses rather than how large its departures are is anti-correlated with them at -0.26, and its frontier holds none of the named timelines and does not hold the perfectly even pattern. Rhythm and metre

The frontier and the ruler

A Pareto frontier is a claim about two quantities and only one of them was measured. The locating cost is exact; the unevenness it was traded against was one choice among several. Under four different rulers the son clave stays on the frontier and the frontier shrinks from twenty-two of 273 patterns to seven, so its membership means more rather than less — but the bembe stops being dominated, the two claves stop being equally uneven, and one plausible measure destroys the result entirely.

 The bias a wall puts in a mean, against the parent it came from. The bias a wall puts into an observed mean, in spreads, for each of eight standardised parents, with the wall exactly on the parent's mean. The censored values run from 0.354 to 0.433, a range of 0.079; the truncated from 0.582 to 1.000, a range of 0.418. The formula now used is the one that hardly depends on a distribution nobody has measured, and the formula it replaced is the one that depends on it a great deal. Rhythm and metre

The parent nobody measured

Every figure drawn behind a wall so far assumed a normal parent, and the assumption turns out to matter in exactly the wrong place. The censored bias is half the parent's mean absolute deviation — a theorem, not a coincidence — so it lands between 0.35 and 0.43 spreads for every distribution tried, and the truncated one runs from 0.58 to 1.00. But the third moment proposed as the test moves 2.9 across parents against 0.65 between the two rules, and a censored sample from a slightly left-skewed parent has a skew of 1.007 where a truncated normal has 0.995. The statistic that does work is a count of ties.

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

The hair runs the wrong way

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

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

What the rosin is worth

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

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

What the second register is for

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

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

What the tongue actually removes

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

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

Read at two different heights

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

 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.

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

The fourth player is a spectrum, not a decision

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

 Which bars the key is decided by, and which bars it is believed on. Every bar of a 32-bar scheme removed in turn, with what its absence costs. The column is how far the passage's mean margin falls without that bar — how much of the model's certainty it supplies. The dot is how many bars are then read as a different key — how much of the answer it supplies. The 18 bars an analysis would point at — a section opening or closing, a dominant, the chord a dominant resolves to — average 0.079 bits of certainty and 0.50 bars moved; the 14 ordinary bars average 0.047 and 0.07. So the structural bars carry 1.7 times as much certainty as the ordinary ones, and 7.0 times of the answer. Those are different quantities, and the second is the one an analysis is about: an ordinary bar can carry a great deal of a passage's certainty and none of its reading. Harmony and voice leading

The bars a key is made of

A discount that treats every bar alike is the wrong shape for a memory, so take each bar away in turn and see what it was worth. The bars an analysis points at carry seven times as much of the answer as the ordinary ones and slightly less of the certainty — and a memory built only of them reads the passage worse than a memory with no structure in it at all.

 The twelve keys a key-finder has never had. Every scheme the key-finder reads, read twice: over the twelve major collections the model has always used, and over twenty-four with the harmonic minor added. The pale bar is the first and the dark one the second. On 5 of the 6 the extra twelve states change nothing a reader would see — the largest loss of certainty is 3.2 per cent, on the rondo — and no bar of any of them is renamed. The exception is the ostinato, every bar of which the twelve-collection model calls E♭ and the twenty-four-collection model calls C minor — the same seven notes, the right name. So the missing states were not costing the key-finder its answers. What they were costing is the ability to say which of a collection's seven degrees is home, and that is a different repair. Harmony and voice leading

The key-finder with no tonic

Thirteen essays of key-finding have run over twelve major collections and not twenty-four keys, so a passage in A minor is read as C. Adding the missing twelve costs almost nothing and fixes almost nothing — because the model has no tonic in it at all, and below three raised sevenths in a passage the extra states are worth exactly zero.

 A long note and a strong note disagree, and the winner is neither. The same 8 notes scored against every triad and seventh at every root, with the weighting run from the metrical one always used to a durational one never drawn. On the left each note counts for its metrical weight; on the right, for how long it is held. The long notes here are on beats 2, 4, 6, 8, which are the weak ones. The two cues point at different chords — C major7 on the left and D minor7 on the right — turning over at a mixture of 40 per cent. And at the crossing the winner is A minor7, which is neither cue's answer — a chord that shares three notes with each and is not the reading either rule asks for. Nothing about the notes changed. What changed is which of two cues a theorist would call obvious is being believed. Harmony and voice leading

The long note and the strong note

The segmentation that produces every object connected here has carried a free parameter since the day it was written: whether a note counts for its metrical weight or for how long it is held. Only the first has ever been drawn. The two name different chords on sixteen per cent of passages where the cues agree about the notes and forty-three per cent where they do not — and where they disagree most sharply a mixture of them picks a third chord neither one asks for.

 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.

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

The scale least committed to its own instrument

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

 6 step patterns, 2 category counts, and only the count matters. Each scale drawn as its own steps across the octave, with the share of trials a listener whose internal noise is 11 cents names correctly — and, beside it, the same figure for a scale of equally spaced degrees of the same size. The two agree to three decimal places on every row, though the narrowest step here is 100 cents on the diatonic major, tempered and the widest is 300. Naming is lost at boundaries, an n-degree scale has n of them wherever they are put, and each costs the same as long as no category is narrow enough for the noise to carry an estimate clean across it — 9.1 standard deviations, at the worst here. Scales and modes

A boundary costs the same wherever it is put

Every capacity figure drawn until now cuts the octave into equal categories, and no scale in the world is equal. Putting the real step patterns through the same model returns exactly the same accuracy to four decimal places — because naming is lost at boundaries, an n-degree scale has n of them wherever they are, and each costs the mean absolute value of the noise. That turns the most-quoted result about hearing from a search into one line: 1504 times one minus the criterion, over sigma.

