A roughness with a rate of its own

Sixteen sweeps against sixteen took two sections of singers an interval apart and let every voice actually sweep, which turned the seventh rung’s beat rate from a number into a distribution. It ended by naming what the site could not then do with the result: the pair it produces crosses the beating–roughness boundary six times a second, and every roughness figure in this collection computes a single number for a steady spectrum.

That is not a small omission. It is an assumption inside the model that has never been stated because there has never been anything to state it against. Plomp and Levelt’s sum takes two frequencies and returns a scalar; a vibrato means there are no two frequencies.

Doing it properly costs one loop. Evaluate the sum at every instant of a cycle instead of once at the nominal pitches, and what comes back is a curve rather than a number.

Why the average of a curve is not the curve of the average #

The reason is one line of mathematics and it is the whole essay.

Roughness as a function of the distance between two partials is a curve: near zero it is small, it rises to a maximum at about a quarter of a critical band, and it falls away again. It is not a straight line anywhere, and a vibrato is a distribution of frequencies rather than a point.

For any curved function, the average of the function over a distribution is not the function of the average. Where the curve is convex — a minimum — the average is higher than the value at the centre. Where it is concave — a maximum — the average is lower. That is Jensen’s inequality, and here it has a musical consequence that is easy to state and was not obvious in advance:

A vibrato makes consonances rougher and dissonances smoother.

The size of the effect is set by how curved the roughness function is where the interval sits, which is why the octave is the extreme case. A pure octave is the deepest and narrowest minimum in the whole dissonance curve — every partial of the upper note coincides exactly with a partial of the lower, so there is almost nothing left to be rough. Move fifty cents either way and coincidences become near-coincidences, which is where roughness lives. The average over the sweep is nineteen times the value at the centre.

At the other end, a major seventh sits near a maximum, and vibrato takes about five per cent off it.

Three quantities where there was one #

The curve has a mean, which is what Jensen’s inequality moves; a depth, peak to peak over that mean; and a rate, at which it fluctuates.

The rate is the interesting one, because it is not necessarily the vibrato’s. With a vibrato on one voice the roughness follows the vibrato, at six hertz. With a vibrato on both — which is the choral case, and the one a section against another section is about — the two sweep past each other twice per cycle, so the roughness fluctuates at twelve.

Six hertz is a rate that can be counted: it is squarely in the range this collection calls beating rather than roughness, which the beat ladder puts at anything under about fifteen hertz. So a sustained interval under vibrato has a roughness that is itself a beating quantity — a sensation with a rate, which is exactly the kind of thing this subject usually reserves for the signal rather than for a percept computed from it.

Twelve hertz, for two singers, is close enough to that boundary to be worth naming as a place where the recursion gets uncomfortable.

How much of this is a real vibrato #

Fifty cents either side is a large vibrato and a real one for an operatic singer; a violinist’s is smaller and a flautist’s smaller still. Sweeping the extent separates the two effects cleanly and shows they scale differently.

The depth rises steadily with extent and is the effect that is always there: at any interval, a wider vibrato swings the roughness by more. The shift depends on where the interval sits. For a major third it goes from one to about 1.1 across the whole range — a correction. For an octave it goes from one to twenty-four.

So on most intervals the fluctuation is the story and the mean shift is a footnote; on the intervals that are supposed to be perfectly smooth, the mean shift is the story.

What this says about who uses vibrato and where #

The result has a direction and the direction matches practice, which is worth stating carefully because it is easy to overclaim.

Vibrato is used on sustained notes and abandoned on fast ones, and it is used most where a line is exposed and least where a chord has to lock. Choral directors ask for straight tone at cadences; barbershop and early-music consorts avoid it entirely; the organ, which cannot do it at all, is the instrument whose tuning is argued about most.

Under this arithmetic all of those are one fact. A vibrato costs a consonance most of what makes it a consonance. The intervals that lose the most are exactly the ones a chord is trying to lock — the octave, the fifth — and the intervals that lose the least are the ones nobody is trying to make smooth anyway.

That does not make vibrato a mistake. What it takes away is the sensory smoothness of a held interval, and what it buys is everything the voice ladder’s other rungs are about: audibility over an orchestra, a source a listener identifies instantly, and a note whose pitch is a running average rather than a target to be hit. A note that is never at its pitch is the essay about that trade from the pitch side, and this is the same trade priced in roughness.

The curve the whole argument is about #

Everything above depends on the shape of one function, and it is worth having it in view rather than described. The dissonance curve for a harmonic timbre is a sequence of minima at the simple ratios separated by broad maxima, and the depth and width of each minimum is what decides how much a vibrato costs there.

The octave’s minimum is the deepest and narrowest because every partial coincides exactly. The fifth’s is next. The major third’s is shallow, because a 5:4 puts the fifth partial of the lower note against the fourth of the upper and everything above that is a near miss rather than a hit. The major seventh has no minimum at all where it sits.

Read the hero figure against this one and the pattern is exact: the intervals with the sharpest minima are the ones the moving average lifts most.

The same effect on a section rather than a pair #

The eighth rung’s figure is the population version of this one and the two say different things, which is worth being clear about.

There, sixteen voices against sixteen give two hundred and fifty-six pairs, each with its own phase, and what the figure reports is the distribution of instantaneous beat rates — how much of the time the pair is inside the beating band and how much inside the roughness band. Here there are two voices and what is reported is the roughness itself.

The relation between them is that a section averages the fluctuation away and keeps the shift. Two hundred and fifty-six pairs at independent phases have a nearly constant total, so a choir does not have a twelve-hertz wobble in its roughness — but every one of those pairs is still contributing its own moving mean rather than its still value, so the shift survives the averaging entirely.

