The mean survives the window

A roughness with a rate of its own took the roughness of an interval and let the vibrato move it, and got three quantities where the ladder had one: a mean, a depth and a rate. It ended with a guess about what a listener does with them:

Running the moving roughness through that window would say which of the three quantities survives to be perceived, and it would probably say that the octave’s factor of nineteen is smaller than it looks.

The window is available and the guess is wrong, for a reason that is arithmetic rather than psychology.

The window is not a free parameter #

A dissonance has to last is the essay that gives this rung its integration time, and the important thing about it is that it does not give a number of milliseconds.

Roughness is itself a fluctuation. Two partials a small distance apart beat at their difference frequency, and the sensation of roughness is what that beating becomes when it is too fast to count. The criterion for whether a dissonance registers is therefore a number of cycles of its own fluctuation, not a duration — which is why a semitone at the bottom of a cello and the same semitone two octaves up are different perceptual objects.

So the window’s length is cycles / rate, and the rate is computed for each interval at each register. A major third on A3 fluctuates at 68 hertz, so four cycles is 59 milliseconds. A semitone at the same pitch fluctuates at 27, so four cycles is 146. An octave fluctuates at 223, so four cycles is 18.

The window is between three and eight times shorter for a wide interval than for a narrow one, and that turns out to decide everything in this essay.

Why the mean cannot move #

The result the ninth rung guessed at is settled by one line of arithmetic, and it is worth stating plainly because the guess was reasonable.

A running average over a window is a linear operation, and its output’s mean over a whole period of a periodic signal is the input’s mean over that period. Exactly. So the average roughness of an interval under a vibrato is the same before and after the window: 0.1487 against 0.1487 for the major third here, and identical at every interval tried.

The ninth rung’s headline was the Jensen factor: because roughness is a curved function of the frequency difference, the average of the curve is not the curve of the average, and an octave — which sits in the deepest and narrowest minimum there is — comes out nineteen times rougher under a fifty-cent vibrato than it is when held still. That factor is a statement about the mean, and the mean is exactly the statistic a running average preserves.

So the octave’s factor stands. A listener integrating over any window at all still gets a mean roughness nineteen times the still-tone value, and the temporal window offers no relief whatever.

That is a genuinely surprising thing for a temporal window to do, because a window’s whole business is smoothing — and it says something worth carrying: smoothing removes movement and cannot remove level. Any argument of the form the ear averages, so the effect is smaller is wrong about a mean and right about a fluctuation.

What the window does take, and where #

The depth is what goes, and how much of it goes is a strong function of the interval — in a direction that inverts the raw result.

The semitone has the largest raw fluctuation and keeps the least of it. Its depth under a vibrato is 0.53 of its own mean, the largest of any interval in the range, and after the window it is 0.07 — twelve per cent survives.

The major seventh is nearly as bad: 0.39 down to 0.08, twenty per cent.

The middle of the range keeps most of it. A minor seventh keeps 85 per cent, a major third 77, a minor third 70.

The octave keeps 70 per cent of a raw depth of 1.24, which is by far the largest absolute fluctuation in the range.

The mechanism is the window’s own length. A narrow interval fluctuates slowly, so its window is long — 146 milliseconds for the semitone — and a 146-millisecond window against a vibrato period of about 170 milliseconds flattens almost everything. A wide interval fluctuates fast, its window is short, and the vibrato passes through it nearly intact.

So: the intervals whose roughness moves most under a vibrato are the ones whose movement a listener is least able to follow, and it is the same quantity — the roughness’s own rate — that produces both halves of that sentence.

What that predicts about a singer #

There is a musical reading of this and it is testable in a way most of this ladder’s claims are not.

A vibrato on a wide interval produces a roughness that visibly moves: an octave’s roughness swings over 70 per cent of its own mean at a rate of twelve hertz, which is well inside the range a listener hears as a fluctuation rather than as a texture. A vibrato on a semitone produces a roughness that is nearly steady, at a level that is high.

So two singers a semitone apart with vibrato should sound rough and not unsteady; two singers an octave apart with vibrato should sound fluttery. That is a prediction about a quality rather than about a quantity, and it fits the way choral directors describe the two cases — a clashing second is described as harsh, and an octave doubling with wide vibrato as wobbly.