 How much of A4's pitch error a key could possibly remove. The largest correlation two notes' pitch errors can have at A4, against how long each note lasts. A note's error has two parts and only one of them is the listener's: the steady-tone limen of 4.04 cents, which context might reduce, and the bound a note of length T puts on its own frequency, which context cannot touch. Taking them in quadrature, the shareable fraction is the curve. At a quarter-second note it is 0.21, so the one-half priced earlier is not available at all until each note lasts 486 milliseconds — which is exactly the crossover found by a different route, because a correlation of a half is the two parts being equal. The step is the convention used here, which takes the larger of the two rather than their sum and therefore says the shareable fraction below the crossover is zero. Intervals and chords

The part of the error a key cannot touch

Four earlier essays turn one dial — the correlation between two notes' pitch errors — and apply it to the whole of a note's limen. Half of that limen is not the listener's: a note of finite length does not carry its frequency more finely than 1/2T, and no context can put information into a signal that is not there. So the correlation has a ceiling, it is 0.21 at a quarter-second note at A4 and 0.07 at A2, and the figure that prices a correlation of one half is drawn where one half is unavailable.

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

One cup and seven lengths

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

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

The collapse belongs to the bass

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

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

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

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

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

The partial that gets louder as it goes sharp

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

 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.

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

An attack time is not an attack

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

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

Which notes go brassy first

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

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

The throat that decides both

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

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

Where a woodwind actually sounds from

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

 Thirty cents out of tune is heard as 8 on a 125 ms note and 25 on a 1 s one. How far out of tune a note sounds against how far out of tune it is, at 4 note lengths, at 440 hertz. The key is treated as a prior over pitch: a mixture of Gaussians on the twelve scale degrees, weighted by Krumhansl and Kessler's probe-tone profile and given the width the degree account already uses. The likelihood is the note's own effective limen, which for a short note is the Fourier bound 1/2T. The estimate is the posterior mean, and the shrinkage toward a prior is one line of arithmetic. At 1 s a thirty-cent mistuning is heard as 25.0 cents and at 125 ms as 7.8. Every curve turns back up near the middle of the semitone, because past there the nearest degree is the other one and the pull reverses. The buttons sound at A4, which is the pitch the figure is computed at, at the shortest note length it draws. Intervals and chords

A short note is heard more in tune than it is

Four earlier essays treat a key as something that reduces the noise in a pitch judgement. Treat it instead as a prior and the prediction changes kind: not a smaller error but a systematic bias, pulling a short note toward the nearest scale degree by an amount the Fourier bound sets. Thirty cents out of tune on an eighth-of-a-second note is heard as eight. And the part the debt got wrong is the part that matters — the bias does not vanish on a long note. It stops at 17 per cent at A4 and at 48 per cent at A2, because the likelihood's width has a floor that no duration removes.

 The mixture at which the chain acquires a tonic. Eight turns of i–iv–v–i in A minor, natural throughout, read at every mixture of the two emissions: nought is the original set overlap against the triad on each degree, one is the measured probe-tone profile rotated to each candidate key. The blocks along the top are the name the model gives, the line below is how far ahead of its best rival that name is. Below a mixture of 0.55 every bar is called C major, which is the collection and not the key; at and above it every bar is called A minor. The margin collapses to 0.33 bits at the crossing and recovers to 4.56 — higher than the 3.38 it started at, because a profile has an opinion about this passage and an overlap does not. Harmony and voice leading

A tonic bought with the function

Putting the measured probe-tone profile inside the ordered key-finder is one term, and it does what was predicted: the natural-minor passage is named A minor at every bar instead of C major. It also does two things nobody predicted. It renames a scheme that has been read in the wrong key at every bar since the day it was written, and it destroys the chord's function while it is buying the key.

 The reading the joint search was never offered. The best chord at each mixture of the two segmentation cues, and what the same weighting gives the same notes shuffled into a different order. Both fall along the axis, and most of the fall is the ruler rather than the music: a metrical weighting over a bar of eight spans a factor of eight and a three-to-one duration spans three, so the weighted note mass is 2.1 times more concentrated at the left of the figure than at the right, and a concentrated mass is easier for four notes to cover. What is not the ruler is the gap. It is widest at a mixture of 0.75, where the reading is D minor7 at 2.15 standard deviations above its own null, against 1.12 for C major7 at a mixture of nought. The joint search holds this axis at nought, so D minor7 is not among the hypotheses it considers. Harmony and voice leading

A fourth decision, and two that were never made

The joint search resolves key, metre and segmentation together and holds the segmentation's cue mixture at zero. Adding the mixture is one loop, and reading the search in order to add it turns up something worse than a missing axis: on the passages it is drawn on, the key it reads is the same key at all forty-eight of its hypotheses and the metre scores every barline identically. The fourth axis then cannot be ranked at all until each reading is measured against its own null, because a mixture changes the ruler and not only the answer.

