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.

17 min read 7 figures Small whole numbers The ear is not a microphone Where the ends are

Assumes: The pair tuned apart on purpose · A dissonance has to last

The pair tuned apart on purpose ended by naming the thing every figure in this ladder has quietly assumed. Every beat computed here — a tuner counting, a cellist’s wolf, a piano unison, a chorused rank — is a beat between two fundamentals. Two real notes are not two fundamentals.

Fig. 1 Two notes at 220 hertz, fifteen cents apart, drawn as the beat rate between each pair of partials rather than as one number. The fundamentals beat at 1.91 hertz and the eighth partials at 15.3, which is past the rate at which a fluctuation stops being counted and starts being heard as roughness. Below the line and above it are the same physical fluctuation in one sound at the same time. The buttons play the pair, then its fundamentals alone, then its eighth partials alone.

A detuned pair has as many beat rates as it has partials, and they are in arithmetic progression. The k-th partials are k times their fundamentals, so the difference between them is k times the difference between the fundamentals, and the ladder is a straight line through the origin whose slope is the detuning.

220 Hz against 221.91 Hz. Two tones 1.91 hertz apart, added. The rapid oscillation is their average; the slow swelling is their difference, heard as 1.91 beats a second and used by every tuner who has ever worked by ear. — Fig. 2 The first figure of all, at the detuning the hero uses: two tones fifteen cents apart at A3, and the envelope their sum has. This is the beat everybody means — 1.91 per second, countable, nullable, and the whole of what the first five essays computed. It is the bottom of the series above and it is one twelfth of the sound.

Where the line crosses

A fluctuation slower than about fifteen per second is heard as a beat: something to be counted, waited out, nulled. Faster than that it stops being a rate at all and becomes the texture the consonance ladder calls roughness. The transition is gradual and the number is soft, and neither of those matters much, because what the crossing decides is a partial number and partial numbers are integers.

Where in the spectrum a beat becomes a roughness. The lowest partial whose own beat rate exceeds 15 hertz, against how far apart the pair is, for a note at 220 hertz. A piano unison at a cent and a half never crosses inside the first 16 partials; a chorused pair at fifteen cents crosses at the 8th; a full chorus at thirty crosses at the 4th. Everything below the curve is heard as a beat and everything above it as roughness, in one sound, at once. — Fig. 3 The first partial whose own beat rate exceeds fifteen hertz, against how far apart the pair is, for a note at 220 hertz, with the ordinary detunings marked on it. A piano unison at a cent and a half never crosses inside the audible spectrum; a chorused pair at fifteen cents crosses at the eighth; a celeste rank at twenty-two at the sixth; a full ensemble chorus at thirty at the fourth. The buttons play four of them.

The curve is one over the detuning, so the interesting part of it is all at the left. Between one cent and five the crossover falls from beyond the audible spectrum to the twenty-fourth partial; between fifteen and thirty it falls only from the eighth to the fourth.

That is why a tuner’s threshold and a chorus designer’s threshold are the same threshold from opposite sides. Under about three cents there is no roughness anywhere in the spectrum and the whole sound is a slow beat that can be nulled. Over about ten there is roughness in the part of the spectrum that carries most of the energy, and no amount of waiting makes it a beat.

Why nobody drew this before

The reason is not an oversight, and it is worth naming because it is a property of how a figure gets made rather than of what anybody knew.

A beat is drawn as an envelope: two tones added, the sum plotted, the slow swell visible. That drawing is only legible for one beat rate, because a picture containing a 1.9-hertz swell and a 15-hertz flutter at the same amplitude scale shows the swell and nothing else. Every figure in this ladder has been an envelope, so every figure in this ladder has necessarily been about one pair of components — and the one pair anybody draws is the fundamentals, because they are the loudest.

The ladder above is not an envelope. It abandons the waveform entirely and plots a rate against a partial number, which is the only view in which twelve simultaneous rates are twelve marks rather than one smear. The result was unavailable because of the drawing, not because of the physics, which is a thing worth watching for in any collection whose unit is the picture.

