A vibrato hides a choir's tuning above four hundred hertz

Every figure in the earlier essays on the singing voice draws a vibrato that was already running when the picture began. A real one is not. A singer starts most notes on a straight tone, and the vibrato grows out of it over the first two or three hundred milliseconds — so the front of every sung note is a different signal from the rest of it: a definite pitch, no sweep, none of the amplitude modulation the partial that gets louder as it goes sharp found the formants imposing.

The rate that does not rise with the partial ended by naming that front edge as the last thing about a singing section that could be computed without a listener. It is the window in which a listener decides a great deal about a note. For a section of singers on one note, one of the things it might decide is how well they are tuned, because the straight start is the one stretch in which their mistuning is a set of steady beats rather than a set of sweeps.

Computing it turned up something larger than the onset. Before asking what the straight start gives back, it is necessary to know what the vibrato takes away — and the answer depends on frequency in a way that divides the spectrum of a choral note in two.

Counting the beats a listener can follow #

Every partial beats at its own rate set out the arithmetic of a mistuned unison: two voices a few cents apart differ by a number of hertz proportional to frequency, so their k-th partials beat k times as fast as their fundamentals. The line between a beat and roughness is put here, as in the earlier essays, at fifteen a second: slower, and the waxing and waning is a fluctuation a listener follows; faster, and it is heard as a texture.

A vibrato changes that in a way a single number cannot describe. With both voices swinging by seventy cents six times a second, independently, the difference between their frequencies is no longer fixed — sixteen sweeps against sixteen found it sweeping through a wide range many times a second, and spending most of its time above fifteen hertz. So the question is not what the beat rate is, but how many beats a listener actually hears.

The count used here is the one a listener could make in principle. The two partials’ sum has a null in its envelope every time the phase between them advances by a half-turn past the last null — that is what a beat is. A null counts as a heard beat only if it happens while the instantaneous beat rate is at or below fifteen hertz. Each voice is detuned from the note by a draw of fifteen cents, has a vibrato of its own rate drawn from the measured 5.5 to 7.5 hertz, and has a random phase. Three hundred pairs are averaged at each frequency.

What the vibrato takes, and what it gives #

The first figure is the census, and it has two halves.

Below about 400 hertz, the vibrato adds beats. At 220 hertz a straight pair gives 2.2 followable beats a second and a pair with vibrato 5.3. The steady difference between two voices at the fundamental is small — a couple of hertz — so their beats are slow and few. A vibrato makes the difference sweep through zero and back many times a second, and each crossing near zero sweeps the phase slowly through a half-turn: a beat at a followable rate, produced by the sweep itself.

Above it, the vibrato removes them. At 880 hertz a straight pair gives 5.5 a second and a pair with vibrato 2.4. At 1,760 the counts are 3.9 and 1.2; at 3,200, the band where one voice stands clear of ninety players, 2.3 and 0.7. Up here the vibrato’s swing in hertz is large — seventy cents at three kilohertz is more than a hundred hertz either side — and it carries the pair’s difference past fifteen hertz for most of every cycle. The beats are still there in the signal; they are going by too fast to be beats.

The two curves cross at 413 hertz. That is close to the frequency at which the earlier essay found a typical pair’s beat rate equal to the vibrato’s own rate, but it is not the same quantity and the agreement is approximate.

The second half needs a word set beside the earlier essay’s account, which said that above 1,217 hertz a section fluctuates by vibrato alone because the mistuning’s beating there has become roughness. That was a statement about the typical pair — two voices differing by the section’s spread times the square root of two. A section’s pairs are spread from nearly unison to twice that, and the closer pairs go on beating at followable rates far above 1,217 hertz: sung straight, a section of fifteen cents’ spread still gives two followable beats a second at 3,500. What the vibrato does in the upper band is take seven in ten of those away. The earlier picture’s division of the spectrum is right in its direction and too sharp at its edge.

The typical pair and the rest of the section #

The arithmetic behind that line is simple enough to redo for the whole section rather than its typical member. Two voices drawn independently from a spread of fifteen cents differ by a draw from a spread of 21.2. A difference of d cents at frequency f beats at about f·d·ln 2 / 1200 hertz, so the pair’s beating is followable at f whenever d is smaller than 15 × 1200 / (f ln 2) cents.