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

An orchestrator doubles a line, not a chord

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

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

The blend arrives before the note does

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

 Where a bar of 25 and a bar of 9 first disagree. The additive metre 2+2+2+3+2+2+2+3+2+2+3 — 25 units, an onset at the head of every group — with the accents it predicts drawn above the accents predicted by reading it as a repeating bar of 9, which is the cut of it that agrees longest. The two rows are identical for 24 consecutive steps and differ for the first time at step 25, where the shorter reading expects an accent and the metre does not supply one. Nothing before that step distinguishes the two hypotheses, so a listener who has not heard 25 consecutive steps has no evidence either way — whatever they are disposed to hear. Rhythm and metre

A twenty-five is a nine until its last unit

Every account of long additive metres says they are heard as groups of shorter ones, and the metre-induction model had never been pointed at the claim. Pointed at it, the model does not prefer the group — it prefers the long bar outright, and would go on preferring it more the longer anybody listened. What it cannot do is start: the evidence that separates a bar of twenty-five from a bar of nine does not exist until the whole bar has been heard, and at the tempo an unequal metre is best played at the psychological present holds sixteen units.

 The longest silence against the spread of the gaps. Every one of the 273 rotation classes of 5 onsets in 16 steps, placed by how uneven its gaps are against how long its longest gap is. The two rank the census together at a Spearman correlation of 0.94, so the longest silence is not a second thing to know about a timeline; it is the same thing measured more crudely. Against the other axis on the frontier — how much of the cycle has to be heard before a listener knows where in it they are — it correlates at -0.02, which is nothing. The named timelines are marked, and all of them sit at 4, the shortest longest-gap any pattern of this size can have. Rhythm and metre

The longest silence is not a third axis

The frontier between evenness and locatability was drawn twice and both times against two quantities, with the third job a timeline does left uncomputed. Computed, neither candidate for it is a third quantity: the longest silence ranks the whole census with the spread of the gaps at 0.94, and syncopation is not a property of a cycle at all — every one of the 273 patterns changes its count when the bar line moves. What the request was actually asking for is a cap rather than an axis, and under the tightest one the census admits thirteen patterns and every named timeline is among them.

 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.

 The same interval, mistuned by the same amount, at each of its two ends. A C to G in the major key, played 25 cents wrong, with the departure carried by the lower note, split between the two, and carried by the upper note. All three are the same interval size; what differs is which note is off the scale. The share of the departure that survives into what a listener hears is 32 per cent when the lower note carries it and 56 when the upper does. The middle bar is the mean of the other two to within a hundredth, so the averaging is linear and the asymmetry is the whole of the effect. Two things produce it: the prior is 9.0 cents wide at the C and 10.0 at the G, and the likelihood is 13.2 cents wide at the lower pitch and 8.8 at the higher. Intervals and chords

An interval is two posteriors subtracted

Treating a key as a prior over one note predicts that an interval's pull is not the single-note pull doubled, because the two degrees are not equally weighted. Half of that is wrong: splitting a mistuning between the two notes gives exactly the mean of what each end gives alone, to a thousandth, at every one of the twenty-one intervals in the scale. What is not the mean is which end carries it — and the pull turns out to be largest not on the shortest notes but on notes of about an eighth of a second, where the likelihood is a quarter of a semitone wide.

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

The setting is not the preference

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

 Two ways to match a tuning note, and they are not the same size. How finely one player can put their A on another's, at 440 hertz on a note of 2 seconds, by each of the two criteria available. Judging one pitch is good to 4.0 cents; comparing two of them adds two errors in quadrature and is good to 5.7. Nulling the beat between them is a different operation altogether — a mistuning of 2.0 cents makes a beat of 0.50 hertz, which shows one full cycle inside the note — and it is 2.9 times finer. The beat criterion is available only when the two tones sound together and share a partial. A player tuning to a note that has already stopped has the coarse one, and so does a singer with nothing to beat against. Pitch and tuning

An orchestra is given a note

Every earlier essay on pitch standards draws a standard as a number an ensemble is at, and no ensemble is at a pitch. It is handed one, by one player, on one note, and everything else is matched to it by ear — so a standard reaches an orchestra through a limen nobody quotes. Matching by comparing two pitches is good to 5.7 cents at A; nulling the beat between them is good to 2.0, and which of the two is available depends on how long the oboe holds the note. The crossover is at about seven tenths of a second, which is shorter than an oboe's A and longer than a plucked one.

 Who listens to whom decides how far apart an orchestra ends up. The spread an ensemble of 60 arrives at, for four ways of passing the tuning note around, with one match good to 2.0 cents. Matching the giver directly leaves every player one match away and a spread of 2.0 cents whatever the size; passing it along a line leaves the last player 59 matches away and a spread of 15.1. Orchestral practice is the middle one — principals to the oboe, sections to their principals — which is two matches and 2.8 cents, and is within half a cent of the best arrangement available at any ensemble size. A convention nobody derived sits one step off the optimum of an arithmetic nobody wrote down. Pitch and tuning

Who listens to whom when an orchestra tunes

One match is good to two cents and an orchestra is sixty of them, arranged in an order that nobody chose deliberately. Passed along a line, the errors accumulate and the last player is fifteen cents from the first; given to everybody at once, nobody is more than two. The convention every orchestra uses — principals to the oboe, sections to their principals — is two matches deep, costs 2.8 cents whether there are four players or a hundred, and sits within a cent of the best arrangement that exists.

 The interval an orchestra tunes on is the least sensitive one it plays. The smallest mistuning each interval betrays, on notes of 2 seconds with the lower note at 440 hertz, taking one full beat cycle as the criterion. A unison shows 1.97 cents; a fifth shows 0.66, a major third 0.39, a minor third 0.33. The ratio is exact and it is the interval's own upper term: the lowest coincidence of a p:q interval sits at p times the lower note's frequency, so the beat runs p times faster. The dashed line is what the same players manage by comparing two pitches instead, at 5.7 cents — coarser than every interval on the axis by between three and seventeen times. Pitch and tuning

The unison is the coarsest thing in the room

An orchestra tunes on a unison and then plays intervals, and the two are not the same test. The lowest coincidence of a p:q interval sits at p times the lower note's frequency, so a mistuning of a given number of cents makes a beat p times faster — a fifth betrays it three times sooner than a unison, a minor third six. The ritual that opens a rehearsal is therefore the least sensitive measurement anybody will make all evening, and every chord afterwards is a finer one.