It is also a prediction about registers, which is the cleaner test. The window’s length depends on the interval’s own roughness rate, and that rate rises with pitch — so the same interval higher up has a shorter window and keeps more of its movement.

The shape of the curve is why the mean is high #

It is worth being explicit about where the Jensen factor comes from, because the window’s failure to touch it is only interesting if the factor itself is real.

Roughness can be computed draws the dissonance of an interval as a function of the interval, and the shape is the whole argument. It has deep, narrow minima at the simple ratios and broad maxima between them, and near a minimum the curve is convex. A vibrato takes the interval on a tour either side of its nominal value, and averaging a convex function over a tour gives more than the function at the centre.

The octave is the extreme case because its minimum is the deepest and narrowest there is: every partial of the upper note lands exactly on a partial of the lower, so the still-tone roughness is nearly zero and any excursion at all is a large multiple of it. A fifty-cent vibrato takes the pair well outside the minimum on both sides, and the average is nineteen times the centre value.

A major seventh is the opposite case. It sits near a maximum, where the curve is concave, so averaging gives less than the centre — five per cent less.

That is why the window cannot help. The factor is produced by the shape of the curve and by where the vibrato takes the interval on it, and both of those are properties of the excursion rather than of time. Nothing that happens after the roughness has been computed can undo an average that was taken before it.

They are not the same number, because roughness is not linear in the interval — so averaging over a sweep is not the same as evaluating at the centre of it, and the difference is the quantity this essay is about. The curve the averaging happens on is what decides the sign of the discrepancy at each interval.

What this does to the ladder’s other claims #

Three rungs of this ladder have results that the window touches, and it touches them differently.

What a choir does that a soloist cannot is about a section of voices whose vibratos are uncorrelated, which produces a much faster and shallower fluctuation than one voice does. A faster fluctuation is more thoroughly flattened by any window, so the choral case loses more of its movement than the solo case — which is consistent with a section sounding smooth where a soloist sounds alive.

A section against another section computes an ensemble’s roughness at an interval as a count of near-coincidences. That is a mean quantity and the window does not touch it.

Sixteen sweeps against sixteen is about partials crossing the beating–roughness boundary six times a second. That is a rate claim, and it survives the window in the same way the depth does: partially, depending on the interval.

So the window sorts the ladder’s results into two kinds. Everything that is a level survives untouched; everything that is a movement is reduced by an amount this figure computes.

It has to be flat, and that is the useful part: the period, the peak’s width and the swing the vibrato puts into it all scale together with the pitch, so the ratio between them cannot depend on register. A result that is invariant by construction is worth drawing precisely so that it is not mistaken for a measurement.

What a window is for, if not this #

The window’s failure here is a good moment to say what it does elsewhere, because it is not a useless instrument.

Its real subject is duration. A dissonance has to last prices how long a rough interval has to be held before it is heard as rough at all, and the answer is a number of cycles of its own fluctuation — which converts directly into a rule about note lengths. A semitone at the bottom of a cello needs 146 milliseconds to complete four cycles, so a semiquaver at a brisk tempo is too short for it to register; the same semitone two octaves up needs 37, and registers easily.

That is a claim about counterpoint. The rule that a dissonance may pass if it is short is a claim about exactly this number, and the ladder’s contribution is to say the rule should be different in different registers — which is what part-writing practice does, without saying why.

Applied to a steady interval the window decides whether the roughness exists. Applied to a moving one it decides only whether the movement is visible. The two uses look like the same operation and answer completely different questions.

Which computation produced the numbers #

The instantaneous roughness is the ninth rung’s: the dissonance of two spectra evaluated at every instant of a vibrato cycle, with the frequencies moving according to the vibrato’s extent and rate rather than being held at their means.

The window is a running mean of length cycles / roughnessRate, where the rate is the amplitude-weighted mean beat frequency over every pair of partials — the quantity a dissonance has to last introduced. The criterion of four cycles is that essay’s own and is asserted there.

The average wraps rather than shrinking at the ends of the sample, because the vibrato is periodic and a shrinking window would report a depth that is an artefact of where the sample was cut.

Depth is peak-to-peak over the mean, both before and after, so the two are directly comparable and the survival is a ratio of two ratios.

