Acoustics & Music Theory
why it sounds like that
Theme

Small whole numbers

Consonance is frequency ratios of small integers, and it has been known since Pythagoras. Almost everything interesting follows from those integers refusing to line up.
Twelve fifths do not make seven octaves. Pitch drawn as a spiral: one turn is one octave, so a note's angle is its pitch class and its radius is how high it has climbed. Twelve pure fifths wind round 7 times and a little further, finishing 23.46 cents past seven octaves — the Pythagorean comma. Nothing in the chain closes that gap; it is what a tuning has to absorb, hide or spread. Pitch and tuning

Twelve fifths and seven octaves, which are not the same thing

Stack twelve perfect fifths and the note that arrives should be the one seven octaves up. It is sharp by about a quarter of a semitone, and the whole history of tuning is a set of decisions about what to do with that.

 The syntonic comma on the lattice. Pitch classes reached by multiplying by three — a fifth, one step east — and by five — a major third, one step north. Two routes to the same note name arrive at different pitches, and the syntonic comma is the size of that disagreement: 21.51 cents. Pitch and tuning

A second comma, arriving by a different road

Four pure fifths ought to land on a pure major third, two octaves up. They miss by 21.5 cents — a different gap from the one twelve fifths leave, produced by a different route, and the two are not the same size.

 Every equal division from 5 to 60, and how wrong it is. For each number of equal steps in the octave, how far its best fifth and its best major third fall from the pure ratios, in cents. The divisions people have actually used are the ones with small errors in both, and no other criterion was applied to pick them out. Pitch and tuning

Nineteen, thirty-one and fifty-three

Divide the octave into a different number of equal parts and different problems disappear. Which ones vanish is not a matter of taste — it can be computed from the prime factors of the intervals, and each division makes a different decision.

 Two tones in the ratio 3 to 2. Two sine tones whose frequencies are in the ratio 3 to 2, and their sum. The combined pattern repeats every time both waves return to the start together, which for a simple ratio is soon — here after 2 cycles of the lower tone and 3 of the upper. Intervals and chords

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

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

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

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

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

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

A string does everything at once

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

 All 55 chords of 3 notes, on both counts. Every 3-note subset of the twelve pitch classes containing the root — 55 of them — with summed Plomp–Levelt roughness on a string spectrum across, and the largest whole number needed to write the chord in just intonation up. The smoothest is 0–5–10, 15:20:27, which is stacked fourths. The simplest is 0–5–9, 3:4:5, which is the major triad. Exactly one chord is in the best 18% on both axes: 3:4:5, the major triad. Intervals and chords

Three is the largest agreeable number

Take every chord of every size that can be built from the twelve, score each one for roughness and for the size of the whole numbers it needs, and ask how many are good on both counts. At two notes the answer is one. At three notes it is still one. At four it is eight, and the question stops having an answer.

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

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

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

 Where a progression in pure ratios ends up. A short chord sequence taken in exact whole-number ratios, with the running difference from equal temperament measured in cents. The sequence returns to its starting chord and the pitch does not, which is the comma arriving in ordinary music. Harmony and voice leading

The progression that never comes home

Just intonation is usually said to fail because a keyboard has only twelve keys. That is not the reason. Take one of the commonest chord sequences in tonal music, play every chord in exact whole-number ratios, and it arrives back on its starting chord a syntonic comma flat — on any instrument, including a voice.

 Two models of consonance, and where they disagree. Every chord scored twice: horizontally by summed Plomp–Levelt roughness, vertically by the largest integer needed to write it as members of one harmonic series. Both are supposed to be measuring consonance and both are computed here from the chord itself. They correlate, but not tightly enough to be the same claim, and the chords furthest from the diagonal are the ones any experiment has to be run on. Perception and the listener

Consonance is half learned, and this is the half

This site's founding claim is that consonance is small whole numbers. Two computable models say so and they disagree about which chords — which is already awkward. The cross-cultural evidence is worse: listeners with little exposure to Western music discriminate roughness exactly as anyone does, match octaves exactly as anyone does, and rate consonant and dissonant chords as equally pleasant.