 The ensemble agrees with itself more and more, about a pitch that is moving. Runs of 16 players each correcting toward the mean of their neighbours, with no term anywhere pulling them back to the note they were given, over 480 corrections. The shaded band is the root-mean-square displacement across all eight runs — the envelope a random walk has — and it grows from 0.84 cents a quarter of the way through to 2.01 at the end, which is the square-root growth a random walk has. Four individual runs are drawn inside it and the furthest of the eight over the top, ending at 4.16 cents. Meanwhile the spread AMONG the players falls from 2.8 cents to 0.5. A consensus with no anchor cannot hold a pitch, and it also cannot lose one quickly: a movement's worth of corrections is a few cents rather than the semitone unaccompanied choirs are said to fall by. Pitch and tuning

A consensus with nothing to hold it

Once the oboe has stopped, no reference is left in the room. Each player corrects toward what they hear around them, which is other players correcting toward them — and a consensus dynamic has a fixed point at every common value, so it pulls the ensemble together and nothing pulls it anywhere in particular. Simulated, the players' spread falls from 2.8 cents to 0.5 while the ensemble as a whole random-walks. The size is the result and it is small: two or three cents over a movement, which is a tenth of what unaccompanied choirs are said to lose.

 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.

 The term that was owed, and the corpus cannot hold it. For each of the three tunes everything here is measured on, how many of its notes have a duration that differs from the gap to the next onset. The answer is none, in 101 notes: these tunes are stored as a list of pitches and lengths with no rests in them, so a note's duration IS its inter-onset interval and conditioning one on the other leaves exactly zero bits. That is a fact about the representation rather than about music. The prediction was that the term would be small, and it could not have been known that the corpus would make it identically zero — which means the prediction cannot be tested here and the exceptions have to be priced directly. Scales and modes

A note lasts until the next one starts

Pricing where a note is against which note it is left duration as the term it had not, with a prediction that it would be small. Measured on the three tunes these readings are built on, it is exactly zero — and it is zero by construction, because those tunes are stored as pitches and lengths with no rests in them, so every duration is its own inter-onset interval. The prediction cannot be tested on the corpus that produced it. Priced directly, a rest costs 0.67 bits a note where a tenth of the notes have one, which is not well under half a bit.

 A tie is charged twice, and the second charge is the larger one. What a tie costs a reader, against the share of noteheads that are the second of a tied pair. The lower curve is the decision itself — is this notehead an event or a continuation? — at 0.52 bits a note where a tenth of them are tied. The upper curve adds what the extra noteheads cost on every other axis: a tied continuation has a pitch and a position and is read like any other notehead before the reader discovers it carries no event, at 4.79 bits each. The total is 1.05 bits a note, which is 2.0 times the decision alone and is a fifth of what a whole note of music costs. A tie is the most expensive mark on the staff per occurrence, and every published account of notational difficulty treats it as a minor one. Scales and modes

The notehead that is not a note

Every quantity so far is charged per notehead, and a tie is the one mark on the staff that puts a notehead on the page carrying no event. Its cost is not the decision that identifies it — that is half a bit where a tenth of the noteheads are continuations. It is the decision plus the whole reading of a notehead that turns out to have been unnecessary, which is 1.05 bits, twice the decision and a fifth of what a note of music costs. Set beside a dot and a longer note value, the tie is five times the price of either and is the only one of the three that can cross a barline.

 Leaps do not fall where offbeats do, and a reader gets the difference free. Where each size of melodic move actually lands in the bar, over the 101 moves of the three tunes measured here. The two axes are priced separately everywhere and they are not independent: the mutual information between them is 0.31 bits a note, which is 20 per cent of the smaller of the two. That is the amount the sum over-charges. A reader who has seen where a note falls already knows something about how far it moved, so the joint cost is 3.47 bits rather than the 3.79 the two axes add to — and every reading load computed so far is high by the difference. Scales and modes

Leaps do not fall where offbeats do

Every reading load computed so far is a sum of two terms priced as though the axes were independent, and an earlier essay named the interaction it could not reach. Measured on the same hundred and one notes every other essay uses, the mutual information between how far a note moves and where it falls in the bar is 0.31 bits — a fifth of the smaller axis, and a sixth of a note's total load. Every reading load published so far is high by that amount, and the quantity saturates at exactly the grid the tunes are notated on, which is the check that it is measuring the music rather than the grid.

 One number a page, and what a hard rhythm buys against a hard tune. Every combination of six kinds of line and seven kinds of rhythm, placed by what each axis costs a reader. The duration term (0.67 bits) and the interaction (0.31) are the same for every cell, so the diagonals are pages of equal difficulty and the exchange rate between the two axes is the slope of one. The pitch axis spans 5.26 bits across the six lines and the position axis 4.46 across the seven rhythms, so a composer choosing between the hardest line and the hardest rhythm is choosing between quantities within 18 per cent of each other. The hardest page is wide leaps in off the beat at 14.0 bits a note and the easiest is a scale on the beat at 4.2. Scales and modes

One number for a page

Four terms and an interaction give a single bit rate per note, and with it the exchange rate a long run of essays has been pointing at. Six kinds of line span 5.26 bits and seven kinds of rhythm span 4.46, so a composer trading a harder tune against a harder rhythm is trading quantities within eighteen per cent of each other — and pages that look nothing alike sit on the same contour. The hardest page on the grid costs 13.96 bits a note and the easiest 4.24, a factor of three and a half, and the subject closes there.