Where the model stops #

A running mean is the crudest possible window. A real temporal integrator is more nearly exponential, with a longer tail and a faster onset, and it would flatten a periodic fluctuation slightly more at the same nominal length. The mean-preservation result is exact for any window with unit gain at zero frequency, which includes every plausible one, so that half is robust; the survival percentages are not.

Four cycles is asserted. It comes from one essay and it is the criterion that whole rung is built on. Both halves of the reassurance that used to sit here — that halving it roughly doubles every survival figure and changes no ordering — turn out to be wrong, and the sweep is worth reporting because the thing that is robust is a different thing.

Halving the criterion does not double anything uniformly. The ratio between the two-cycle and four-cycle figures runs from 1.07 to 4.73 across the twelve intervals: the semitone nearly quintuples and the minor seventh moves by seven per cent. And the ordering does change — the major second is ninth of twelve at four cycles and fourth at two, and at eight cycles the octave is top of the list rather than middle.

What survives every setting is the result the essay is named for. The mean before the window and the mean after agree to 2.8 × 10⁻¹⁶ — machine precision — at one cycle, two, four and eight, because that is an identity rather than a computation. And the pair at the bottom is stable at 2, 4 and 8: the semitone and the major seventh keep least of their movement wherever the criterion is put, which is the claim the musical reading rests on.

The vibrato is a sinusoid. A real singer’s vibrato is not quite sinusoidal, not quite periodic and not quite steady in rate, and a note that is never at its pitch is the rung about how much.

And roughness is one model. Everything here is the Plomp–Levelt kernel with amplitudes from a stated spectrum, which is one of several published accounts of sensory dissonance and the one this collection uses throughout.

What the picture cannot show #

It cannot show what a listener does with a fluctuation. The window says how much of the movement reaches whatever comes next; it says nothing about whether a fluctuating roughness is judged as more unpleasant than a steady one of the same mean, and that is the question a musician would ask.

Nor can it show the pitch. A vibrato is heard as a pitch modulation as well as a roughness modulation, and the pitch percept has its own integration — a listener hears a steady pitch under a wide vibrato, which is a much larger act of averaging than anything here.

It cannot show the critical band moving. The roughness of a pair depends on how far apart they are relative to the auditory filter at that place, and a third is rougher in the bass is the essay about how much that varies. A vibrato moves both partials, so it moves them slightly within the filter as well as relative to each other, and only the second is modelled.

And it cannot show two vibratos at different rates. Two singers with vibratos at six and seven hertz produce a roughness with a beat in it at one hertz, which is far too slow for any window to flatten. That case is on the ladder already and this figure does not reach it.

Whose singing, and when #

The vibrato modelled is the Western operatic one: a rate around six hertz and an extent of a few tens of cents either side, which is the range measured across recordings of the twentieth century and which the fourth rung of this ladder sets out.

That is a specific practice with a date. Renaissance and early baroque singing treatises describe vibrato as an ornament applied deliberately rather than as a continuous property of the tone, and much recorded early-music singing since the middle of the twentieth century uses very little. On a straight tone the whole of this essay collapses: there is no fluctuation, the mean is the still value, and the Jensen factor of nineteen does not arise.

So this rung, like the ninth, is about a particular way of singing rather than about singing. A choir singing straight and a choir singing with operatic vibrato differ in the roughness of an octave by a factor of nineteen, and neither of them is doing it wrong.

Where this ladder goes next #

Ten rungs. The larynx as a reed; two mechanisms and a seam; one voice over ninety players; a note that is never at its pitch; the sound a listener knows best; what a choir buys; two sections beating between partials; those partials moving; the roughness that motion produces; and now which of its three statistics a listener can have.

What is owed after this is the pitch. Everything in the last three rungs treats a vibrato as a modulation of a sensory quantity, and the reason singers use one is that it does something to the note — a listener hears a steady pitch, more presence, and a voice that carries through a texture. This collection has an account of how a pitch is extracted from a set of moving partials, on the missing-fundamental ladder, and it has never been asked what that extractor does with a signal whose partials are all sweeping together at six hertz. That is the same integration problem as this rung’s, one level up, and it is where the reason for the practice actually lives.