Timbre and acoustics

A section against another section

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

Assumes: What a choir does that a soloist cannot · Every partial beats at its own rate

What a choir does that a soloist cannot is a rung about a unison. Sixteen singers on one note beat against each other at a hundred and twenty rates, no one of which is countable, and what the section buys is the disappearance of any single beat. That rung ended by naming the question one level up: what a section does against another section, where the interval is large and the beating is between partials.

The answer starts with which partials, and it is decided by arithmetic rather than by the singers.

How far equal temperament puts each interval's coincidence out. For each interval, the pair of partials it brings together and how many cents equal temperament mistunes that coincidence by. A fifth's third-against-second is out by 2 cents and a major third's fifth-against-fourth by 14 — so the same temperament that is inaudible on a fifth produces, at 220 hertz, a beat of 8.7 per second between two sections singing a third, with every singer in both of them perfectly in tune. — Fig. 1 For each interval, the pair of partials it brings together and how far equal temperament mistunes that coincidence. A fifth’s third-against-second is out by two cents; a major third’s fifth-against-fourth by fourteen; a minor third’s sixth-against-fifth by sixteen. The right-hand column turns each into a beat rate at A3, and the range is from three quarters of a beat per second to fourteen.

Two sections singing a tempered major third beat 8.7 times a second with every singer in both of them perfectly in tune. The rate is not a property of the singers, of the hall or of the vowel. It is the syntonic comma, arriving as a fluctuation because the fifth partial of one voice and the fourth of the other are fourteen cents apart and fourteen cents at that height is nine hertz.

Why it is a partial and not a note

Two notes a third apart are not close enough to beat. Their fundamentals are 220 and 277 hertz, fifty-seven hertz apart, which is not a fluctuation at all but two separate pitches.

What is close enough is a pair of partials. A harmonic spectrum has components at every multiple of its fundamental, and a just major third is the ratio 5:4 precisely because at that ratio the lower voice’s fifth partial and the upper voice’s fourth land on the same frequency. Take the interval pure and they land on it exactly; take it tempered and they land fourteen cents apart.

Fig. 2 The series an earlier essay on beating established, at the detuning a tempered third puts on its coincidence. The pair in question is not the fundamentals: it is one specific pair of partials, and the beat between them is the whole audible consequence of the temperament on this interval. What the series adds is that once the coincidence is mistuned, the next coincidence up — the tenth against the eighth — is mistuned by the same cents and beats twice as fast.

This is not a new observation about temperament. Somebody has to pay the comma is exactly the essay about where fourteen cents goes, and every tuner in the collection’s history has heard it as beating.

What is new is what it does when there is more than one singer on each side.

The product, not the sum

Fig. 3 Every beat rate between the fifth partial of one section and the fourth of the other, for sixteen singers against sixteen with a fifteen-cent intonation spread: 256 pairs. The line is the rate two perfectly in-tune sections would still have. The distribution is spread over the whole audible range of beating and a fifth of it is past the point where a beat stops being a beat. The buttons play the tempered interval and the pure one.

Sixteen against sixteen is 256 beating pairs, where sixteen singers within one section is 120. Two sections do not have twice a section’s problem; they have rather more than twice, and the pairs are spread around a rate that is not zero.

Fig. 4 The same two sections as they grow. The pair count under the axis is the product and the two curves are what fraction of those pairs are fast enough to be roughness and slow enough to be counted. At one against one there is a single rate and it is countable. At twenty-four against twenty-four there are 576 rates, two fifths of them past the roughness boundary, and nothing countable is left.

That is the same shape as the unison rung’s finding and with one thing added. A unison section’s beat rates are distributed around zero — the more singers, the more thoroughly the beating smears into a texture with no rate in it. A section interval’s rates are distributed around the temperament error, so the smearing happens around a centre that is 8.7 hertz away from where beating stops.

A larger section does not fix a tempered third. It converts a beat into roughness.

It is worth being careful about what the growth curve is and is not saying. The share of pairs on the roughness side rises with section size only because a larger draw has a wider range — the distribution of rates is the same at every size, and what changes is how thoroughly it is filled in. Two singers against two have four pairs drawn from that distribution and might get four slow ones; twenty-four against twenty-four have five hundred and seventy-six and will get some of everything. So the finding is about reliability rather than about severity: a big section reliably produces the whole distribution, and a small one is a lottery.

That is the same argument the unison rung made about why a choir sounds different from a soloist, and it is worth noticing that it is an argument about sampling rather than about acoustics. Nothing in it needs the singers to interact.

The instruction every choir gets

It also puts a number on something choral directors say and cannot quantify, which is that a section sounds worse in a chord than in a unison. Sixteen singers in unison have 120 pairs beating around zero; the same sixteen against another sixteen at a third have 256 beating around nine hertz. The count more than doubles and the centre moves from a place where a wide spread is harmless to a place where it is not.