The register result, which is the surprise

The crossover is a partial number and a partial number is not a fixed property of a detuning, because a detuning in cents is a different number of hertz at every pitch.

Fig. 4 The identical fifteen-cent detuning two octaves higher. The fundamentals now beat at 7.7 hertz, which is fast enough to be a tremolo rather than a countable beat, and the crossover is at the second partial — the hero figure’s eighth divided by the four octaves’ worth of frequency. Almost the whole spectrum is on the roughness side, and 34 per cent of the energy is with it.

The same tuning is a different device in each register. At A2 a fifteen-cent pair is a slow beat with a faint high roughness on it. At A5 it is a rough sound with a fast flutter under it. Nothing about the tuning has changed; the detuning was specified in cents, which is the unit that holds the interval constant and lets the hertz run.

This is the same trap a tuning is not a table of cents records for a gamelan, arriving from a different direction. There the point was that a table of cents does not say what a paired tuning sounds like; here it is that a table of cents does not say what a paired tuning is, because the crossover it produces moves by three octaves across a keyboard.

It also says something about where chorusing is used. An organ celeste is voiced in the middle and lower middle of the compass and is rare at the top; a twelve-string guitar doubles its lower courses at the octave and its upper ones in unison; an ensemble’s unison width is a thing string players complain about in the high positions and not in the low ones. All three are the crossover moving.

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. — Fig. 5 The width the roughness half of the crossing is set by. A fluctuation is heard as roughness when the two components are inside one critical band and fast enough to flutter; the band is thirty-five semitones wide at the bottom of a piano and under three at the top. Fifteen cents is inside the band everywhere, at every partial, so the crossover above is entirely about *rate* and not at all about whether the pair is resolved — which is the one thing that makes the series a straight line rather than a curve.

Before that, the practical question the whole rung implies: if every partial beats at its own rate, what should a tuner hold constant?

Two ways to detune a pair, and they disagree by an octave. A pair of instruments tuned deliberately apart, drawn across 4 octaves. Holding the detuning at 10 cents gives a beat rate that rises from 0.64 to 10.20 beats a second — a factor of 16, one doubling per octave, because a fixed ratio is a growing number of hertz. Holding it at 3 beats a second instead gives a flat rate and an interval that shrinks from 46.6 cents at the bottom to 2.9 at the top. A tuner has to choose, the choice is audible across the range, and a table of cents can only write down the first of the two. — Fig. 6 A pair of instruments tuned deliberately apart, across four octaves, under the two policies available. Holding the detuning at ten cents gives a beat rate rising from 0.64 to 10.20 a second — a factor of sixteen, one doubling per octave, because a fixed ratio is a growing number of hertz. Holding it at three beats a second instead keeps the rate and gives up the ratio.

The two policies disagree by an octave’s worth of anything, and neither is wrong: one keeps the interval and one keeps the sound. That is the choice this ladder has been assuming away every time it quoted a detuning in cents.

What this does to the ladder’s own results

Two of the earlier rungs need re-reading and one of them comes out stronger.

The piano unison survives intact. Three strings and the note that comes back found the unison detuned by one to two cents, below the bifurcation, bought for sustain. At a cent and a half the crossover is beyond the sixteenth partial at every pitch on the instrument, so there is no roughness anywhere in the sound and the whole bargain is exactly what that rung said it was.

The chorused pair does not. The fifth rung’s finding was that past the bifurcation the sustain is flat, so the entire cost of chorusing is paid in the first four and a half cents and everything beyond is free. That is true of the sustain and it is not true of the sound: past about ten cents the pair is buying roughness in the upper half of its spectrum, and the amount is a computable share of the energy.

Computed, on this site’s own string spectrum to twelve partials, as the share of the energy at or above the crossover:

detuning	A2	A3	A4	A5
1.5 cents	—	—	—	—
5	—	—	0%	4%
15	—	1%	11%	34%
30	1%	11%	34%	100%

And the width at which roughness first takes a tenth of the energy runs 58.1, 29.3, 14.7 and 7.4 cents at those four pitches — halving exactly with every octave, because the crossover partial is inversely proportional to both the detuning and the fundamental while the energy share depends only on the partial. So there is one law here rather than a table: the usable chorus width halves with every octave, and a designer choosing a detuning in cents has chosen four different sounds across four octaves.