 Helmholtz motion, bowed at 9% of the way from the bridge. Above: the string at 5 instants of one period. It is two straight lines meeting at a corner, and the corner travels round the string rather than the string swinging. Below: the resulting force on the bridge, a sawtooth whose two segments are in the ratio 0.09 to 0.91 — the bow's own position. A sawtooth contains every harmonic at exactly one over n, so the spectrum barely changes with bow position even though the waveform plainly does. Instruments and their design

The bow makes a corner

A bowed string is not a string swinging. It is two straight lines meeting at a single sharp corner that travels round the string once per period, triggering the slip that keeps it going. The force on the bridge is therefore a sawtooth, and a sawtooth is every harmonic at exactly one over n — which is why a bowed string is the most nearly perfect harmonic series in the orchestra.

 Every delay, scored — a round and a tune that is not one. Mean Plomp-Levelt roughness of the simultaneities a tune makes against a copy of itself entering a whole number of bars later, for Frere Jacques and Twinkle, twinkle. The two do not overlap: the worst delay of the first is smoother than the best delay of the second. The model sees roughness and nothing else — no voice leading, no parallels, no distinction between a passing dissonance and a structural one. Harmony and voice leading

A melody that can accompany itself

Whether a tune works as a round is decidable before anyone sings it. Score every simultaneity it makes against a delayed copy of itself, at every delay, with the roughness model already in use, and two nursery tunes separate completely — every delay of one is smoother than every delay of the other. What the scan does not do is pick out the entry the tradition uses, and that refusal is the most informative thing in it.

 Every keyboard temperament before 1800, on one line. The fifth's departure from pure against the major third's, for a temperament that narrows every fifth by the same amount. The relation is a straight line — a third is four fifths — so the systems that are usually presented as rival schemes are points on one continuum, and the parameter that indexes them is a single number. Pitch and tuning

A fraction of a comma

The tuning systems of three centuries are usually presented as rival schemes with names and dates. They are points on a line. Narrow every fifth by the same fraction of a comma and both errors that matter are linear in that fraction — and the system named as the break with the tradition turns out to be a member of it, at one part in eleven.

 The comma is conserved; only the place it is paid is chosen. Every policy available to a continuous-pitch ensemble on one line: drift per circuit along the bottom, worst mistuned interval up the side. The line is straight and its intercepts are fixed, because the 4 moves of the pump have to absorb 21.51 cents between them however they are shared. Spreading it evenly puts 5.377 cents on each interval — the same narrowing quarter-comma meantone applies to every fifth. Harmony and voice leading

Somebody has to pay the comma

A choir has no frets and no keys, so it can tune every chord exactly — and a common progression sung that way arrives a comma flat, every circuit. The freedom a keyboard lacks turns out to be a freedom about where the error goes rather than whether there is one, and the two costs always add to the same number.

 Three spectra, and where each one's consonances fall. The positions of the roughness minima for 3 partial sets, over 12 semitones from the same fundamental, found by scanning each curve rather than by marking them. A harmonic tone — 498 cents at 3.1 per cent prominence, 702 cents at 37.1 per cent prominence, 884 cents at 6.2 per cent prominence. Partials stretched by 2.1 per octave — 533 cents at 3.7 per cent prominence, 751 cents at 40.1 per cent prominence, 947 cents at 7.3 per cent prominence. A bar's partials — 1, 2.76, 5.40, 8.93 — 694 cents at 5.6 per cent prominence, 870 cents at 14.2 per cent prominence, 982 cents at 1.0 per cent prominence, 1166 cents at 14.7 per cent prominence. The minima are computed by the same well-finder the dissonance curve uses, which requires a minimum to have one per cent of the curve's range on both sides of it before it counts. Intervals and chords

A spectrum chooses its own scale

The roughness curve's dips are always described as landing on the small whole numbers. They do not land on numbers. They land where partials coincide, and stretching a spectrum by seven per cent moves every one of them by exactly seven per cent — so the scale a set of intervals belongs to is a property of the instrument's spectrum rather than of arithmetic.

