Acoustics & Music Theory
why it sounds like that
Concept

Entrainment — where it appears

The locking of an internal periodicity onto an external one, which is what a listener does on finding the beat. Modelled as an oscillator that corrects its period and phase toward arriving onsets, it coasts through silence at whatever rate it had.

Named by 15 essays across 4 fields — each of them below, with the objects they name alongside it.

3 against 2. Two evenly spaced pulses over the same span, one in 3 and one in 2. They agree only where both grids land together, and the gap between agreements is the least common multiple — which is what makes one polyrhythm sound like a shape and another like two unrelated things.

Two clocks at once, and where they agree

Three against two repeats every six subdivisions and feels like a figure. Seven against five repeats every thirty-five and feels like weather. The difference is one number.

rhythm · Polyrhythm One pattern, four metres. The same 12-step onset pattern read under 4 candidate metres, each scored by a preference rule set: 3 for a strong position that carries an onset, -2 for one that does not, -1 for an onset that lands off every strong position. The scores are 4/4 -4, 3/4 12, 6/8 12, 12/8 as 3+3+3+3 12, so 3/4 and 6/8 and 12/8 as 3+3+3+3 tie and the model does not choose. Nothing about the sound differs between these readings; the bar line is supplied by the listener.

The beat is inferred, and sometimes wrongly

A bar line is not in the sound. One sequence of onsets supports a reading in four, in three and in six, and which one a listener settles on comes from a small set of preference rules rather than from anything in the signal — which is why the same recording can be heard two ways and why nobody can be talked out of either.

rhythm · Metre induction The metrical levels of 120 bpm, against the window. The range of inter-onset intervals that can be heard as a beat at all, from about 100 to 2000 milliseconds, with the preferred rate near 550. Each mark is one metrical level of a piece at 120 beats a minute. Which of them a listener taps is decided by which falls nearest the preferred rate, not by which one the notation calls the beat.

The beat has a preferred rate, and it is not the notation's

Metre is a hierarchy of levels and only one of them is tapped. Which one is decided by the listener's own clock rather than by the time signature: the window in which a series of events can be a beat at all runs from about a tenth of a second to two seconds, with a marked preference near half a second — so at the extremes of tempo the notated beat and the felt one part company, predictably.

rhythm · Metre induction A cycle has no ending to compute, so it uses density instead. 5 layers over 32 cycles of a 12-step pattern, with each layer's entry and exit marked, and the onsets per step summed underneath. No chord changes and no cadence occurs; the closure vector is zero throughout, and every change a listener hears is a change in how many things are playing.

A cycle cannot cadence

Every component of closure is defined by a first time and a last time. Music built on a repeating cycle has neither, so the whole apparatus returns zero on it — not a small value, zero, at every setting. What such music uses instead is how many things are playing, and that is a curve which can be computed from the onsets and nothing else.

form · Cyclic rhythm One pattern, four metres. The same 6-step onset pattern read under 3 candidate metres, each scored by a preference rule set: 3 for a strong position that carries an onset, -2 for one that does not, -1 for an onset that lands off every strong position. The scores are the 3 is the beat 8, the 2 is the beat 4, every step is a beat 8, so the 3 is the beat and every step is a beat tie and the model does not choose. Nothing about the sound differs between these readings; the bar line is supplied by the listener.

Which of the two is the beat

A polyrhythm is notated as a bar of two with three laid across it. Run the metre rules used here over the composite and they choose the three — at every ratio tried, without exception. Add the one other rule the rules have, and the answer changes hands at a bar of 1,013 milliseconds, which is 118 to the minute.

rhythm · Polyrhythm Where the two accounts part company. The spread of the error between the hands, in milliseconds, against bar number, averaged over 120 seeded runs of each model at 100 bars a minute. With one timekeeper it is flat at about 9 ms after 24 bars; with two it reaches 106 ms and is still climbing, because a random walk has nothing to return to. Measured players hold 3 against 2 inside about 25 ms indefinitely.

One player is not two clocks

Two accounts of a pianist playing three against two, simulated from the same noise. With a timekeeper in each hand the hands drift apart by a hundred milliseconds inside two dozen bars. With one timekeeper they never drift at all. The measurement everybody cites as evidence for a timekeeper turns out to be the same number under both accounts.

rhythm · Polyrhythm One pattern, four metres. The same 16-step onset pattern read under 4 candidate metres, each scored by a preference rule set: 3 for a strong position that carries an onset, -2 for one that does not, -1 for an onset that lands off every strong position. The scores are downbeat on step 1 0, downbeat on step 2 -12, downbeat on step 3 -12, downbeat on step 4 0, so downbeat on step 1 and downbeat on step 4 tie and the model does not choose. Nothing about the sound differs between these readings; the bar line is supplied by the listener.

The beat that is never sounded

A listener who has heard four bars of a groove and then hears two bars with the downbeats taken out does not move the downbeat. This site's rule set does, every time, on every pattern tried — and the direction it moves in says exactly what kind of model would be needed instead.

perception · Metre induction A tracker following a tempo that will not stay still. The tracker's error as a fraction of a beat, against beat number, for 3 rates of tempo change with a period-correction gain of 0.2. It settles at 0.025 of a beat behind at 0.5 per cent a beat, 0.092 of a beat behind at 2.0 per cent a beat, 0.206 of a beat behind at 5.0 per cent a beat. It does not lose the beat; it lags, by very nearly the rate divided by the period-correction gain, and the lag reaches a quarter of a beat at 6.3 per cent a beat — at which point the tracker is nearer the wrong onset than the right one.