 One noise everywhere, and the noise each boundary actually gets. The seven boundaries of the tempered diatonic scale, with the noise the earlier model gives each of them — 11 cents, the same everywhere — and the noise the harmonicity model gives them instead, which runs from 11.0 cents to 35.6. The error rate is the sum of these over the octave rather than seven copies of one, so it rises from 5.1 per cent to 11.8. And the moment the boundaries differ, where they are put matters: a scale that moved its degrees would move its boundaries onto different intervals and would pay a different sum. That is the earlier null broken by the assumption its own last section named. Scales and modes

A boundary beside a fifth

A closed form established earlier says an n-category division costs n·σ·√(2/π) whatever the widths are, so a boundary costs the same wherever it is put and the step pattern cannot matter. Its own last section named the assumption that produces the null: one σ, applied to every boundary in the octave. Let σ follow how securely each interval is held and the formula becomes a sum over boundaries rather than n copies of one — the diatonic's error rises from 5.1 per cent to 11.8, and where the degrees are put matters again.

 Each tradition's own steps against the equal division of the same size. Six scales, each drawn against the equal division into the same number of degrees, under both models of how securely an interval is held. On the harmonicity model the tempered diatonic is 11 per cent better than seven equal steps; on the profile model the same comparison is 0.9 per cent, which is nothing. The two models disagree about the one comparison anybody would want the measure for, and only one of them is free of circularity: the probe-tone profile was measured on listeners raised inside the diatonic tradition, so using it to explain why the diatonic is well chosen assumes the answer. The harmonicity model assumes only that a simple ratio is easier to hold than a complicated one. Scales and modes

The unequal scale that is easier to name

The whole of the earlier result was that a scale's step pattern cannot matter, and every tradition it drew agreed with the equal division of its own size to three decimal places. With one noise per boundary the comparison is live again, and the tempered diatonic beats seven equal steps by eleven per cent — while the pentatonics gain nothing and the maqam scales gain two tenths of one per cent. Only one of the two security models produces the effect, and it is the one that was not measured on listeners raised inside the tradition it is being used to explain.

 Every question is answered in a corner, not on a ridge. The degree share times the minor share, over the plane of two cues: how much of the emission is the probe-tone profile, across, and how much a bass note is worth, down, with every chord's root in the bass and a bass rule that rewards the triad rooted on the bass. It runs from 0% at a profile share of 0 and a bass weight of 0 to 87% at 0.6 and 1.5. Bass 0: 0%, 0%, 0%, 0%, 41%, 32%, 46%. Bass 0.25: 0%, 0%, 0%, 0%, 52%, 55%, 44%. Bass 0.5: 0%, 0%, 0%, 0%, 68%, 62%, 51%. Bass 1: 0%, 0%, 0%, 0%, 78%, 77%, 77%. Bass 1.5: 0%, 0%, 0%, 0%, 87%, 80%, 86%. Bass 2: 0%, 0%, 0%, 0%, 87%, 87%, 87%. Scales and modes

Two cues meet in a corner

The profile finds a key's tonic and the bass finds a chord's degree, and until now each was swept with the other held at nothing. Swept together across 42 settings, the plane they make is not the ridge that was predicted. The tonic is a step in one direction, at a profile share of 0.55, and the bass cannot move it; the degree is a slope in the other, rising to 89 per cent as the bass is weighted, and the profile barely touches it. Every question is answered only in a corner of the plane — and the one place the two cues overlap is the one piece of music both can rescue.

 How often the metre, the chords and their product find the barline, chords at 1. Constructed passages of four bars of eight quavers, 100 at each setting, with the barline at the first slot. Rhythm regularity is how much likelier a note is on a strong slot than a weak one; chord regularity is how much likelier a note is to be a tone of its bar's chord than a random scale tone. rhythm 0: metre finds it 10%, chords find it 41%, product finds it 16%; rhythm 0.25: metre finds it 34%, chords find it 35%, product finds it 56%; rhythm 0.5: metre finds it 49%, chords find it 21%, product finds it 66%; rhythm 0.75: metre finds it 50%, chords find it 17%, product finds it 56%; rhythm 1: metre finds it 50%, chords find it 11%, product finds it 45%. Harmony and voice leading

The chords never move the barline

Every hypothesis the joint search had drawn was one bar long, and on one bar with a note in every slot the metre cannot choose a barline at all. Four bars with rests in them make the barline a decision the metre and the chords both have an opinion about, and the prediction was that the chords would move the barline more often than the barline moves the chords. It is the other way round, completely: whenever the two prefer different barlines the search takes the metre's, on up to 72 per cent of passages, and in fifteen hundred passages the chords never once move it. What the chords decide is the one thing the metre cannot see — whether the bar starts on the downbeat or half a bar later — and they decide it right a little over two times in three at best.