The remedy is the one choral directors have given for four hundred years and which this collection has otherwise treated as a matter of taste: sing the third pure.

Fig. 5 The identical two sections with the interval taken pure rather than tempered. The nominal rate is now zero — the coincidence is exact — and the distribution collapses toward the left. The roughness share falls from 22 per cent to 14, and the median rate falls with it. What does not go away is the spread itself, which is the singers and not the tuning.

Tuning the third pure removes the centre and cannot remove the spread. The 8.7 hertz goes; the fifteen-cent intonation of sixteen humans does not, and it leaves a distribution of rates around zero that is exactly the unison rung’s distribution transplanted onto a partial five times higher.

That is worth stating as a limit on the advice. A pure third from two sections is not a pure third; it is a third with no systematic beat and the same random one it always had, scaled up by the partial number. The reason expressive intonation can be argued about at all is that the systematic part is the only part anybody controls.

There is a second reading of the pure-third figure that is not about choirs at all. The rate a listener would have to tune away is 8.7 hertz at A3 and it scales with the pitch: the same tempered third an octave up beats at 17.5, which is past the boundary altogether, and an octave down at 4.4, which is a comfortable countable beat. So the same interval in the same temperament is a beat in the bass, a fast flutter in the middle and a roughness in the treble — which is the register result the partial-beat ladder found for a chorused unison, arriving here for a tempered interval and for the same reason.

Which interval is worst, and why the fifth is free

The hero figure’s ordering is the practical content and it is not what a table of temperament errors would suggest.

Equal temperament’s fifths are two cents narrow and its thirds fourteen cents wide, a ratio of seven. But the beat rate is the error in cents converted to hertz at the coinciding partial’s frequency, and the third’s coincidence is at the fifth partial while the fifth’s is at the third. So the third beats at nearly twelve times the fifth’s rate rather than seven times, and the minor sixth — sixteen cents at the eighth partial — is worse than either at fourteen hertz.

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. 6 The other reason this is about intervals rather than about notes. Two partials beat only when they are inside one critical band; fourteen cents at the fifth partial of A3 is well inside it at every pitch on the figure. So the coincidence’s beating is not in competition with anything — the two components are unresolved by construction, and the whole effect is about rate.

This gives an ordering — sixth, minor third, major sixth, major third, then fourth and fifth — and it is very nearly the ordering in which choral traditions worry about intonation. Fifths and octaves are treated as things that are either right or wrong; thirds and sixths are treated as things to be placed.

Fig. 7 The worst of the common intervals: a tempered minor third, whose sixth-against-fifth coincidence is sixteen cents out. The nominal rate is 11.9 hertz — already at the edge of countable — and 53 per cent of the pairs are past the boundary. A large choir singing a sustained tempered minor third in the middle of its range is producing roughness rather than beating, and there is no rate in it to tune away.

Which computation produced the numbers

The coincidence for an interval is found by enumeration: for each pair of partial numbers up to ten, how many cents the lower voice’s p-th partial is from the upper voice’s q-th, keeping every pair inside sixty cents and ranking them by 1/(pq), which is the product of the two partials’ amplitudes in a 1/n spectrum.

The nominal beat rate is the frequency difference between the two coinciding partials at the interval’s tempered size. Nothing is fitted; it is the temperament’s own error read at the height where it lands.

The distribution is a draw: each singer gets a deviation from a normal distribution of fifteen cents, which is the spread the choir rung measured and used, and every one of the nA × nB pairs contributes the difference between its two partials. The seed is fixed, so the figure is the same picture every time it is drawn.

The roughness share is the fraction of pairs beating faster than fifteen hertz, which is the same soft boundary the partial-beat ladder uses and says is soft. Moving it to ten or twenty moves the shares by a few points and moves no ordering.

Where the model stops

The spectrum is a bare harmonic series, and putting a real one under it is the one caveat here that can be settled rather than stated. A voice is a glottal source through a vocal tract, so the fifth partial’s amplitude is whatever the formants make it, and a vowel is two resonances that move with the word being sung. Passing a six-decibel-per-octave source through a three-formant bank at Peterson and Barney’s measured vowel centres gives the amplitude of the beating partial in each voice, and with it the two quantities the rest of this essay does not have: how much of the singer’s energy is in the pair that beats, and how deep the resulting fluctuation is.

vowel	share of the lower voice’s energy in its fifth partial
“ah”, hod	25.6%
“uh”, hud	9.6%
“oo”, hood	6.3%
“eh”, head	1.4%
“ee”, heed	1.0%
“oo”, who’d	1.2%

The vowel changes the beating partial’s share of the energy by a factor of twenty-five, and the strongest is “ah”. The coincidence for a major third on A3 sits at 1100 hertz; that vowel’s second formant is at 1090, twenty hertz away and well inside its bandwidth, so the resonance is sitting on the exact frequency the temperament error is expressed at. A quarter of the lower voice’s partial energy and a third of the upper’s is in the pair that beats 8.7 times a second.