At 1,217 hertz that limit is 21.3 cents, which is the typical pair’s difference: by construction, the typical pair sits on the line. But 68 per cent of all pairs differ by less than that and beat followably there. At 1,760 hertz the limit is 14.8 cents and 51 per cent of pairs still beat followably. At 3,200 hertz it is 8.1 cents and 30 per cent do.

So above the earlier essay’s line the section’s beating does not stop; it thins. Three pairs in ten are still beating slowly enough to follow at the singer’s formant, sung straight, and the 2.3 beats a second in the first figure are theirs. What the vibrato does there is sweep even those close pairs’ difference past fifteen hertz for most of every cycle. That is the precise sense in which a vibrato hides a section’s tuning in the upper band: the beats it removes are the ones made by the best-matched pairs, which are the ones a listener judging the section’s tuning would most need.

The front edge of a note #

Now the onset. Let the vibrato’s extent grow in a straight line from nothing at the start of the note to its full 71 cents at 250 milliseconds, and count the followable beats in each tenth of a second.

The onset does what was expected, in the direction the census predicts. At 880 hertz the first tenth of a second carries 4.0 followable beats a second, the second 3.4, and from there on the rate settles near 2.3. At 1,760 the first two tenths carry 1.9 and the rest about 1.2; at 3,200, 1.3 in the first tenth and about 0.6 thereafter. Below the crossing the order reverses: at 330 hertz the first tenth carries 2.4, the next 5.8, and the rest about 4.4, because the arriving vibrato adds beats there rather than removing them.

But look at the scale. Even the first tenth of a second never reaches the straight tone’s rate — 5.5 at 880, 3.9 at 1,760, 2.3 at 3,200 — because the vibrato begins to grow at once, and a few cents of swing are already enough to move a high partial’s beat rate out of range.

Less than one beat #

A rate is not yet a count. What a listener gets from the front edge is the number of beats it contains, and a beat is a slow event.

Over a 250-millisecond onset a pair produces, on average, 0.86 followable beats at 880 hertz, 0.44 at 1,760 and 0.21 at 3,200. Over a slow, 400-millisecond onset those become 1.33, 0.81 and 0.42. Sung straight for the same stretches the counts would be roughly twice as large, and still small: 0.58 at 3,200 hertz for a quarter of a second.

So the front edge of a note is not where a section’s tuning is heard in the singer’s formant band. It holds the beats at a higher rate than the rest of the note, but it is too short to hold more than a fraction of one. A listener judging a choir’s tuning by beating in the band where the voices carry has, in the first quarter-second, mostly nothing to judge by — and after it, a vibrato that hides seven in ten of what a straight section would give.

That reverses the guess the earlier essay’s lead was built on. The straight start looked like a window in which none of the sweeping and modulation of the earlier essays applied, and so a window in which the section’s tuning would be exposed. It is a window, and none of that applies in it, but at the frequencies where a choir’s tuning would matter most it is far too narrow to expose anything as slow as a beat.

Where the line falls for a better or worse section #

The crossing frequency is not fixed. It depends on the section’s own tuning, because what matters is the pair’s steady difference in hertz, and that is the product of the spread and the frequency. A tight section, spread by five cents, keeps the vibrato from hiding its beats up to 787 hertz; one spread by twenty-five cents loses them above 300. The product of the two runs from about four to seven thousand rather than being exactly constant, because the vibrato’s own sweep enters too.

Either way the singer’s formant is above the line. At three kilohertz, for every section from the most precise to the most ragged, a vibrato makes the mistuning harder to count as beats. The tighter the section, the more of its spectrum — fundamental and first few partials — keeps the beats the vibrato adds; the more ragged, the less.

What a choir does that a soloist cannot found that a unison’s fluctuation does not fall as voices are added, because its beat rates are all different. This is the companion statement about the vibrato: what a vibrato does to that fluctuation is not uniform up the spectrum, and where it changes from giving to taking depends on how well the section sings together.

How much the width of the vibrato hides #

The measured range of vibrato extent is wide — 34 to 123 cents across ten tenors — and the width matters as much as the presence. At 3,200 hertz a straight section gives 2.3 followable beats a second; a narrow 34-cent vibrato leaves 1.2; the typical 71 leaves 0.7; 90 leaves 0.5, and beyond that the count stops falling. At 440 hertz, just above the crossing, a narrow vibrato adds beats — 5.3 a second at 34 cents against 4.2 straight — and only a wide one takes them away.