 The product and the sum of standard scores, finding the barline, chords at 1. Constructed passages of four bars of eight quavers, 100 at each setting, with the barline at the first slot. Rhythm regularity is how much likelier a note is on a strong slot than a weak one; chord regularity is how much likelier a note is to be a tone of its bar's chord than a random scale tone. rhythm 0: product finds it 16%, sum of z finds it 22%, metre finds it 10%, chords, z 35%; rhythm 0.25: product finds it 56%, sum of z finds it 58%, metre finds it 34%, chords, z 38%; rhythm 0.5: product finds it 66%, sum of z finds it 61%, metre finds it 49%, chords, z 19%; rhythm 0.75: product finds it 56%, sum of z finds it 58%, metre finds it 50%, chords, z 14%; rhythm 1: product finds it 45%, sum of z finds it 48%, metre finds it 50%, chords, z 12%. Harmony and voice leading

The chords are a weak witness to the barline

Scaled by its own range, the metre overrules the chords every time the two disagree about where a bar begins. The obvious repair is to score each reading against its own chance — the metre against the same number of notes placed at random, the chords against the passage's notes shuffled across its bars — and add the standard scores. It changes very little: the search finds the barline within six points of where the product found it, and the chords gain the power to move the barline only on passages whose rhythm says nothing, where random notes move it nearly as often. The null's real result is the size of the two witnesses. At the written barline the metre stands up to 5.9 standard deviations above chance, and the chords, with every note a tone of its bar's chord, stand 1.55 above it at best.

 5 censuses: the floor of the longest gap, the ceiling of the syncopation total, and how few patterns reach both. For each census of k onsets in n steps, counted by rotation class: 3 in 8: patterns 7, floor of longest gap 3, at the floor 1, ceiling of total 14, at the ceiling 2, at both 1; 5 in 8: patterns 7, floor of longest gap 2, at the floor 2, ceiling of total 12, at the ceiling 2, at both 2; 5 in 12: patterns 66, floor of longest gap 3, at the floor 6, ceiling of total 28, at the ceiling 6, at both 6; 7 in 12: patterns 66, floor of longest gap 2, at the floor 3, ceiling of total 20, at the ceiling 38, at both 3; 5 in 16: patterns 273, floor of longest gap 4, at the floor 13, ceiling of total 79, at the ceiling 9, at both 9. The rotation-total of syncopation is taken against a metre of 2 × 2 × 2 for 8 steps, 2 × 2 × 3 for 12 steps, 2 × 2 × 2 × 2 for 16 steps. In every census drawn the evenly spaced pattern is at the floor and at the ceiling. Rhythm and metre

The other censuses keep evenness, not locating

Five onsets in sixteen suggested two rules: a named timeline's longest gap is the shortest its census allows, and its syncopation totalled over every rotation is the largest. Nine of ten named timelines across five censuses meet both — but so does the evenly spaced pattern, in all twenty-one censuses from two onsets in eight to eight in sixteen, and in four of those five censuses the named timeline is the evenly spaced pattern. What the rules describe is evenness. The quantity the frontier was built on points the other way: the evenly spaced pattern is the slowest of its census to locate in nineteen of twenty-one, the standard bell pattern is the slowest of all sixty-six patterns of seven in twelve, and among the orders of its own gaps a named timeline is usually both the most even and the slowest to locate. The son clave, whose most even order is also the fastest, is the exception the locating story was told about.

 An accent moves the longest separable bar from 16 units to 18, and no further. For every bar length from 9 to 25 units, over all 1820 arrangements of twos and threes that are not a repeat of a shorter bar, the fewest and the most consecutive steps before the whole bar beats every shorter cut of it. 9: onsets alone 9 to 14, long beat predicted 7 to 12, downbeat predicted 7 to 8; 10: onsets alone 10 to 13, long beat predicted 8 to 11, downbeat predicted 8 to 9; 11: onsets alone 11 to 18, long beat predicted 9 to 16, downbeat predicted 9 to 10; 12: onsets alone 12 to 17, long beat predicted 10 to 15, downbeat predicted 10 to 11; 13: onsets alone 13 to 22, long beat predicted 11 to 20, downbeat predicted 11 to 12; 14: onsets alone 14 to 23, long beat predicted 12 to 21, downbeat predicted 12 to 13; 15: onsets alone 15 to 26, long beat predicted 13 to 24, downbeat predicted 13 to 14; 16: onsets alone 16 to 25, long beat predicted 14 to 23, downbeat predicted 14 to 15; 17: onsets alone 17 to 30, long beat predicted 15 to 28, downbeat predicted 15 to 16; 18: onsets alone 18 to 29, long beat predicted 16 to 27, downbeat predicted 16 to 17; 19: onsets alone 19 to 34, long beat predicted 17 to 32, downbeat predicted 17 to 18; 20: onsets alone 20 to 35, long beat predicted 18 to 33, downbeat predicted 18 to 19; 21: onsets alone 21 to 38, long beat predicted 19 to 36, downbeat predicted 19 to 20; 22: onsets alone 22 to 37, long beat predicted 20 to 35, downbeat predicted 20 to 21; 23: onsets alone 23 to 42, long beat predicted 21 to 40, downbeat predicted 21 to 22; 24: onsets alone 24 to 41, long beat predicted 22 to 39, downbeat predicted 22 to 23; 25: onsets alone 25 to 46, long beat predicted 23 to 44, downbeat predicted 23 to 24. A present of 3.5 seconds holds 16.0 steps at 218 milliseconds a step, so the longest bar some arrangement of which separates inside it is 16 units on onsets alone, 18 with the long beat predicted and 18 with the downbeat predicted. Rhythm and metre

The accent buys two units, however loud it is

A twenty-five cannot be told from a group of shorter bars on its onsets until more steps have gone by than a listener's present holds, and the obvious objection is that nobody plays an aksak bar as bare onsets: the long beat is louder, and the bar's first beat is marked. So how loud does an accent have to be? The question has a surprising answer. Loudness is not the variable. The existing accent cue changes nothing, and delays the answer where it changes anything. An accent that a reading has to predict works at any strength at all, and at no strength does more than a fixed amount: on the long beat it buys the two steps of a short beat, and on the downbeat it takes every arrangement to one floor — the bar less its last beat — which no cue carried by the notes can break. The longest bar that can be heard as one moves from sixteen units to eighteen.

