Acoustics & Music Theory
why it sounds like that
Concept

Microtiming — where it appears

The differences of a few tens of milliseconds between where a note is notated and where it is actually played. They are systematic rather than random, and they are the same order of size as the gaps a complex polyrhythm depends on.

Named by 10 essays across 3 fields — each of them below, with the objects they name alongside it.

How much swing there is, against how fast the music goes. The ratio of the long note to the short note of a swung pair, as a function of tempo, derived from the finding that the short note holds a roughly constant hundred milliseconds. The notated readings are horizontal lines, and each is correct at exactly one tempo. Above about three hundred beats a minute the ratio reaches one and the swing has gone.

Swing is a ratio, and it is not two to one

Swung quavers are notated as a triplet figure, which is a ratio of two to one. Measurements of what drummers actually play give anything from three and a half to one down to one to one, and which of those it is depends almost entirely on the tempo.

rhythm · Microtiming Where the beats actually fall. Measured timing deviations from a strict grid, in milliseconds, for three published profiles. The right-hand column converts each deviation into the note value it would have to be written as, at three tempi — and because a fixed number of milliseconds is a different fraction of the beat at every tempo, no single notated rhythm describes any of these.

The milliseconds that are the groove

A Viennese orchestra plays the second beat of a waltz about fifty milliseconds early, every bar. A jazz soloist sits thirty behind the ride cymbal. Neither deviation can be notated, and not because notation is coarse — because it measures the wrong quantity.

rhythm · Microtiming Two kinds of systematic timing, which share a word. Each measured profile split into a constant offset from the grid and a pattern that varies by position in the bar. Viennese waltz, second beat is −10.0 ms of offset and 33.4 ms of pattern; jazz soloist against the ride is 28.8 ms of offset and 1.9 ms of pattern; quantised is 0.0 ms of offset and 0.0 ms of pattern. A motor deviation anywhere in the 8 to 20 ms range published for skilled performers leaves 74–95% of Viennese waltz, second beat's variation systematic, 1–5% of jazz soloist against the ride's variation systematic. The two quantities are independent and no single deviation figure distinguishes them.

The deviations are not noise

Two published timing profiles, split into the quantities they actually carry. A jazz soloist thirty milliseconds behind the ride is almost pure offset and has no pattern at all. A Viennese second beat is almost pure pattern and has no offset. They are different things, they are reported under one word, and no figure of total deviation tells them apart.

rhythm · Microtiming 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 The swing ratio, against the categories it passes through. The same swing curve read against the boundaries between duration categories rather than against notated values. A category's centre is a simple ratio — 1:1, 2:1, 3:1 — and the boundary between two of them is the midpoint, which is arithmetic. The curve crosses 2 of them: out of 2:1 and into 1:1 at 240 beats a minute, out of 3:1 and into 2:1 at 171 beats a minute. So the same notated figure is, by the categorical criterion, a different rhythm at each end of an ordinary tempo range, and the notation says triplet feel throughout.

How late is a different note

A deviation of thirty milliseconds is expression and a deviation of two hundred is a wrong note, so there is an edge. The edges in time are arithmetic — the midpoints between the simple ratios — and the swing ratio crosses two of them as the tempo rises, at 171 and at 240 beats a minute, while the notation says triplet feel throughout.

perception · Microtiming A duration category has a tempo range of its own. Each simple ratio's short note is the beat divided by one more than the ratio, so at a high enough tempo it falls under the fastest interval that can be a beat at all — 100 milliseconds. Each bar here runs from the slowest tempo at which the ratio's long note still belongs to a beat to the fastest at which its short note is still a note: 1:1 ends at 300 bpm, 2:1 ends at 200 bpm, 3:1 ends at 150 bpm, 4:1 ends at 120 bpm. The line is this site's swing curve, and where it crosses a category boundary the category it is leaving has already ceased to exist.

