Sixteen sweeps against sixteen

The seventh rung of this ladder computed what two sections of a choir do to each other at an interval: not the fundamentals, which are far apart, but the partials the interval’s ratio brings together — a major third puts the fifth partial of the lower voice against the fourth of the upper, and in equal temperament those are fourteen cents apart, which at that height is nine beats a second. The number of beating pairs is the product of the section sizes, so sixteen against sixteen is two hundred and fifty-six of them.

Every one of those two hundred and fifty-six was computed as a pair of steady frequencies. Its own last paragraph said what that was worth:

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.

What a vibrato does to a beat #

Two steady tones a few hertz apart beat at their difference, and the difference is a constant. Two tones each of which is sweeping is a different object: the difference is itself a function of time, and it goes wherever the two sweeps take it.

If both voices swept in phase, the difference would be constant — the two would move together and the beat would not know. They do not sweep in phase. A choir’s vibratos are uncorrelated, so at any moment one voice is at the top of its excursion and another at the bottom, and the difference between a partial of one and a partial of the other runs from very nearly zero to the sum of the two excursions.

For a hundred-cent vibrato at the fifth partial of a 220-hertz note, the excursion is a hundred cents of 1,100 hertz, which is 64 hertz peak to peak. Two of those, uncorrelated, put the difference somewhere between nothing and about 128 hertz — against the nine hertz the steady calculation gives.

Beating is not roughness, and this pair does both #

The distinction the figure turns on is one this site has carried since beats are arithmetic: below about twenty hertz a difference is heard as a fluctuation, a wavering of one sound. Above it, the two components are inside one critical band and moving too fast to follow, and what is heard is roughness — a single harsh sound with no countable rate.

The steady calculation puts the pair at nine hertz, well inside the beating region, and the previous rung’s whole account is about a fluctuation: a choir’s major third wavers, and that is why choirs argue about thirds and not about fifths.

With vibrato the pair crosses that boundary constantly. At a hundred cents of extent it is inside the beating region twenty per cent of the time and above it for the other eighty. At fifty cents — a restrained ensemble vibrato — it is beating thirty-four per cent of the time. At twenty cents, which is about as straight as a trained voice gets, sixty-six.

Those three shares are all at A3, and the section on the boundary below shows they are strongly register-dependent — 35 per cent in the bass and 4 in the treble at the same hundred cents. So the numbers in this paragraph describe a tenor or an alto and not a choir.

So the previous rung’s nine hertz is not wrong; it is a limiting case at an extent nobody uses. What a choral third actually presents is neither a beat nor a roughness but an alternation between them at the vibrato rate — a pulsing that goes rough and smooth six times a second, which is a sound with no name in the acoustic vocabulary and a completely familiar one in a concert hall.

The thing this predicts about straight tone #

The useful consequence is not about opera. It is about the repertoires that ask for no vibrato at all, and it is the reverse of the usual explanation.

Renaissance polyphony sung with straight tone is routinely described as sounding purer, and the usual account is about intonation: without vibrato a section can tune a third exactly and the beats disappear. That account is available and this collection has the arithmetic for it.

The figures here say something else is happening as well, and it goes the other way. Straight tone does not remove the beat between the two sections’ near-coinciding partials; it makes that beat audible for the first time, because it stops the pair spending most of its time in the roughness region. A straight-toned equal-tempered third does not sound pure; it wavers at nine hertz, steadily, and it is the one condition under which the previous rung’s number is a description of an experience.

Which is why straight-tone ensembles tune. Not because vibrato hides bad intonation — the usual and slightly sniffy version — but because straight tone exposes a specific quantity, the difference between the fifth partial of one part and the fourth of another, which vibrato was smearing into a texture. An ensemble that has removed its vibrato has acquired a beat it now has to do something about, and what it does about it is sing the third fourteen cents flatter.

Where the boundary itself comes from #

The crossing this essay is about needs a boundary, and it is worth being explicit about how soft the boundary is, because the headline percentages are shares of time on either side of it.