A metre has to be able to change its mind

Replace the scoring function with a phase-corrected oscillator and the two failures reported earlier separate. The beat now survives two silent bars, drifting 23 milliseconds of a 560-millisecond beat. But it does not lose a moving tempo so much as lag behind it, by the rate over the correction gain — and a cadential ritardando that halves the tempo in eight beats is faster than the model can follow.

form · Metre induction Two players, and the correction that keeps them together. The spread of the asynchrony between two players, in milliseconds, against beat number, for 3 correction gains, averaged over 120 seeded runs each. It reaches 120 ms after 64 beats at a gain of 0, 27 ms after 64 beats at a gain of 0.1, 20 ms after 64 beats at a gain of 0.3. With no correction at all the asynchrony is a random walk and grows without bound; with any correction it settles at a fixed spread within a few beats and stays there. Two people cannot share a timekeeper, so the fact that ensembles do not come apart is itself the evidence that they are correcting.

Two players and no clock

Two people cannot share a timekeeper, and two independent ones drift a hundred and twenty milliseconds apart inside a minute. Ensembles do not, so something is correcting — and the measurement everybody reaches for recovers the pair's total responsiveness exactly and cannot tell which of the two is doing it. Four tenths from one player and two tenths each give the identical number.

instruments · Microtiming A silence measured in two ways that were not chosen to agree. How far a carried beat drifts during a silence, in fractions of a beat, for beat periods of 200 ms, 550 ms, 2000 ms and a tempo estimate 5 per cent wrong — which is the published discrimination limen rather than a figure chosen here. Half a beat of drift is where the metre coming out of the silence is no longer the one that went in, and it is reached after 10 beats whatever the tempo, which is 2.0 seconds at 200 ms, 5.5 seconds at 550 ms, 20.0 seconds at 2000 ms. The shaded band is the psychological present, 2 to 8 seconds, measured by a completely different literature and used in this collection to bound a phrase. The two answers overlap: a silence under about 2 seconds is a rest inside something and one over about 5.5 is after it.

A silence long enough to be an ending

Four of the five closure components are present or absent. Silence is the one with a continuous scale, so it is the one that can be given a threshold — and the threshold arrives from two literatures that were not chosen to agree, landing between three and a half and five and a half seconds. In a large hall, the room's own decay uses up most of it.

form · Closure 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 A cycle whose position is in the instrumentation. 3 isochronous layers over a cycle of 16 steps, at periods 16, 8, 4. Every layer on its own is perfectly symmetric and tells a listener nothing about where they are; the combination gives 4 distinct signatures over 16 steps, and hearing one of them leaves 2.81 bits unknown. The information is in which instruments sound rather than in where the onsets fall, which is a different answer from the one a single timeline gives — and it needs no asymmetry anywhere. The cost is 7 strokes a cycle, 0.44 to the step, spread over 3 players.

A cycle that says where it is

Euclidean timelines were asked how quickly they tell a listener where in the cycle they are, and answered it with rotational asymmetry: a symmetric pattern never locates at all. A colotomic cycle answers the same question with nothing asymmetric in it. Several isochronous layers at nested periods — a gong every sixteen, a kempul every eight, a kenong every four — put the position in which instruments sound, and the position is legible from a single stroke.

rhythm · Cyclic rhythm 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 Where a cycle of 16 at 16, 8, 4 outruns the listener's memory. The residual uncertainty a listener is left with once the evidence has stopped accumulating, against how long one turn of a 16-step cycle takes. The listener's memory of a step halves after 3.5 seconds throughout; what changes is how many steps that is. At a cycle of 1.6 seconds it is 35 steps and every design reaches certainty, which is the regime a clave is played in. At 60 seconds it is 0.93 steps and none of them does: layers at 16, 8, 4 settles at 1.89 bits, son clave settles at 2.27 bits, the bossa-nova pattern settles at 2.38 bits, the best single line of 7 settles at 1.85 bits. That is the range a gong cycle occupies, and it is the design that wins there.

The cycle that outruns the memory

A timeline and a colotomy were compared at equal strokes and the comparison had no clock in it. A memory span is a number of seconds and a cycle is a number of steps, so the two only meet through a tempo — and at a clave's two seconds a listener's memory covers twenty-eight steps and forgets nothing, while at a gong cycle's forty it covers 1.4 and forgets almost everything. The single line is the better locator up to twenty-three seconds a cycle and the layered code is better after it, which is very close to where each is actually used.

rhythm · Cyclic rhythm A cycle already known, against a cycle just arrived at. How many bits of uncertainty about position a listener has, against how long the cycle takes, for a single timeline and for a layered colotomy — each drawn twice, once as a listener arriving and once as a listener who has been hearing it long enough to settle. At 1.6 seconds a cycle the timeline goes 0.94 bits arriving and 0.00 settled, and the colotomy 0.71 and 0.00; At 16 seconds a cycle the timeline goes 1.10 bits arriving and 0.08 settled, and the colotomy 0.93 and 0.23; At 60 seconds a cycle the timeline goes 2.47 bits arriving and 2.27 settled, and the colotomy 2.14 and 1.89. The gap between each pair is what the repetitions are worth, and it narrows as the cycle slows. The two designs are drawn at their own step counts rather than at equal strokes, so the levels here are not the earlier ones and the gaps are.

Repetition buys least where it is needed most

Every locating figure so far is a listener arriving — the uncertainty averaged over the first cycle heard. Cyclic music comes round dozens of times, and the same model already carries the answer for a listener who has settled: a floor of uncertainty that nothing had read. At two seconds a cycle the repetitions close the whole gap. At sixty they close eight per cent for a single timeline and twelve for a layered code. A slow cycle is worse on the first hearing and gains less from the second, and the two disadvantages compound.

rhythm · Cyclic rhythm

Named alongside it

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

MetreOnset patternTempoBeatClosureInformationMotor delayPolyrhythmAttack transientCycleCyclic formEnsemble asynchrony

All concepts