 The least rough 7 notes of the twelve, at 262 Hz. Every 7-note selection from the twelve that contains C, scored for total Plomp–Levelt roughness at a root of 262 Hz, ranked. The best 8 are shown with the spread between them. The major scale ranks 5 of 462; the first selection that is a mode of the diatonic set ranks 1. Change the root and the ranking changes, because roughness is a fact about frequencies and a scale is not. Intervals and chords

Roughness cannot choose a scale

Score all four hundred and sixty-two seven-note selections from the twelve for roughness and the answer at middle C is a mode of the diatonic set, first out of four hundred and sixty-two. Ask the same question an octave lower and the same selection ranks a hundred and forty-first. The model is not wrong; it is answering a question about frequencies, and a scale is not one.

 Every voicing of a major triad, least rough first. All 27 arrangements of the same three pitch classes within 3 octaves from 131 Hz, scored for roughness. The best is spaced 19 then 9 semitones — wide below, close above — and the worst is the chord in close position at the bottom of the range, 6.6 times rougher with exactly the same notes in it. Harmony and voice leading

Where to put the third

Take three pitch classes, four octaves to put them in, and score all twenty-seven arrangements. The smoothest is root, fifth an octave up, third two octaves up — which is partials one, three and five of the harmonic series — and the roughest, at every register tried, is the chord in close root position at the bottom of the range. Every orchestration manual states that rule and none of them derives it.

 The thirteen intervals at C4, ordered by roughness, against Fux's species, 1725. Every interval within an octave above 262 Hz, ordered by the Plomp–Levelt roughness of a string spectrum — from 0.0029 for the unison to 0.2635 for the minor second — beside whether Fux's species, 1725 counts it a consonance and how many of the thirteen may follow it under that rule set. The best single threshold on roughness misclassifies 2 of the thirteen, among them the major third and the minor third, and the gap it falls in is 4.2 per cent of the roughness either side of it. Harmony and voice leading

A dissonance is what has to be resolved

The perfect fourth is a consonance between two upper voices and a dissonance against the bass, and the same three notes are involved either way. Score both arrangements with the roughness model and the one the rules call a dissonance comes out thirty per cent smoother. Whatever the rule is tracking, it is not the sound.

 3 against 2, and the line it adds up to. Two pulse trains over one bar, 3 against 2, and beneath them their union on the common grid of 6 steps. The composite has 4 onsets — 3 + 2 − 1, because the two layers share the downbeat — and its gaps run 2, 1, 1, 2 steps, which uses 2 distinct lengths and reads the same in both directions. No single division of the bar produces that sequence. Rhythm and metre

The third pattern nobody played

Two players play two even pulses and a third rhythm arrives that neither of them played. It has a + b − 1 onsets, its gaps read the same forwards and backwards, and it uses exactly min(a, b) different lengths — which is a better account of why 3:2 is a figure and 7:5 is weather than the number used for the last three essays.

 How far a chain of fifths is from closing, after each number of fifths. The distance from a whole number of octaves after each number of pure fifths, in cents, out to 60. It is never zero, because a fifth is 3/2 and no power of 3 is a power of 2. The running minima are 1, 2, 5, 12, 41, 53 fifths, and the residue at each is that division's comma: 23.46 cents at 12, 19.84 cents at 41, 3.62 cents at 53. A 3-against-2 polyrhythm asked the same question closes exactly and at once — 3 pulses of the 3 are 2 of the 2, to the last microsecond — because 3/2 is a ratio of whole numbers and log2(3/2) is not. Pitch and tuning

A comma is a polyrhythm that never closes

Seven earlier essays have rested on a common grid. Two cycles have one when their ratio is a ratio of whole numbers, and the fifth against the octave is not — provably, from unique factorisation. So there is no grid, no composite and no least common multiple, and what is left over instead is the sequence 2, 5, 12, 41, 53 with a comma attached to each.