There are two boundaries here and it is easy to run them together, so it is worth separating them before the register question.

The critical band decides whether two components interact at all. Two inside one band interfere and produce a single fluctuating or rough sound; two in different bands are simply two sounds. At the fifth partial of A3, 1,100 hertz, that band is about 143 hertz wide.

A rate of about fifteen to twenty hertz decides whether the interference is countable. Below it the fluctuation is a beat; above it the same interference is too fast to follow and is heard as roughness. That limit is temporal and is very nearly fixed in hertz, because what sets it is how fast a listener can track an amplitude envelope rather than anything about frequency.

Substituting one for the other is a mistake with a visible consequence. Re-running the whole computation with the critical band as the boundary instead of the rate limit gives a beating share of 100 per cent at every register and every vibrato extent, because the excursions never exceed 143 hertz and everything counts as “beating” — an answer that says a hundred-cent operatic vibrato produces a countable pulse, which it does not. The rate limit is the right boundary and the band is the wrong one.

With that settled, the register question the essay owes has an answer, and the register dependence is large:

The share halves with every octave, because the boundary is fixed in hertz and the excursion is fixed in cents — so a hundred cents is twice as many hertz an octave up and spends twice as long past the limit. The twenty per cent this essay quotes is an A3 number and nothing more general: two basses singing a third are beating a third of the time and two sopranos four per cent of it.

Which is a prediction about who complains. A choral third in the bass presents an audible waver even with full vibrato; the same third in the treble presents roughness almost continuously. That is the opposite of the register effect the roughness ladder usually reports, and it is because this one is about a rate rather than about a bandwidth.

So the band that decides everything here is not a fixed width in hertz and the window is not a fixed length in time. Both scale with the register, which is why one number for “how rough a fourth is” cannot be right and why this essay’s answer has to be a curve up the compass rather than a value.

What a soloist does instead #

One voice with vibrato against an accompaniment is a different problem from two sections, and it is worth separating because the same hundred cents produces the opposite effect.

A soloist’s vibrato has nothing to beat against except the accompaniment, and the accompaniment’s partials are fixed. So the difference between the singer’s fifth partial and an instrument’s fourth sweeps back and forth through zero rather than between two moving values — it passes through unison twice a cycle, every cycle, regardless of how the singer is tuned. A vibrato of a hundred cents makes the intonation of a soloist’s third almost undiscoverable, which is what the fourth rung found and is the reason a singer can be a semitone’s worth of vibrato wide and still sound in tune.

Two sections do not get that. Each section is a spread of moving voices against another spread of moving voices, and there is no fixed reference for either to pass through. What the ensemble presents is a distribution, and a distribution does not have an intonation in the way a single sustained interval does.

There is one thing a section retains that a soloist does not, and it follows from the same asymmetry. A soloist’s sweep passes through exact coincidence twice a cycle whatever the tuning is, so tuning changes nothing about whether the null happens and only when. Sixteen voices against sixteen have 256 differences and no reason for any of them to reach zero, so a section’s mistuning cannot be cancelled by anybody’s vibrato phase — it sits in the distribution’s centre, which is where the previous rung put it. Vibrato hides a soloist’s intonation and does not hide a section’s; it only makes a section’s harder to hear by moving most of the evidence past the countable rate.

Inside that band the peak keeps most of its height, which is the answer to the question the crossover raises: a pitch extractor handed sixteen sweeping voices still finds a pitch, and it finds it because real vibrato is narrow enough to leave the periodicity intact. The extent singers actually use is not an arbitrary point in the range.

Which computation produced the numbers #

The near-coinciding partial pair is found by the same enumeration the seventh rung used: for an interval of r, look for small whole p and q with p close to q·r, and take the closest. For four semitones that is five against four, fourteen cents apart in equal temperament.

Each singer is a phase — a uniform random offset into the vibrato cycle, from a seeded generator so that the figure is the same every time it is drawn. At each of three hundred instants the deviation of every voice is evaluated, its partial’s frequency computed, and the absolute difference taken over all sixteen-by-sixteen pairs. The band drawn is the tenth and ninetieth percentiles of those 256 numbers and the line is the median.

