Acoustics & Music Theory
why it sounds like that
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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.
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.

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. 17 of its 17 placements carry sound, built from the same numbers as the drawing, offering these buttons: 109 bpm — 3.50:1, 110 bpm — 3.50:1, 131 bpm — 3.50:1, 132 bpm — 2.50:1, 160 bpm — 2.75:1, 171 bpm — 2.51:1, 185 bpm — 1.49:1, 220 bpm — 1.73:1 and 6 more.

Called by 6 essays

the blast radius of changing 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 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 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 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

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