 Four bars read by where the chords change, barline by barline. A constructed passage of four bars of eight quavers, its barline at the first slot and its chords C, F, Em, Dm. Notes: slot 1 C, slot 3 E, slot 4 G, slot 5 E, slot 6 C, slot 9 C, slot 10 C, slot 11 F, slot 13 A, slot 16 C, slot 17 B, slot 19 B, slot 20 E, slot 21 E, slot 24 G, slot 25 F, slot 29 F. For each of the eight places the barline could fall: as written metre, z 3.97, chords, z 2.00, change, z 4.30, metre + change, z 8.27; 1 quaver late metre, z -2.45, chords, z 2.14, change, z -0.87, metre + change, z -3.32; 2 quavers late metre, z -1.38, chords, z 2.15, change, z -0.54, metre + change, z -1.93; 3 quavers late metre, z -0.31, chords, z -0.27, change, z -2.64, metre + change, z -2.95; 4 quavers late metre, z 3.97, chords, z -1.63, change, z -4.30, metre + change, z -0.33; 5 quavers late metre, z -2.45, chords, z 1.27, change, z 0.87, metre + change, z -1.58; 6 quavers late metre, z -1.38, chords, z 1.18, change, z 0.54, metre + change, z -0.84; 7 quavers late metre, z -0.31, chords, z 1.05, change, z 2.64, metre + change, z 2.32. Best metre, z: as written and 4 late. Best chords, z: 2 late. Best change, z: as written. Best metre + change, z: as written. Harmony and voice leading

The chords mark the barline by changing there

Read bar by bar, the chords stood barely above chance at the barline and broke the metre's half-bar tie two times in three at best. Read instead by where they change — how different the chords are across a candidate's barlines against how different they are across the middle of its bars — the same notes break the tie right on 81 to 96 per cent of passages, and added to the metre they find the barline on up to 89 per cent against 61. The weakness was the question the old reading asked, not the harmony.

 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.

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

One note in the compass loses its pizzicato

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

 The golden section and an equal division are one judgement. Where a boundary falls in a piece, as a share of its length, with the band a listener cannot tell from the golden section shaded. A stretch of minutes is judged with a Weber fraction of about 38%, so one criterion's worth of ratio spread around 0.618 covers everything between 0.492 and 0.730 — a quarter of the piece wide, and containing the halfway point. 1 : 1 and 4 : 3 and 3 : 2 and golden section and 2 : 1 are inside it. A claim that a climax falls at the golden section rather than at the middle is, at this resolution, not a claim about anything a listener could hear. Form and structure

A golden section is a coin toss with six coins

An analysis that reports a climax at 0.618 of a piece has not tested one prediction; it has looked at a piece with several defensible boundaries and reported whichever landed nearest. The rate at which that happens under no hypothesis is one line of arithmetic, and the tolerance it needs is not a number chosen on the page — it is the blur a listener's own timing puts on the judgement. Over a stretch of minutes that blur covers everything from 0.492 to 0.730 of the piece, which contains the halfway point, and six candidate boundaries produce a hit eighty per cent of the time.

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

The ceiling everybody names is the loose one

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

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

The quartet settles at two pitches, not four

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

 The change reading follows the chords, not the bar. How far above the other candidates the true barline stands, in standard units, for the reading that scores how much the pitch-class content changes at each candidate — at three harmonic rhythms. At 2 chords a bar the margin is 0.12 and the reading finds the barline 12 per cent of the time; At 1 chord a bar the margin is 1.66 and the reading finds the barline 42 per cent of the time; At a chord every two bars the margin is 0.47 and the reading finds the barline 27 per cent of the time, against a chance rate of 13 per cent. The passages read earlier all changed chord once a bar, which is the middle column and the only one where the reading has anything. Two chords a bar puts a change at the half-bar as well and the reading cannot tell the two apart; a chord every two bars leaves half the barlines with no change at all and the margin halves exactly. Harmony and voice leading

The change reading follows the chords, not the bar

Every passage read until now changes chord exactly at the barline, which is the one harmonic rhythm at which 'the chords change here' and 'the bar starts here' are the same sentence. Pull them apart and the reading goes with the chords: at one chord a bar it stands 1.52 standard units above the other candidates and finds the barline half the time, at two chords a bar it stands 0.01 above them and is at chance, and at a chord every two bars its margin is exactly half — because half the barlines then carry no change at all.

 Asked for the rate, it answers a multiple of it. The change reading asked its own question — what period do the chords change at — over passages built at three harmonic rhythms, with its standardised score for each candidate period. Given 2 chords a bar it recovers the rate 33 per cent of the time and answers too slow 65; Given 1 chord a bar it recovers the rate 58 per cent of the time and answers too slow 38; Given a chord every two bars it recovers the rate 93 per cent of the time and answers too slow 0. It never errs fast in the way it errs slow, and the reason is structural: a chord change every four slots also produces a change at every eighth slot, so a slower grid inherits a faster rate's evidence and a faster grid cannot inherit a slower one's. That ambiguity is why the reading looked like a barline detector in the first place — the bar is a multiple of every harmonic rhythm that fits inside it. Harmony and voice leading

Asked for the rate, it answers a multiple

A reading that follows the chord rate rather than the bar can be asked what the rate is, and the shape of its errors is the whole of why it looked like a barline detector. Given two chords a bar it returns the right period a third of the time and something slower two thirds; given a chord every two bars it is right nine times in ten. It errs slow and essentially never fast, because a change every four slots also falls on every eighth slot and a slower grid inherits a faster rate's evidence — which is the same asymmetry that makes a pitch detector report an octave too low.

