Acoustics & Music Theory
why it sounds like that
Concept

Ensemble asynchrony — where it appears

The spread between the moments at which an ensemble's players actually begin a note they mean to begin together. It is thirty to fifty milliseconds in professional groups, and part of it belongs to the instruments rather than to the players, because a slow attack is heard late.

Named by 8 essays across 3 fields — each of them below, with the objects they name alongside it.

One envelope, and the three places a listener might be said to hear it. The amplitude envelope of a note with a 90 millisecond exponential attack, with the three criteria the literature offers drawn across it. The heard moment is 8.0 ms at the detection criterion, 28 ms at the perceptual-onset criterion and 94 ms at the perceptual-attack criterion. The physical onset is at zero on this axis and no criterion puts the heard moment there. The buttons play this attack against a two-millisecond one, started at the same instant.

A note is heard after it starts

Every rhythm essay until now has treated a note's onset as the moment it happens. It is not: the instant a listener aligns a note with a beat is later than its physical start by an amount the note's own attack decides, and for a sung or bowed note that amount is about thirty milliseconds — the size of the whole quantity six essays on microtiming set out to measure.

rhythm · Perceptual-centre Which ensembles have this problem and which do not. The width of the heard-moment spread built into 7 standard instrumentations, before any player does anything. An ensemble drawn from one attack family has a spread of zero — every note is heard the same distance after it is started, so a common onset is a common heard moment, and this is true of a string quartet and of a gamelan for the same reason and in the same amount. A mixed ensemble carries between 11 and 33 milliseconds of it.

The players who have to be early

Ensembles have been measured for fifty years and found to be about forty milliseconds out of alignment, which has always been reported as the limit of human precision. Part of it is not: an ensemble that mixes attack families carries a heard-moment spread of ten to thirty-three milliseconds before anybody plays a note, and an ensemble drawn from one family carries none at all — which is true of a string quartet and of a gamelan for the same reason.

instruments · Perceptual-centre The heard moment against the pitch, on an instrument whose own attack is 8 ms. A note cannot establish an amplitude in less than 4 of its own cycles, so the attack has a floor of 4 periods — 145 milliseconds at A0 and 1.9 at C7. Below A4 the floor is longer than the instrument's own attack and the pitch decides the heard moment; above it the instrument does. The lag runs from 46.0 milliseconds at the bottom to 2.5 at the top, a spread of 44 milliseconds that no player can play their way out of.

A low note cannot start on time

Three earlier essays have held the pitch at one value. A note cannot establish an amplitude in less than a few of its own cycles, so the attack has a floor that rises as the pitch falls — 146 milliseconds at the bottom of a piano and three at the top. On an instrument whose action takes eight milliseconds everywhere, that is a forty-three millisecond spread across the keyboard from the period alone, and no player can do anything about it.

rhythm · Perceptual-centre Which notes of a scored chord have to be played early. Four parts of one chord, each with its own instrument, its own pitch and its own dynamic, and the perceptual centre that comes out of all three. piano, sforzando on E1: an attack family of 8 milliseconds against a pitch floor of 97, so the pitch is what limits it, shortened by the dynamic to 65, heard 20.5 after it starts and needing to be played 12.0 early; flute, quiet on A5: an attack family of 60 milliseconds against a pitch floor of 5, so the instrument is, shortened by the dynamic to 69, heard 21.8 after it starts and needing to be played 13.2 early; violin, mezzo forte on E4: an attack family of 90 milliseconds against a pitch floor of 12, so the instrument is, shortened by the dynamic to 90, heard 28.5 after it starts and needing to be played 19.9 early; trumpet, forte on A3: an attack family of 30 milliseconds against a pitch floor of 18, so the instrument is, shortened by the dynamic to 27, heard 8.6 after it starts and needing to be played 0.0 early. The spread is 19.9 milliseconds, which is well above the two or three a listener resolves, so a conductor asking for these four to sound together is asking for four different physical onsets.

Which notes have to be played early

There are three separate contributions to one quantity — the instrument's attack family, the dynamic it is played at, and the note's own period — and every figure so far varies one and holds the others. Added together for a real scoring they do not add: a sforzando low piano note is pitch-limited to a hundred-millisecond attack and the sforzando shortens it back to sixty-five, so flattening the dynamics makes the ensemble's spread larger rather than smaller.

rhythm · Perceptual-centre An ensemble finding an asynchrony nobody told it about. An earlier essay produced a map of required leads — which notes of a scoring have to be played early, and by how much — and nothing tells the players those numbers, because they are a property of the instruments' attacks rather than of the music. So an ensemble has to find them, and the mechanism is already here: each player hears sounds rather than onsets and moves their next onset toward the mean of the others'. The spread of arrival times starts at 20 milliseconds and settles at 4, crossing 5 milliseconds after 5 beats — about 1.3 bars of four. The leads it converges on match that map to within 0.2 milliseconds, which is what makes this a convergence rather than a coincidence: the fixed point of players listening to each other is every player leading by their own attack.