The table is also worth reading down its diagonal, where the same figures repeat: 11 per cent at A3 with thirty cents and at A4 with fifteen, 34 at A4 with thirty and at A5 with fifteen. Doubling the pitch and halving the detuning is the same sound, exactly, which is the register result stated as an invariance rather than as a warning.

One number in this essay does not survive that arithmetic. The A5 figure is captioned as putting the crossover at the fourth partial; the crossover is inversely proportional to the fundamental, so two octaves above the hero figure’s eighth it is the second — and the site’s own function agrees, giving 15.3 hertz at partial two against a threshold of fifteen. The energy share that goes with it is 34 per cent rather than the 27 quoted. The direction of the register result is unaffected and its size is larger than the essay said.

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. — Fig. 7 The other half of the crossing, established by the essays on consonance: a dissonance has to last, because roughness is a fluctuation and a fluctuation needs cycles to exist. The same requirement applies to the series above from the bottom: a partial beating at fifteen hertz completes two cycles in a semiquaver at 120 and the fundamental beating at two completes a quarter of one. **A short note has no beats in it at all** — only whatever roughness the top of its spectrum can establish in the time available.

The same halving explains the one thing about the table that looks like a mistake: at A2 a fifteen-cent pair has no crossover inside twelve partials at all, so its roughness share is not small but absent. A chorus in the bass is a beat and nothing else, at any width a builder would use — the first detuning that puts a tenth of a bass note’s energy into roughness is 58 cents, which is over a quarter-tone and is not a chorus but two notes. The device changes kind rather than degree across the compass: beat-only at the bottom, mixed in the middle, roughness-dominated at the top.

There is a third reading that is really a warning. Every published tolerance for a unison — a piano technician’s, a choir director’s, an organ builder’s — is quoted as a number of cents, and this rung says that a tolerance in cents is a tolerance that means different things at the two ends of the instrument. A one-cent unison tolerance at the top of a piano is 0.6 hertz at the fundamental and 7 hertz at the twelfth partial; the same tolerance in the bass is a twentieth of that. A technician working to a fixed number of beats per second rather than to a fixed number of cents is doing the thing this arithmetic recommends, and that is how the trade has always described the work.

The recommendation has a limit worth naming, though. A fixed beat rate at the fundamental holds the bottom of the ladder still and lets the top run, since the k-th partial beats k times as fast whatever the fundamental does — so a technician working to a constant beat rate has fixed the crossover partial rather than the crossover frequency, which is the right invariant if the spectrum is the same length everywhere and is not if it is not. On a piano the treble strings carry fewer usable partials than the bass ones, so a constant beat rate leaves rather less roughness at the top than the ladder alone would predict. The trade’s convention is better than a cents tolerance and it is not exactly right either.

Which computation produced the numbers

The beat rate between two partials is the difference of their frequencies, and that is the whole model. The k-th partial of a harmonic spectrum at f is kf, so a pair at f and f’ has k-th partials at kf and kf’, differing by k(f’ − f).

The detuning in cents becomes a difference in hertz through the site’s own pairBeats, which is the function the paired-tuning rung was written around: a fixed number of cents gives a beat rate that doubles every octave, and a fixed number of hertz gives a flat rate and a shrinking interval.

The crossover is the smallest k for which k(f’ − f) exceeds fifteen hertz. The fifteen is not derived and the figures say so; it is the customary boundary between a fluctuation that is counted and one that is not, and the roughness model this site uses has its maximum at a quarter of a critical bandwidth, which at these frequencies is a few tens of hertz. Moving the boundary to ten or to twenty moves every crossover by one or two partials and moves none of the orderings.

The energy share is the sum of the squared amplitudes above the crossover over the sum of all of them, on this site’s own string spectrum. Amplitude squared rather than amplitude, because roughness is quadratic in the amplitudes of the two partials involved.