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

The third the model has no opinion about

A neutral third — halfway between major and minor, and a scale degree in most of the music between Morocco and Iran — is nothing at all on an ordinary roughness curve. It becomes a well at 347 cents, which is exactly 11:9, only for a spectrum whose ninth and eleventh partials are at least three times their natural strength. No instrument has that spectrum.

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

The spectrum that was supposed to explain the gamelan

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

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

The inversion that cannot end a phrase

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

 a major triad, C–E–G. The partials are at 261.6, 329.6, 392.0 Hz. Each row is a harmonic series they are consistent with to within 30 cents: the filled dots are the partials in their assigned slots and the open dots are slots the series predicts that nothing occupies. There are 2 such series with harmonic numbers up to 16, the best fitting them to 10.1 cents with a fundamental of 65.54 Hz and no unoccupied slot. The rest sit at one half of it and their arithmetic is exactly as exact; what separates them is the count of empty slots, which is why a residue pitch built from few partials is reported an octave out by a minority of listeners and not by the rest. Intervals and chords

The root an ear supplies

A major triad's notes fit 4:5:6 with nothing missing and a fundamental two octaves below the bass. A minor triad's fit two different series with two different answers a major sixth apart, and the model cannot choose between them. The ambiguity theorists argued about for two centuries is a computable quantity, and the spectrum invented to remove it is one no object produces.

 The same intervals, played higher and higher. Sensory roughness for four fixed intervals as the pair is transposed up five octaves, computed from the same Plomp–Levelt model as the dissonance curve. Every one of them falls as it rises, and the small intervals fall furthest — so how consonant an interval is depends on where it is played, not only on what it is. Intervals and chords

A chord is a register

The same three pitch classes are five times rougher at the bottom of a piano than in the middle, and the arrangement that minimises the roughness over a low bass turns out to be the bass's own fifth and sixth partials. The orchestration rule about low thirds is not a convention. It falls out of the width of a critical band, exactly.

 Every 3-note stack of one interval. Each chord built by repeating a single interval 2 times from the root, scored for summed roughness on a string spectrum and asked whether one low fundamental accounts for all its notes within 30 cents. 8 of the 11 are smoother than the major triad, which scores 0.288: stacked major thirds (an augmented triad) at 0.279, stacked fourths at 0.218, stacked tritones at 0.144, stacked fifths at 0.140. And 3 of them have no fundamental at all — stacked minor sixths, stacked major sixths, stacked major sevenths — meaning no series of harmonics up to the 16th accounts for their notes to within a 30-cent tolerance. Harmony and voice leading

A stack that is not thirds

Build every chord that repeats a single interval and score them all. Eight of the eleven three-note stacks are smoother than the major triad; the four-note stack of fourths is smoother than a dominant seventh by a third and has no fundamental any matcher can find. Two absences, both computable, and both are exactly what the chord was adopted for.

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

The register where the series becomes a scale

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

 The gap that decides whether two clocks are two. The closest approach of the two streams in each ratio, at 100 to the minute in 4-time — a bar of 2.4 seconds. It is the bar divided by the product of the two numbers, which is an identity and is checked here against the measured minimum. 9:5 is the last ratio whose onsets are securely separate at this tempo; past it the two streams' events fall inside the window in which the ear cannot put two onsets in order, and what is heard is one irregular pattern rather than two clocks. The bound has a product in it, which is why 3:2 and 4:3 are everywhere and 11:7 is a notation. Rhythm and metre

The ratio that stops being two

Eight earlier essays have taken two clocks at a rational ratio to be a thing a listener can hold. There is a ratio past which it is not, and the bound is not where anyone would look for it: the cycle is exactly one bar long at every ratio, so the length of the pattern separates nothing. What separates them is the closest the two streams ever come, which is one part in their least common multiple — 400 milliseconds for three against two, and seventeen for thirteen against eleven, which is not two events at all.

