Field

Intervals and chords

Two notes at once, why some of them beat, and the geometry of the ones that do not.

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.

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

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

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

Beats are arithmetic that anybody can hear

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

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.

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

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

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

The note that is not there

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

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

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

A dissonance has to last

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

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.

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.

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

The same chord is harsher when it is louder

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

Which rotations have a fifth above their own tonic. Each rotation drawn by scale degree, with the degree a perfect fifth above the tonic marked. 1 of the 7 does not have one: Locrian. A tonic without a fifth above it has no triad of its own, which is a stronger disqualification than any preference.

The mode with no fifth

Six of the seven rotations have a perfect fifth above their own first degree and one does not, which is a structural disqualification rather than a preference. And the set's single tritone sits on a different pair of degrees in every rotation — on the fourth and seventh in Ionian, on the tonic itself in two others — which decides most of what each mode can do.

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 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.

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.

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.

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.

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 same distance, under two names

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

The series has three tops

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

How 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.

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.

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.

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 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.

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.

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”.

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.

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

The fourth top is the maker's

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

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

Sixteen sweeps against sixteen

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

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.

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.

A roughness with a rate of its own

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

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

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 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.

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.

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.

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.

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.

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.

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.

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.

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

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.

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.

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.

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 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.

Which intervals in a key can be mistuned invisibly, and from which end. Every interval between two degrees of the major scale, ranked by how differently its two ends treat a 25-cent departure on notes of 0.25 seconds. The C to B is the most lopsided, at 47 points: a mistuning on its upper note reaches the listener nearly 2.5 times as strongly as the same mistuning on its lower one. The D to E is the most even, at -0. A negative bar is an interval whose LOWER note is the one that carries a mistuning into the listener, which happens whenever the lower degree is the less specified of the two. No account of interval perception predicts a table like this, because an interval is usually treated as one quantity rather than as a difference of two estimates.

Which end the mistuning is on

Twenty-one intervals in the major scale, each with two ends, and the same twenty-five cents reaches a listener at anywhere between 32 and 79 per cent of its size depending on which of the two notes carries it. The most lopsided is the tonic to the leading note, where a departure on the upper note arrives two and a half times as strongly as the same departure on the lower. Two mechanisms produce it and they can be separated by one flag: two thirds of the asymmetry is register and one third is the key.

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.

The setting is not the preference

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

A 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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Three harmonics of the bass arrive before the bass

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

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

The ghost bass drops a twelfth at a forte

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

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.

The played notes already name the ghost bass

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

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

A firm touch buys beats until the aftersound sinks with it

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

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