A note sung at 440 Hz, drawn in cents
Drawn above with its standard settings, which is almost never how an essay draws it: an essay states the numbers it is arguing about, so the figure a reader meets there is about that argument rather than about the drawing in general. Every option a placement passes is checked against the ones this function actually reads, because an option it does not read is silently ignored and the figure quietly draws what is above instead.
It makes a noise. 83 of its 83 placements carry sound, built from the same numbers as the drawing, offering these buttons: "heed" at 220 Hz, straight, "hod" at 220 Hz, straight, "hod" at 330 Hz, straight, 220 Hz, steady, 220 Hz, ±71 cents at 6.0 Hz, 220 hertz held perfectly steady, 440 Hz, steady, 440 Hz, ±100 cents at 6.0 Hz and 61 more.
Called by 13 essays
the blast radius of changing it
A note that is never at its pitch
Eleven essays in this field argue about differences of one to twenty-four cents. An ordinary operatic vibrato is a hundred and forty cents wide and completes six excursions a second, so every one of those distinctions fits inside a single sung note several times over — and the beat rate a tuner would null passes through zero eleven times a second.
What a choir does that a soloist cannot
Two singers on one note produce one beat and it can be counted. Sixteen produce a hundred and twenty at once, and the amplitude still fluctuates by as much as it did — a choir is no steadier than a duet. What has gone is not the fluctuation but its rate: the modulation energy that sat in a single line at two voices is spread across a band at sixteen, with no line in it. That is the choral sound, and it is also why the just-intonation drift measured for tuned singing describes only ensembles that hold their pitch still.
Sixteen sweeps against sixteen
Every intonation figure about the voice treats a singer as a frequency. A singer is a frequency being swept a hundred cents wide six times a second, and two sections singing an interval are two hundred and fifty-six pairs of sweeps. The beat rate between the partials the interval brings together stops being a number and becomes a function of time — and the pair spends four fifths of its time above the rate at which beating is beating at all.
A roughness with a rate of its own
Every roughness figure so far computes one number for a steady spectrum. Evaluate the same sum at every instant of a vibrato and there are three numbers instead — a mean, a depth and a rate — and the mean is not the roughness of the mean frequency. On an octave it is nineteen times it, because an octave sits in a deep narrow minimum and a vibrato smears it out of one.
The mean survives the window
A roughness that moves has a mean, a depth and a rate — all three of which a listener could only have through a temporal window. Applying the window already to hand settles which of the three survives, and the answer refutes the guess: a running average cannot change an average, so the octave's factor of nineteen stands and the movement is what goes.
The pitch that does not wobble
Three earlier essays have treated a vibrato as a modulation of roughness. The reason singers use one is what it does to the note, and there is an extractor here that turns a set of partials into a pitch and has never been asked what it does with partials that will not hold still. The period survives, at a cost that rises with the extent — and the practice stops within a hair of where the cost becomes total.
The partial that gets louder as it goes sharp
Eleven earlier essays sweep a set of partials and hold their amplitudes still, and a real tract does not move with the fundamental. Put the formant account and the sweeping account together and every partial acquires an amplitude modulation at the vibrato rate: 0.07 decibels on the fundamental and 8.6 on the twelfth partial of the same note. They are in phase below a formant and anti-phase above one — not ninety degrees apart — and the whole note swings 0.82 decibels, because they cancel.
The rate that does not rise with the partial
Twelve earlier essays give every vibrato the same six hertz, and the measured spread is 5.5 to 7.5. Putting the two fluctuations a choir contains on one axis shows why the rate matters: the beating between mistuned voices rises with the partial and leaves the range a listener follows as fluctuation at 1,217 hertz, while the vibrato's own modulation is six hertz at every partial. Above that frequency a section fluctuates by vibrato alone — and if every singer had the same rate, it would barely fluctuate at all.
A vibrato hides a choir's tuning above four hundred hertz
Thirteen essays drew a vibrato that had always been running; a real one grows out of a straight tone over the first quarter-second of a note. Counting the beats a listener can follow between two voices of a section shows what that front edge is worth. A vibrato takes most of a section's countable beats away above about 400 hertz — seven in ten of them at the singer's formant — and adds some below. The straight start gives them back only while the vibrato is still growing, and in the band where a voice carries that is less than one beat.
A choir's vibratos must lock to two milliseconds
A vibrato hides a section's mistuning from a listener counting beats, and the obvious way it could stop hiding it is for the singers' vibratos to move together. They would have to move very nearly together. Half-entrained vibratos leave a section as hidden at the singer's formant as independent ones; locked to nine-tenths they expose three-quarters of what a straight section does; only past nineteen parts in twenty does the section's tuning show as if they sang straight. At six cycles a second that is two vibratos peaking within two milliseconds of each other — and the better a section is tuned, the tighter the lock has to be.
Singers who hear each other late lock their vibratos in opposition
A choir's vibratos give back the section's tuning at the singers' formant only if they are locked to within about two milliseconds. Singers who lock do it by hearing each other, and they hear each other late. Two vibratos pulled towards what each hears of the other settle near in step while the delay is under about forty milliseconds, and even then come within two milliseconds only if each singer corrects a phase error in about thirty. From forty to a hundred and twenty milliseconds late — a third to two-thirds of a vibrato cycle — they lock in opposition, peaks seventy to ninety milliseconds apart, and hide the section's tuning more completely than vibratos that never locked at all.
Sixteen singers cannot all oppose each other
Two singers who hear each other a third of a vibrato cycle late lock their vibratos in opposition and hide their tuning better than if they had not listened at all. Sixteen cannot do it, because no arrangement puts every one of them opposite every other. At the latencies where a pair opposes, a section of sixteen does not lock at all: each singer keeps a vibrato rate of its own, and a listener hears the section's tuning exactly as through independent vibratos. Outside that band the section locks in step and gives most of its tuning back. A section that listens to itself can expose its tuning or leave it as it was, but never hide it.
A row that hears its neighbours moves its tuning, it does not hide it
Sixteen singers who all hear each other alike cannot lock in opposition, because opposition between every pair is impossible. A row that hears mainly its neighbours has no such obstacle — opposition along a chain is consistent — and at the latencies that make a pair oppose, it half-alternates: neighbours opposite each other, every second singer nearly in step. That hides the tuning between neighbours as well as a duet hides it, and exposes it between every second singer more than independent voices would. Over the whole row the two cancel to the hundredth. With no latency, the same local hearing stops the row locking at all, and hides nearly half of what a whole-row lock would show.