That corrects the guess this section previously made, which named “ee” as the strong case: “ee” is the weakest of the nine, at one per cent, because its second formant is at 2290 and the coincidence falls in the valley beneath it.

Depth and audibility are different quantities and the vowels separate them. The modulation index — how far the sum dips at each beat — is near total for “ee” as well, because its two partials, though both tiny, are nearly equal in amplitude. So “ee” gives a complete cancellation of a component carrying one per cent of the sound, and “ah” gives a complete cancellation of one carrying a quarter of it. A figure of rates cannot tell those apart and neither can a figure of depths; it takes both.

One thing about that index should be said plainly, because it is the model and not the world. Each voice’s spectrum here is normalised to its own loudest partial, and on “ah” the beating partial is the loudest one in both voices — so the two amplitudes come out equal by construction and the dip is a complete null. Whether the real modulation is total depends on the two sections being balanced in level, which nothing in this collection has measured. What survives the normalisation is the share, and the share is the number that varies twenty-five-fold.

And the worst vowel is a function of the pitch, so the effect never goes away. Sweeping the same computation from A2 to G4, the vowel with the most energy in the beating partial changes as the coincidence rises past each formant — the back vowels below about 250 hertz, where their second formants are, then the front vowels above — but at seven of the eight pitches tested there is always some vowel putting between 13 and 27 per cent of the voice into the pair. A choir cannot sing its way out of this by choosing a word. It can only change which word does it, and the sustained syllables of the repertoire the effect belongs to — Gloria, Sanctus, Amen — are the open “ah” that maximises it.

Every singer is one steady frequency. They are not: a note that is never at its pitch established that an ordinary vibrato is a hundred cents wide and sweeps the beat rate through zero six times a second. A section with vibrato has no fixed beat rates at all, and the distributions above are of a quantity that is being modulated faster than it can be counted.

And the two sections are assumed independent. Singers listen and adjust, which is the entire mechanism of ensemble intonation; a section that is tuning toward the other section is not drawing from a fixed distribution.

Whose music, and where the temperament is not there

The arithmetic is about any two harmonic sources. The claim about temperament is a claim about a practice, and the practice is narrower than it looks.

Unaccompanied choral music is not sung in equal temperament and never has been. What it is sung in is a matter of long argument, but every account of it — pure thirds, drifting reference, adjusted-by-ear vertical intervals — has the property this rung explains: the vertical coincidences are placed rather than tempered. The tempered third is what happens when a choir is accompanied, because the organ or the piano is not adjusting and the singers must match it or beat against it.

That gives the effect a specific home. A cappella Renaissance polyphony has no tempered thirds in it; a nineteenth-century oratorio with organ is full of them, at nine beats a second on every sustained third in the middle of the range. The complaint about the “muddiness” of large choral forces with organ is usually put down to the building, and part of it is arithmetic that would happen in a dry room.

The other tradition worth naming is the one where it does not arise. A gamelan’s paired tunings beat by design and the table of cents does not record what the tuning is; there is no temperament being conformed to and no coincidence that is supposed to be exact, so there is nothing here for the arithmetic to be about.

What the picture cannot show

Whether 8.7 hertz is heard as a beat or as part of the sound. The vowel computation above supplies the depth and the share, which is what the figures alone do not, and it still does not answer this: a total modulation of a component carrying a quarter of the energy is a large signal, and whether a listener parses it as a fluctuation in the chord or as a quality of the chord is a question about hearing rather than about spectra.

The sections are assumed to be on one note each. Real choral writing puts the altos on a note and the sopranos on another and then moves both, so the coincidence being measured changes at every chord — and a voice is a stream rather than a sustained partial. The figures are of a held chord, which is the case the argument is easiest to state for and not the case that is usually being sung.

And nothing here is a claim about which sounds better. A beating third is a different object from a still one and both have been wanted; the celeste and the chorus exist because a fluctuation is sometimes the point.

Where this ladder goes next

Seven rungs. The larynx is a reed; two mechanisms meet at a seam; one voice can be heard over ninety players; a sung note is never at its pitch; the voice is the sound a listener knows best; a choir buys the disappearance of a beat rather than a level; and now, two sections at an interval beat between the partials the interval’s ratio brings together, at a rate the temperament sets and a count that is the product of the sections.

The rung after it is the one the vibrato limitation names, and it is the one this ladder has been avoiding since its fourth rung. Every intonation figure here treats a singer as a frequency, and a singer is a frequency being swept a hundred cents wide six times a second. A section of sixteen such sweeps, against another section of sixteen, has a beat rate that is a function of time rather than a number — and whether the result is a texture, a shimmer or nothing at all is a computation this collection can run and has not.