How many bars an ensemble needs

The map of required leads is something nobody tells the players, because the leads are a property of the instruments' attacks. So an ensemble has to find them, and the mechanism is the one the microtiming essays describe: each player hears sounds rather than onsets and moves toward the others. It converges on the map to within a fifth of a millisecond, in five beats, and there is a best correction gain.

rhythm · Perceptual-centre What a slow start costs a trumpet, in its own settling time. The amplitude of a trumpet's A4 resonance — the 4th impedance peak, of Q 37, time constant 27.1 milliseconds — driven from rest by a pressure that rises over 10, 40, 80, 140 milliseconds, against the step every earlier figure has assumed. The step reaches 90 per cent of its final amplitude in 62.5 milliseconds. A ramp of 140 takes 157.8, which is 95.3 more — and that excess is 68 per cent of the ramp's own length. Across the whole range a player works in the excess is a little over half the ramp: 0.52, 0.56, 0.61, 0.68 at 10, 40, 80, 140 milliseconds. So the tongued attack is not something added to the note. It is the step, which is what has been computed all along, and what has a price is its absence.

What the tongue actually removes

The question this essay was written against asked for an impulse: a tongued attack, a martelé stroke and a struck key all deliver one before the steady drive begins. The arithmetic refuses the framing. A tongue release does carry energy at the note's own frequency, and it is worth 0.17 milliseconds on a trumpet against a settling time of 62 — capped at about 1.13 over the resonance's Q. What articulation is worth is the ramp it removes, and that is about half the ramp's own length: 44 milliseconds, and very nearly the same 44 on every wind instrument in the collection.

rhythm · Onset time The census with the criterion moved under it. Every instrument's speaking time in milliseconds, against the fraction of the steady amplitude counted as speaking. The criterion is in the wind instruments alone: a resonance takes −ln(1−p)·Q/(πf) to reach a fraction p, so those lines rise across the whole picture, while a bow's capture and an exciter's contact contain no criterion at all and are flat. Every earlier figure sits at 0.9, where the wind instruments are the slowest things in the collection by a factor of 20.5. At 0.05 they are the fastest: a violin's G3 string is the slowest at 15.5 milliseconds and a trumpet takes 1.4. The three clusters cross at a criterion between 0.18 and 0.39, which is inside the range the perceptual measurements work in — their three named criteria are 15 decibels below peak, 6 decibels below peak, and ninety per cent — and the settling figures have only ever used the third of them.

Read at two different heights

Ten placements of these figures, one value: the settling criterion is nine tenths in every one of them, and nothing is measured behind it. It is a multiplicative constant only inside the mechanism that has it — a bow's capture and an exciter's contact contain no criterion at all — so moving it rescales one of three clusters against two that stand still. The most-quoted number here, a factor of sixty-nine between the instrument's account and the listener's, is 5.9 at the criterion the listener's own measurements use, and the ordering an earlier essay was written about does not exist below a fifth.

perception · Onset time One attack time, three shapes, 48 ms of disagreement. Three amplitude envelopes with the same 90-millisecond attack time, which is the only quantity the published tables report. a resonator from a step rises as one minus a decaying exponential; an excitation ramping rises in a straight line; a ramp through a resonator is a raised cosine. Each is normalised so that its own 10-to-90 per cent rise takes exactly 90 milliseconds, so all three are the same measurement. The horizontal rules are the three criteria a heard moment is read off. At 6 dB below peak the three shapes put the heard moment at 28.5, 56.4, 76.3 milliseconds — a spread of 48, on one attack time, from a property nothing in the table records.

An attack time is not an attack

Eight earlier essays read a heard moment off an envelope, and every one of them used the same envelope shape without saying so: the source table records one curve for all nine of its families, and the map's own arithmetic does not carry the parameter at all. A published attack time fixes a ten-to-ninety time and nothing else. Under the two other shapes the same measurement admits, every millisecond computed so far doubles — and the constructed passage called inaudible earlier becomes three times a listener's threshold.

perception · Perceptual-centre

Named alongside it

The objects these essays reach for when they reach for this one.

Attack transientOnsetPerceptual-centreEnvelopeTiming deviationTransientEntrainmentHelmholtz motionQuality factorRegisterAsynchronyBrass

All concepts