The vibrato itself is the site’s own model from the fourth rung: a sinusoidal deviation in cents, so the excursion is a constant interval and grows in hertz with the partial number — which is why the fifth partial of a note swings five times as far in hertz as the fundamental does, and why this effect is a partial-pair effect rather than a fundamental effect.

The boundary between beating and roughness is taken at the site’s own constant, and it is a soft boundary being used as a hard one. That is stated rather than hidden: the percentages quoted move by several points if the boundary is moved by a few hertz, and the ordering — more vibrato, less time beating — does not.

Where the model stops #

Uncorrelated is an assumption and it is the load-bearing one. Singers in a section listen to each other, and there is a published literature on whether vibratos in an ensemble entrain. If they did, even weakly, the difference between two voices would fluctuate less and the pair would spend more time near its steady rate. Nothing here measures it, and the whole result is a claim about what happens when sixteen oscillators are independent.

The vibrato is a sine and a real one is not. Measured vibrato has an asymmetric cycle, a rate that varies within a note, and an onset that takes a few hundred milliseconds. The onset in particular matters here: a note that begins straight and acquires vibrato begins in the beating regime and moves out of it, which would be a structure within the note that this model, running at a constant extent, cannot show.

And amplitude is not modelled at all. A real vibrato modulates loudness and spectrum as well as pitch, so the two partials whose beating this essay is about are also varying in strength — and a beat between components of unequal and varying amplitude has a varying depth as well as a varying rate. What is drawn is the rate alone.

Nor is anything here a measurement. Every number is a model of a vibrato evaluated at published parameters. The measurement that would settle it is a recording of two sections at a third with the individual voices separable, which is not a thing this collection has or can make.

Whose singing, and when #

The hundred-cent, six-hertz vibrato is a specific object with a date: it is the operatic voice of the twentieth century, and the extent in particular is much larger than what is documented in nineteenth-century treatises, which describe vibrato as an ornament rather than as a default.

That makes the hero figure a claim about a repertoire rather than about voices. Verdi and Wagner in a large hall with modern voices is the far right of the extent curve — a third that is rough four fifths of the time, with the roughness itself pulsing. Palestrina with a straight-toned consort is the far left, where the same third is a steady nine-hertz waver unless the ensemble flattens it. Those are not two performances of the same acoustic object.

And there is a middle that is worth naming because it is where most choral singing lives. An amateur choir has neither a trained hundred-cent vibrato nor the control to sing straight; what it has is a spread of intonation between singers, which is the seventh rung’s other variable and produces a static spread of beat rates rather than a moving one. Three different acoustic situations, one written interval.

What the picture cannot show #

What any of it sounds like. The figure draws a rate against time and a listener does not hear a rate. What a listener hears is a texture whose harshness fluctuates, and whether that is heard as one sound with a character or as a defect is a question about auditory scene analysis that this collection is only two rungs into.

And it cannot show the other three hundred pairs. The figure takes the closest coinciding pair, which is the loudest contributor and not the only one. A major third also brings the tenth partial against the eighth, the fifteenth against the twelfth, and so on, each with its own rate and each smeared by the same vibratos. The total roughness is a sum over all of them and this figure is the largest term in it.

Where this ladder goes next #

Eight rungs. The larynx is a reed; two mechanisms meet at a seam; one voice carries 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; two sections beat between partials at a rate the temperament sets; and now those partials are moving, and the rate is a distribution rather than a number.

The rung after it is the one the alternation names. What this figure produces is a pair crossing the beating-roughness boundary six times a second, and the site has no model of what a listener does with a roughness that is itself modulated — every roughness figure in the consonance ladder computes a single number for a steady spectrum. A time-varying roughness has a mean, a depth and a rate, and the rate here is the vibrato’s own; whether a listener integrates it, tracks it, or hears the depth as a quality of the chord is a question the machinery can pose and the arithmetic alone cannot settle.