So a narrow vibrato and a wide one are not the same thing at different strengths. A narrow one adds beats in the lower partials and halves them in the upper; a wide one takes them away nearly everywhere.

The model the counts rest on #

Two voices, each a pure partial at the compared frequency moved by a fixed detuning drawn from a normal distribution of fifteen cents and by a sinusoidal vibrato in cents, of a rate drawn about 6.0 hertz with a spread of half a hertz, clipped to the measured 5.5 to 7.5, and of a uniformly random phase. The vibrato’s extent rises linearly from nothing to its full value over the onset time and then holds; the established figures are counted over a second beginning 600 milliseconds into the note. The envelope of the pair’s sum has a null each time the phase between the two partials passes an odd multiple of a half-turn, found at half-millisecond steps; a null is counted when the instantaneous difference of their frequencies is at or below fifteen hertz. Three hundred pairs are averaged at each point, with a fixed seed so that the figures are the same pairs on every reading.

The fifteen hertz is a convention for where a fluctuation stops being followed, used throughout these essays, and the counts move with it. The linear growth of the extent is the simplest shape that starts at nothing and arrives; a vibrato that began only after a delay, rather than growing at once, would give a straight stretch with more beats in it, and the figures would be the lower bound on what an onset can hold.

What the counts leave out #

The onset’s shape and length. The earlier essays on the voice have quoted “a few hundred milliseconds” for a vibrato to establish itself, and no measurement of the shape of the onset is used here. The onset time is swept from 100 to 400 milliseconds rather than fixed for that reason, and the conclusion that the front edge holds less than one beat in the formant band holds across the whole sweep.

Every voice modulating one partial, with the formants still. The amplitude modulation a formant imposes on a partial sweeping past it, at the vibrato rate, is not in these sums. It would add a fluctuation of its own at six hertz, which the rate that does not rise with the partial counted, and it would not change the count of beats between the two voices’ frequencies.

Two voices, not a section. A section of sixteen is a hundred and twenty pairs, all beating at once, and a listener does not count nulls in any one of them. The pair count is a measure of how much beating the mistuning produces at a frequency, not of what a listener hears in the sum. A roughness with a rate of its own and the mean survives the window are the account of what a listener’s window makes of the moving sum.

What none of these figures show #

The pitch and the vowel. The question the earlier essay asked was what fraction of a note’s identification happens before the vibrato arrives, and identification is mostly about what the note is — its pitch, its vowel, its singer. None of that is computed here. The pitch that does not wobble found the period surviving a full vibrato at a cost, so the pitch is available on both sides of the onset; whether a listener weighs the straight start more heavily in naming the note is a question for an experiment.

And whether beats are how a listener hears a section’s tuning at all. The counts are a statement about how many followable beats the signal contains. Listeners may judge a choir’s intonation by its roughness, by the spread of pitch they hear, or by something else entirely; what is shown is only that if beating is the cue, the vibrato removes most of it where the voice carries, and the straight start does not put it back.

Whose singers #

The vibrato rates and extents are the measured values of twentieth-century operatic soloists; the fifteen-cent spread is the value used for choral unisons since the choir essays. A chorus singing with a reduced vibrato sits between the straight and the vibrato curves, and a chorus singing straight tone — much of the early-music repertoire, and many school and church choirs — sits on the straight one, with three or four times the followable beats at the singer’s formant. On this arithmetic such a chorus’s tuning is far more exposed in the band where its voices carry, in both directions: a well-tuned straight-tone choir is audibly well tuned there, and a poorly tuned one audibly not.

Still open: whether a section’s vibratos lock #

Every count here draws each voice’s vibrato rate and phase independently. Singers listening to each other may entrain, and two voices whose vibratos moved in step would sweep their partials together, so the difference between them would stay near its steady value and the beats would come back. The earlier essay named entrainment as the most important unmeasured quantity on the page. The computation is this one with a coupling between the two voices’ phases, swept from none to complete, and it would say how far a section has to lock before its tuning is as exposed at the singer’s formant as a straight-tone section’s — which turns a question nobody has measured into a number a recording of two singers could check.