Unison — where it appears
Named by 13 essays across 5 fields — each of them below, with the objects they name alongside it.
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.
Every partial beats at its own rate
Five earlier essays have drawn one beat rate per figure, and every one of them is the rate between two fundamentals. Two real notes beat between all of their partials at once, the k-th pair beats k times as fast, and somewhere up the spectrum the rate passes the point at which a beat stops being a beat — so a chorused note is a beat at the bottom of itself and a roughness at the top, simultaneously, with a crossover partial that is arithmetic.
A beat has a depth, and six essays held it at one
Every beat figure so far adds two tones of equal amplitude, which is the single ratio at which the trough of a beat is a true null — and a null is what a tuner is actually listening for. Vary the ratio and the picture changes: at two to one the dip is nine and a half decibels, at ten to one it is under two, and the interval that gives the shallowest null of all is the octave.
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.
Twelve violins are more punctual than one
Every essay until now treats a part as one player, and an orchestral part is a dozen. Sectioning does two things at once and only one of them was expected: it pulls the part's heard moment forward, by four milliseconds against a map spanning twenty-six, and it makes the part's arrival more accurate by very nearly the root of the number of players. So the map of required leads applies to an orchestra better than it applies to a quartet, and the case where it fails is three trumpets rather than fourteen violins.
The octave is counted best over a unison left two and a half cents wide
A firm touch buys a mistuned octave's beats only if the aftersound's drop does not deepen with the strike, and the drop was left as a number to be measured. It has a mechanism, and the mechanism has been known since the essay on three strings: the unison's two coupled modes. Computed from them, the drop does not depend on the strike at all. It depends on how wide the tuner left the unison — sixty-five decibels at a quarter of a cent, twenty at three — and the octave's beats peak, at 6.8, over the same two-and-a-half-cent unison that makes the longest note.
The plane of the blow matters only where the unison is clean
A piano string vibrates in two planes, and the one parallel to the soundboard gives even a single string a second, slower decay. A hammer tilted off the vertical feeds it. That was the last way left for a firm touch to cost a tuner beats: if harder blows came down straighter. They would have to straighten by half for every seven decibels, and even then only on a single string or a unison tuned clean. Over a unison left two and a half cents wide, no tilt of the blow reaches the count.
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.
Two strings undo what two quills were regulated for
A harpsichord's two eight-foot registers each pluck their own string of a unison, and the unison is never exact. Two cents apart, the second string's partials turn against the first's at a rate proportional to their number, so within a quarter of a second at C5 the pair has drifted past the regulation tolerance the jacks were voiced to, and through the rest of the note it passes again and again through every interval a regulator could have set. The pair's fixed spectrum becomes one that breathes, its own holes become flickers a few per cent of a cycle long, and averaged over the note the second register is worth a decibel and a half at the fundamental, not four.
Named alongside it
The objects these essays reach for when they reach for this one.
BeatingModulationVibratoFormantPartialRoughnessTuning by earChorus effectCoupled oscillatorCritical bandwidthDecayHammer