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

The tempo moves a scale further than the touch

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

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

The scale belongs to the ringing instrument

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

 Given the bar in octaves, the cue gets the degree back. How often the reading names the right scale degree, against how much the bass cue is worth, under three rules for what a bass note rewards. the bass is the root: 68 per cent at best; the bass is some chord tone: 67 per cent at best; the bar in octaves: 92 per cent at best; the bass is the root, on a root line: 89 per cent at best. The published cue rewards the triad rooted on the bass, which is right only when the chord is in root position; rewarding any triad containing the bass is right always and rewards three chords a bar instead of one. The third rule is not a bass cue at all — given the bar voiced in octaves a listener knows which three pitch classes are sounding, and that names the chord outright. It reaches 92 per cent on a real line against the published cue's 68 on the same line, and the dashed curve is what that cue manages on a line made entirely of roots — 89 per cent. Scales and modes

Given the bar in octaves, the degree comes back

A bass note is a bare pitch class in the usual figure, and a register in any realisation anybody plays. Voice each bar in octaves and a listener knows which three pitch classes are sounding and which is at the bottom, which names the chord outright — and the rule that uses it reads the right scale degree in 92 per cent of bars on a real bass line, against 68 for the published cue on the same line and 89 for that cue on a line made entirely of roots. The question was whether the register recovers the 89. It recovers it and passes it.

 A sharper cue is worth nothing to a reading that will follow it anywhere. How often the reading names the right scale degree, against how much it costs to change key, at a bass worth 1.5 on a real bass line. the bass is the root: 20 per cent at a key cost of 0.5 and 68 at 4; the bass is some chord tone: 47 per cent at a key cost of 0.5 and 59 at 4; the bar in octaves: 46 per cent at a key cost of 0.5 and 91 at 4; roots, and a line of roots: 49 per cent at a key cost of 0.5 and 91 at 4. Where a key change is cheap the rule that names the chord outright reads no better than the rule that names three — 46 against 47 per cent — because a reading that will move key for one bar's evidence follows a sharp cue wherever it points. The sharper cue's whole advantage appears only once the reading is reluctant enough to stay put, and by a key cost of 2.2 it is 31 points ahead. Scales and modes

A sharper cue is worth nothing to a reading that moves

The rule that names the chord outright reads 92 per cent of scale degrees right where the published bass cue reads 68 — at the key cost these readings have always been run at. Sweep that cost and the advantage is not a property of the cue. Where a change of key is cheap the sharp rule reads 46 per cent and the vaguest rule 47, because a reading that will move key on one bar's evidence follows a sharp cue wherever it points. The cue's whole value is borrowed from the model's reluctance to be moved.

 Which tuning a piece wants, and when the answer is one that has a name. Three short pieces, each a list of fretted chords with durations, scored by how far their intervals sit from just on average — and each played three ways: with every string on its fret, with every adjacent interval of the tuning set pure, and with all six strings free and searched. open G blues costs 5.74 cents on the frets, 0.00 on the pure chain and 0.00 at its own optimum; D A D G A D air costs 1.79 cents on the frets, 0.60 on the pure chain and 0.60 at its own optimum; standard song costs 5.97 cents on the frets, 13.80 on the pure chain and 2.05 at its own optimum. The two pieces whose tuning closes gain nothing from the search: the named tuning already is the optimum, to a thousandth of a cent. The piece in standard tuning, whose chain falls a syntonic comma short, gains 3.92 cents on a tuning that has no name. Pitch and tuning

A tuning is right for some chords and wrong for the rest

An earlier essay asked which tuning minimises a piece's total departure from just, and guessed the answer would be a few cents off a named one. Searched over all six strings, two of three pieces get back the tuning they were already in — because their chains close and there is nothing to improve. The third lands on a tuning nobody names, three and nine tenths of a cent better than anything a player would have tried, and it is not a compromise: two of its three chords are exactly just and the third is abandoned by thirteen cents.

 The two parameters are not one, and the reason is a ceiling. The plane of the two parameters these readings have been swept one at a time: how much weight the bass cue carries, against what a change of key costs. Every cell is how often the reading names the right scale degree, and the lines are the contours of equal share. along the 50 per cent contour the product of the two coordinates runs from 0.10 to 0.50; along the 60 per cent contour the product of the two coordinates runs from 0.43 to 1.84; along the 70 per cent contour the product of the two coordinates runs from 0.66 to 2.56; along the 80 per cent contour the product of the two coordinates runs from 1.85 to 4.91; along the 90 per cent contour the product of the two coordinates runs from 2.71 to 15.20. If the two multiplied cleanly those products would be constant and the contours would be hyperbolae. They are not: every contour turns upward and then vertical, because past a bass weight of about 3 more of the cue buys nothing at all and only reluctance is left to buy anything with. The key cost has an interior best, at 3 on this grid, where the reading names 93 per cent of degrees — so a reading that will not change key at all is worse than one that will, which no sweep of a single parameter had found. Scales and modes

The two parameters turn out to have a ceiling between them

The essay before this one asked whether the bass cue's weight and the cost of changing key are one quantity with two names, and said the test was a contour: if they multiply, the curves of equal degree share are hyperbolae. They are not. Along the ninety per cent contour the product of the two runs from 2.7 to 15.2, because past a bass weight of about one and a half the reading saturates and more cue buys nothing. And the sweep finds something no single-parameter sweep here could: the key cost has a best value, and a reading that will never change key is worse than one that will.

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

A wrong bar beats the same on either instrument

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

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