That is the useful consequence for a choir. A section’s vibrato costs it the sensory smoothness of its consonances and does not give it an audible fluctuation in exchange, which is a worse bargain than a soloist’s and is close to what choral practice has always said.

Which computation produced the numbers #

The roughness is dissonanceSpectrum, unchanged — the same double sum over pairs of partials this collection has used since roughness can be computed, with a string spectrum of eight partials. Nothing about the model has been altered; it is evaluated four hundred times over a second instead of once.

The vibrato is vibratoCents, the voice ladder’s own: a sinusoidal modulation of the log frequency, so the excursion is the same number of cents at every partial and a different number of hertz at each. The extent is stated as cents either side of the nominal, which is the convention the fourth rung established, and the typical values are that rung’s.

The mean is the arithmetic mean of the instantaneous roughness over whole cycles. The depth is the peak-to-peak range over that mean, so it is dimensionless and comparable between intervals whose absolute roughnesses differ by two orders of magnitude. The still value is the same function evaluated once at the nominal frequencies.

Sampling four hundred times a second over one second is far more than the six or twelve cycles need; halving it changes the reported means by under a tenth of a per cent.

Where the model stops #

A roughness is not something a listener has a time series of. The whole apparatus computes a sensory quantity instant by instant and then reports statistics of it, which assumes the sensation follows the signal without delay. It does not: roughness is a percept built over some tens of milliseconds, and a dissonance has to last is this collection’s own essay about exactly that integration time. A twelve-hertz fluctuation has a period of eighty milliseconds, which is the same order, so the smoothing is not negligible and is not here.

The vibrato is sinusoidal and a real one is not. Measured vibrato is closer to a triangle with an uneven duty cycle, and its rate wanders. A triangle spends more time at its extremes than a sinusoid does, which would make the shift larger and not smaller.

And there is no amplitude modulation. A singer’s vibrato drags every partial past a fixed set of formants, so partials get louder and quieter as they sweep — which changes the roughness through the amplitudes as well as through the frequencies, and is a second modulation with the same period.

What the picture cannot show #

It cannot say whether nineteen times is audible. The number is a ratio of two model outputs. What a listener does with a roughness that has a mean, a depth and a rate is precisely the question the eighth rung named and this one has not answered — it has only turned one number into three, which is the arithmetic half.

Nor can it show what happens in a chord. Two notes are two notes; a chord of four voices with independent vibratos has six pairs, each with its own phase relationship, and the total is not the sum of six independent fluctuations because the phases are not independent within a singer.

And it cannot separate the mean from the depth as things a listener uses. They are two statistics of one curve and nothing here says a listener has access to either. The honest statement is that the still value — the number every roughness figure on this site reports — is not one of the three.

Whose singing, and when #

The extents used here are those of Western operatic and orchestral practice since roughly the middle of the nineteenth century, which is when continuous vibrato becomes the default rather than an ornament. Before that it is an ornament, named as one and written as one, and the string treatises of the eighteenth century treat it exactly the way they treat a trill.

That history and this arithmetic fit together in a way worth noticing. Continuous vibrato arrives at the same time as the large orchestra, the large hall and equal temperament as a practical standard — and it arrives after the repertoire whose whole argument is about the exact tuning of sustained consonances. A sixteenth-century consort holding a pure major third is holding an interval whose sensory smoothness is the point, and this figure says a vibrato would spend it. The practice and the arithmetic agree about which repertoires can afford one.

The prediction this essay was going to end on #

The obvious next move is to run the moving roughness through the temporal window that a dissonance has to last established, since both statistics above are things a listener could only have through such a window. The expectation before doing it was that both would shrink: the octave’s factor of nineteen smaller than it looks, and the twelve-hertz fluctuation smaller still.

Both halves of that are wrong, and the first is wrong for a reason that needed no computation at all. The window is a running mean, and a running mean cannot change a mean. Averaging the trace over cycles of the fluctuation leaves the average over whole vibrato cycles exactly where it was — the drift is zero to four decimal places at every interval tested. The Jensen shift is not a fragile artefact of computing a sensory quantity at a rate no listener has access to; it is the one part of this essay that survives the integration untouched.

The second half is wrong in an interesting direction. What the window does destroy is the depth, and how much it destroys depends on the ratio of two times — the window’s length against the fluctuation’s period — and the window’s length is set by the roughness rate, which is the beat rate between the nearest pair of partials.

The ordering inverts. The intervals whose fluctuation is smoothed away are the semitone and the major seventh, which lose nine tenths of it; the octave keeps three quarters. That is the opposite of what the shift does, and it happens because a fast roughness rate buys a short window: the octave’s partials beat at two hundred and twenty-three hertz, so four cycles of that is eighteen milliseconds, which is a ninth of the vibrato’s period and smooths almost nothing. The semitone’s partials beat at twenty-seven, so its window is a hundred and forty-six milliseconds and swallows most of a cycle.

So the honest summary is that a vibrato costs a consonance its smoothness and leaves the resulting wobble audible, while on a dissonance it changes little and the wobble is averaged out. The mean survives the window is the rung that does this properly, across registers as well as intervals.

Where this ladder goes next #

Nine rungs to here. 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; and now the roughness that motion produces.

What was owed was the integration, and the section above is the part of it that belongs here: the check that this rung’s own numbers survive being read through a window. They do, and the reason the mean survives is arithmetic rather than a finding about hearing.

What is owed now is the amplitude side. Every figure above modulates frequency and holds the partial amplitudes fixed, and a real vibrato drags every partial across a fixed formant, so the amplitudes modulate too — at the same period, with a phase set by where each partial sits on the formant’s flank. That is a second modulation of the same sum, and nothing in this collection has computed it.