The short note is sitting on the floor

Swing is modelled here as a short note of constant absolute length, which was measured from drummers and left as a fitted parameter. The tempo window's fast edge — the shortest interval a series of events can be a beat at — was measured from listeners tapping. Both are a hundred milliseconds, and if that is not a coincidence then the swing ratio has no free parameter in it at all: the short note is not held constant, it is resting on the floor.

rhythm · Microtiming The distribution that can be measured is not the one the player has. A player aiming at a short note of 100 milliseconds with a standard deviation of 15, against a floor at 100. The pale curve is what the player is doing and the heavy one is what can be recorded, because the 50 per cent of the parent below the floor arrives at the floor instead. The measured mean is 112.0 milliseconds rather than 100 and the measured standard deviation is 9.0 rather than 15. At 160 beats per minute that turns an intended swing ratio of 2.75 into a measured 2.35.

A quantity resting against a wall

The short note of a swung pair was found sitting exactly on the fast edge of the tempo window. If that edge is a floor rather than a fitted number, then every swing statistic computed until now was computed on a censored sample — and a censored sample has a mean that is 0.80 standard deviations too high, a spread that is 40 per cent too low, and a correction gain that can come out twice what the players actually have.

rhythm · Microtiming How far a wall moves the mean, in spreads. The bias a floor puts into an observed mean, in units of the parent's own spread, against how far the parent sits above the floor. With the mean exactly on the wall the truncated bias is 0.797 spreads and the censored bias is 0.399 — half of it, exactly, because half the mass sits at one point and the other half is an upper half-normal. The earlier essay used the upper figure for a process that produces the lower one, so every bias it quoted is twice what a floor on execution actually causes.

The shape a wall leaves behind

A floor on execution biases every statistic computed on swing timing, and the bias was priced with Pearson's truncated-normal formulas. Those are the formulas for a sample with everything below the wall thrown away. A player who cannot execute a short gap does not throw the attempt away — it comes out at the floor. That is a censored sample, its bias is exactly half, and its skew is two thirds larger.

rhythm · Microtiming The passage that separates them, and a listener cannot hear it. A scoring changes at the halfway bar, and the two maps of required leads differ by 18.1 milliseconds at their widest. An ensemble that has internalised the map applies the new one on the first note of it and its spread never leaves zero. An ensemble that is listening to each other has to re-converge: its spread jumps to 11.8 milliseconds and takes 3 beats to get back under 5. The dashed line is twenty milliseconds, which is what a listener notices — and the disagreement never reaches it. So the two accounts are separable on a recording and very nearly not separable by ear, which is why nobody has noticed the distinction and why the measurement is worth making.

The passage that separates two players

An ensemble that has learnt where the asynchronies are applies them; one that is listening discovers them. In steady state the two are identical, which is why nobody has separated them. Change the scoring mid-phrase and they are not: one ensemble is wrong by twelve milliseconds for three beats and the other is not wrong at all — and twelve milliseconds is under what a listener notices and far above what a microphone resolves.

rhythm · Perceptual-centre The bias a wall puts in a mean, against the parent it came from. The bias a wall puts into an observed mean, in spreads, for each of eight standardised parents, with the wall exactly on the parent's mean. The censored values run from 0.354 to 0.433, a range of 0.079; the truncated from 0.582 to 1.000, a range of 0.418. The formula now used is the one that hardly depends on a distribution nobody has measured, and the formula it replaced is the one that depends on it a great deal.

The parent nobody measured

Every figure drawn behind a wall so far assumed a normal parent, and the assumption turns out to matter in exactly the wrong place. The censored bias is half the parent's mean absolute deviation — a theorem, not a coincidence — so it lands between 0.35 and 0.43 spreads for every distribution tried, and the truncated one runs from 0.58 to 1.00. But the third moment proposed as the test moves 2.9 across parents against 0.65 between the two rules, and a censored sample from a slightly left-skewed parent has a skew of 1.007 where a truncated normal has 0.995. The statistic that does work is a count of ties.

rhythm · Microtiming

Named alongside it

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

SwingTempoTiming deviationGrooveCensoringMotor delayBeatCategorical perceptionMotor noiseNotationStatisticsSubdivision

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