 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.

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

The series has three tops

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

 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.

 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.

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

The partial the lips cannot reach

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

 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.

 Which note a chord would rather have twice. Every complete four-part voicing of each chord inside the SATB ranges, grouped by which member sounds twice and scored for roughness — 480 voicings for a triad. The order for a major triad is root < fifth < third, which is the rule every part-writing treatise states. For a minor triad it is fifth < root < third, which is not. The numbers printed under each bar are the mean error, in cents, with which the four sounding notes fit a single harmonic series, and that measure separates the three far more sharply than roughness does. Intervals and chords

The note that sounds twice

A triad has three notes and a four-part texture has four voices, so one note is doubled — and the voicing model used here leaves the choice free because the rules have an opinion about it. Asked properly, the arithmetic agrees with the treatises for the first time in nine essays: root, then fifth, then third. For a minor triad it does not agree, and for a symmetric chord it correctly has nothing to say.

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

Which player on which note

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

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

Every partial beats at its own rate

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

 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 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

Surprise is a number

The chord that did not come was described rather than measured. Its measure is the information content of what did arrive, and a model of the probability has been to hand since the key-finding essays — eight root-motion weights, ordinal and stipulated. Reading them as a distribution prices a deceptive cadence at 2.71 bits against a perfect one's 1.85, and turns up the fact that the largest of the eight had never been read by anything.

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

A roughness with a rate of its own

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

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

The flare that makes a series harmonic

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

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

The bell is tuned for the cup

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

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

Two players on one note

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

 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 number nobody has moves the size and not the order. The mean total surprise per chord change, against how much the two surprises share. At zero they are independent and the total is their sum; at one they are the same event and the total is the larger of the two. The mean falls by a factor of 1.53 across that whole range, which is the size of the thing a corpus would settle. The ordering of the events by total surprise does not move at all until the very end: 5 of the 6 correlations swept give exactly the ordering independence gives, and only perfect dependence changes it, by 3 places out of 7. The most surprising event in the passage is the same one at every correlation. So the corpus three separate accounts have recorded wanting would change what this figure reports and not what it concludes. Harmony and voice leading

Two surprises and one event

A chord change is surprising twice over — in which chord it is, and in when it comes — and a listener meets one event. Adding two surprises needs to know how much they share, which is a fact about a repertoire nobody has. Sweeping it instead: the total moves by half, the ordering does not move at all, and the most surprising moment in a passage is the same one whatever the answer turns out to be.

 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.

 Second order and fourth order, from the same sequence of wavenumbers. Frequency against wavenumber for a string and for a bar, with each object's own wavenumbers marked on the axis. A string's equation is second order in space, so frequency goes as wavenumber and the straight line carries its modes at 1, 2, 3, 4. A bar's is fourth order, so frequency goes as the SQUARE of wavenumber and the parabola carries the same kind of sequence to 1, 2.757, 5.404, 8.933. Both sequences of wavenumbers are arithmetic — a string's are the even multiples of π/2L and a bar's the odd ones, except that the first is 3.011 rather than 3 — and squaring an arithmetic sequence is the whole of why one object has a fundamental and the other has none. Every number is derived from the boundary condition; none is measured. Instruments and their design

A bar's partials are the odd numbers, squared

A string's wave equation is second order in space and a bar's is fourth, so a string's frequencies go as its wavenumbers and a bar's go as their squares. Both objects fit a nearly arithmetic sequence of wavenumbers between their ends; squaring one is the whole of why a marimba key has no note in it. The ratios are 1 : 2.756 : 5.404 : 8.933, which are the odd squares over nine, uniformly 12.9 cents flat, and the 12.9 cents is one root that refused to be 3π/2.