Where the model stops

Real strings are not harmonic and the ladder is not straight. The piano is tuned wrong on purpose because a stiff string’s k-th partial sits at k·f·√(1 + Bk²) rather than at kf. Two strings of the same design share B, so the stretch multiplies both partials and the difference is multiplied too — by 1.05 at the eighth partial of a middle-C string and 1.10 at the twelfth. The ladder curves upward and the crossover comes one partial sooner in the treble. That is a small correction and it goes in the direction that makes the register result stronger.

The two notes are assumed equally loud. Roughness between two partials is proportional to the product of their amplitudes, so a chorused pair in which one voice is much quieter has a much weaker upper roughness and the same beat rates. The figures draw the bar width at the partial’s own amplitude for that reason, and the energy share is the honest version of the crossover.

And a detuned pair is not two independent sources when they share a bridge. Three strings and the note that comes back established that a coupled pair has normal modes rather than two lives, and everything above treats the pair as two independent tones added together. That is right for a chorus, an organ celeste and two players; it is wrong for a piano unison, which is the one case where the crossover conclusion is safe anyway because nothing crosses.

Whose music, and where the crossover was found by ear

The arithmetic is about a listener and applies everywhere. The devices are a claim about repertoires, and there are three worth naming.

The organ celeste is a rank tuned sharp of another and drawn with it, and the tuning is specified by builders in beats per second at a stated pitch rather than in cents — which is exactly the unit that holds the crossover still rather than letting it run. A celeste laid out in beats per second at a fixed rate has a crossover partial that rises with pitch instead of falling, which is the opposite of the figure above and is very probably why the convention exists.

The Javanese and Balinese paired tunings are the case this collection has already argued about. A gamelan’s paired instruments are tuned by ear to a beat rate that the players describe and no table records, and the paired-tuning rung found the two policies — constant cents and constant hertz — differing by a factor of eight over three octaves. This rung adds that the choice is not only about the beat rate but about how much of the spectrum is on the roughness side.

And the European string section’s unison is a chorus nobody designed. Its width is whatever the players’ intonation spread happens to be, which is around fifteen cents, and the register result says the same section is a different sound at the top of its range for a reason that has nothing to do with the players.

What the picture cannot show

How much of the difference is audible at all. The energy above the crossover in a chorused pair at fifteen cents is one per cent of the spectrum’s total, which is a small share of a sound to be making a claim about. The share reaches eleven per cent at thirty cents and twenty-seven at the same fifteen cents two octaves up, and those are large. Between them is a range where the arithmetic is confident and the audibility is not.

Whether a listener separates the two. The whole claim is that one sound contains a beat and a roughness at once, and nothing here shows that a listener hears two things rather than one compound texture. The crossover is a fact about the signal.

The number of voices is one pair. What a choir does that a soloist cannot is about sixteen voices rather than two, and the beating there is between every pair at once — which is a hundred and twenty pairs with a hundred and twenty rates, each of which has its own ladder. The figures here are the two-voice case because it is the one with a single answer.

And the amplitude of a beat is not drawn. A pair of equal partials beats to full cancellation; an unequal pair modulates by less. The figures give a rate per partial and not a depth, and the depth is what decides whether a beat at a given rate is noticeable at all.

Where this ladder goes next

Six rungs. Beats are arithmetic anybody can hear; a tuner counts them; a cellist’s wolf is the same arithmetic on a coupled pair; a piano’s unison is a detuning below a bifurcation; a chorus is the same pair past it; and now, a pair has one beat rate per partial and crosses into roughness at a partial the detuning and the register decide together.

The rung after it is the one the crossover makes available and this rung has no way to reach. Everything above is one pair on one note. Two sections of an ensemble singing or playing an interval are many pairs at once, on notes that are not the same note, and the partials that nearly coincide are the ones the interval’s ratio brings together — a major third brings the fifth partial of the lower voice against the fourth of the upper. So an ensemble’s roughness at an interval is a count of near-coincidences times the number of ways to pick one player from each section, and both of those are arithmetic this collection already has.