 Where one arch can put a bar's second and third partials. The plane of the second and third partials, with the path a single arch traces through it as it is cut deeper, for 4 arch lengths from 50 to 100 per cent of the bar. Every path starts at the plain bar's 2.756 and 5.404 and runs up and to the right. The xylophone's target of 3 and 6 lies on them: reach the second partial and the third is there. The marimba's target of 4 and 10 lies above every one of them — the best any of these arches manages while holding the second partial at 4 is 9.176, which is 149 cents short. A marimba's third partial is not something one cut can place, and that is a statement about the shape of the cut rather than about the maker. Instruments and their design

One cut cannot place two partials

An arch cut into a bar's underside is one knob, and a maker aims it at two targets. Swept across every arch length, the path it traces through the plane of the second and third partials passes through the xylophone's 3 and 6 — dead through it, within a tenth of a cent at the natural arch length — and under the marimba's 4 and 10 by between 149 and 326 cents. Hitting one target means missing the other, and that is why a marimba bar's underside is not a simple arch.

 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.

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

The third sound magnifies cents, not hertz

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

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

The tone on the root changes hands at the fifth

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

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

A major triad's combination tones are its own notes

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

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

The bass line under a passage in thirds

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

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

The tuning a string quartet cannot change

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

 An inversion lasts as long as its outer sixth. The six three-note voicings of a major and a minor triad, each over a bass of C3 struck at 80 decibels, with how long the strongest partial coincidence of each of its three intervals survives the strike. The interval that goes first is marked, and its time is how long the chord keeps the evidence of all its intervals at once. major, root position: major third 5:4 1.26 s, minor third 6:5 1.03 s, fifth 3:2 2.14 s; the chord 1.03 s. major, sixth chord: minor third 6:5 1.04 s, fourth 4:3 1.62 s, minor sixth 8:5 0.74 s; the chord 0.74 s. major, six-four: fourth 4:3 1.60 s, major third 5:4 1.27 s, major sixth 5:3 1.26 s; the chord 1.26 s. minor, root position: minor third 6:5 1.04 s, major third 5:4 1.27 s, fifth 3:2 2.14 s; the chord 1.04 s. minor, sixth chord: major third 5:4 1.26 s, fourth 4:3 1.63 s, major sixth 5:3 1.26 s; the chord 1.26 s. minor, six-four: fourth 4:3 1.60 s, minor third 6:5 1.03 s, minor sixth 8:5 0.74 s; the chord 0.74 s. The major six-four lasts longest and the major sixth chord shortest; every root position is held to its minor third's life. Timbre and acoustics

An inversion lasts as long as its outer sixth

A struck interval keeps the partial coincidence that names it for a time set by its ratio, and a chord is three intervals at once. Voiced over one bass and struck on a piano, a triad keeps the evidence of all three only as long as its weakest one lasts, and for an inversion that is the sixth on the outside: a major sixth lasts as long as a major third, a minor sixth dies first. So the major six-four and the minor sixth chord are the most durable voicings of their triads and the major sixth chord and the minor six-four the least — and unlike a dyad, a triad's inversions keep their order by roughness through almost the whole decay.

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

The smoothness is in the skips

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

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

A combination-tone bass needs a forte

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

 Opening a triad changes which interval goes first. The six voicings of a major and a minor triad over C3, each close and with its middle note raised an octave, struck at 80 decibels, with how long each keeps the coincidences of all three of its intervals and which interval goes first. major root position: close 1.03 s, held by its minor third 6:5; open 1.26 s, held by its major tenth 5:2. major sixth chord: close 0.74 s, held by its minor sixth 8:5; open 0.47 s, held by its minor tenth 12:5. major six-four: close 1.26 s, held by its major sixth 5:3; open 0.74 s, held by its eleventh 8:3. minor root position: close 1.04 s, held by its minor third 6:5; open 0.47 s, held by its minor tenth 12:5. minor sixth chord: close 1.26 s, held by its major third 5:4; open 1.26 s, held by its major sixth 5:3. minor six-four: close 0.74 s, held by its minor sixth 8:5; open 0.74 s, held by its minor sixth 8:5. Intervals and chords

An open triad lasts as long as its tenth

Close, every triad's weakest link is a third or a sixth. Raise its middle note an octave and the link becomes a compound interval, and compound intervals do not last alike: a major tenth, 5:2, lives as long as a major sixth, while a minor tenth, 12:5, needs the twelfth partial and lives 0.47 seconds. So the spacing orchestration manuals recommend for a major chord in the bass is the longest-lived voicing a struck triad has, and the same spacing halves the life of a minor chord — and the six-four's advantage reverses.

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

A scale is committed to how long its instrument rings

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

 Both qualities reach the same ceiling. The longest-lived spacing of a major triad and of a minor one, over four basses, taken over every arrangement of the three pitch classes within 2 octaves. They are the same number at every bass — 1.16 seconds over C2, 1.26 seconds over C3, 1.26 seconds over C4, 1.41 seconds over C5 — and at each bass 2 major and 2 minor spacings are tied at it. The faint line is the worst a minor spacing can do, which is 3.5 times shorter. So the asymmetry found earlier is a fact about the minor tenth rather than about the minor triad: a minor chord has a spacing that avoids it, and that spacing is its first inversion, where the minor third between two of its notes appears as a major sixth instead. Intervals and chords

A minor triad can be spaced to last

Two spacings of each triad, drawn side by side, say that opening lengthens a major chord and halves a minor one. Drawn over every arrangement of the three pitch classes within two octaves of a fixed bass, the asymmetry disappears: a minor triad reaches 1.26 seconds over C3 and so does a major one, with two spacings of each tied at the top. The short-lived chords were never the minor ones. They were the ones with a minor tenth on the outside, and a minor triad has three spacings that avoid it.

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

Three harmonics of the bass arrive before the bass

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

 One of these tunings has no comma to place. Five guitar tunings, each with every adjacent interval set to the pure ratio nearest the interval it spans, drawn as how far each string then sits from where the frets put it. standard, spanning 5–5–5–4–5 semitones, closes 21.5 cents short and spreads its strings over 21.5 cents; drop D, spanning 7–5–5–4–5 semitones, closes 17.6 cents short and spreads its strings over 19.6 cents; D A D G A D, spanning 7–5–5–2–5 semitones, closes exactly and spreads its strings over 3.9 cents; open G, spanning 5–7–5–4–3 semitones, closes exactly and spreads its strings over 15.6 cents; open D, spanning 7–5–4–3–5 semitones, closes exactly and spreads its strings over 15.6 cents. Standard tuning's four fourths and a third fall a syntonic comma short of two octaves, which is what was found earlier. D A D G A D's fifth, two fourths, a tone and a fourth multiply to exactly four — two octaves — so nothing has to be put anywhere. Pitch and tuning

One tuning has no comma to place

Standard guitar tuning is four fourths and a third, and tuned pure it closes two octaves exactly a syntonic comma short — so tuning by ear decides where the comma goes rather than whether. An open tuning replaces the interval list. D A D G A D's fifth, two fourths, a tone and a fourth multiply to exactly four, so its chain closes and there is nothing to place: its six strings sit within four cents of the frets. The open-chord tunings close too, and pay for it with one string fifteen cents flat — which every chord barred straight across inherits exactly, and which is why every such chord is precisely as pure as the open one.

 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.

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

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

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

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

The played notes already name the ghost bass

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

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

A seventh chord cannot be spaced to last like a triad

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

 A 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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