How late is a different note

Everything in this ladder so far has been a displacement measured in milliseconds. A jazz soloist thirty behind the ride; a Viennese second beat fifty-five early; a duo’s residual asynchrony twenty. None of them is heard as a wrong note.

Two hundred milliseconds late, at the same tempo, is a wrong note. So there is an edge, and the question is where.

The categories in time are the simple ratios #

The ear sorts pitch into categories, the categories are the intervals, and a note thirty cents sharp is a third that is out rather than a different interval. The same argument runs in time and the categories are the ratios in which one duration divides another: 1:1, 2:1, 3:1, and the small handful beyond.

That is not a claim about hearing so much as an observation about notation. Every notation system that exists writes durations as simple ratios of one another, and the ones it writes are 1:1, 2:1, 3:1 and 1:3 — with everything else expressed as a combination of those or not expressed at all. A system of writing that has been arrived at repeatedly and independently is at least evidence about what is being written down.

The boundaries follow immediately. Expressed as the first note’s share of a pair, 1:1 is 50 per cent, 2:1 is 66.7 and 3:1 is 75. The boundary between two adjacent categories is the midpoint: 58.3 per cent between the first two, 70.8 between the second two.

Every one of those numbers is arithmetic. Nothing has been fitted, and nothing here depends on how sharp the boundary is — which is fortunate, because that is the one quantity in this essay that is not well measured.

A deviation is a distance inside a category #

At 200 beats a minute the beat is 300 milliseconds and a pair of quavers is 150 each. Move the second of the pair thirty milliseconds late and the split goes from 150-and-150 to 180-and-120 — a first-note share of 60 per cent, which is past the 58.3 boundary and into the two-to-one category. Thirty milliseconds is a category change.

Move the whole part thirty milliseconds late and nothing of the sort happens. Both notes move together, the split stays at 150-and-150, and the share stays at 50.

That distinction is the whole of why an offset is never heard as a different note value, however large it is. An offset moves everything; a ratio change moves one thing relative to another. Categories are about ratios, and an offset does not change any ratio at all.

The two measured profiles split into offset and pattern make the distinction the categories need: an offset is a constant displacement of a whole part and a pattern varies by position in the bar, and neither of them is a change of ratio. Both are deviations inside a category; the swing ratio is not.

A pattern does change ratios. The Viennese waltz’s second beat, 55 milliseconds early against a beat of about a second, moves the first beat’s share of the first two beats from 50 per cent to about 47. The nearest boundary is at 58.3. It is well inside, and it stays inside at every waltz tempo, which is why nobody has ever transcribed a Viennese waltz as anything but three equal beats.

And the swing ratio is not inside anything #

The swing ratio is the odd one out among the three kinds of systematic timing, because it is a ratio. It has no protection from the argument above.

The site’s own swing model — the short note holds a roughly constant hundred milliseconds, so the ratio falls as the tempo rises — puts the ratio at 3.5:1 at slow tempos and 1:1 at about 300 beats a minute. Between those it passes through every value, including both boundaries.

It leaves the 3:1 category at 171 beats a minute and the 2:1 category at 240.

So a jazz quaver pair at 150 beats a minute and the same pair at 260, played by the same musician with the same intention and written identically, are on opposite sides of a category boundary. By the criterion that makes a third a third, they are different rhythms.

What that does and does not claim #

It does not claim that a listener hears a rhythm change at 240 beats a minute. Category membership and perceived identity are related and not the same, and the honest position is that the categorical argument has been run properly in the pitch domain and not properly here.

What it does claim is more useful and is checkable. The notated category and the played category come apart, and the tempo at which they do is computable from the swing model alone. A performance notated as a triplet figure occupies the 3:1 category below 171 beats a minute, the 2:1 category between 171 and 240, and the 1:1 category above — three categories, one notation.

That is a stronger version of what rung one of this ladder said. It said the notated ratio is correct at exactly one tempo. This says the category is correct over a range, and the range has ends.

A swung bar at 240 beats a minute has a ratio of 1.5 to one, which is between the 1:1 and 2:1 categories rather than inside either — the case the whole argument turns on.

The other direction: how far is too far #

The category picture also puts a number on the question this rung opened with, and it is a different number at every tempo.

A note is a wrong note when it has crossed into a neighbouring category. For an isolated pair, the nearest boundary from the equal category is at a share of 58.3 per cent, which is a displacement of 8.3 per cent of the pair. At a pair of 300 milliseconds that is 25 milliseconds; at a pair of 1,000 it is 83.

So the tolerance is proportional, and at fast tempos it is smaller than the deviations performers routinely produce. A 25-millisecond tolerance sits below a measured jazz offset and at the same size as a duo’s residual asynchrony.

That would be a contradiction if offsets and asynchronies changed ratios, and they do not. It is a real constraint on the third kind of timing — the deliberate subdivision — and the constraint is that at fast tempos the room to swing at all is very small, which is exactly what the swing curve says by another route.

The categories crowd, and that is why nothing beyond three is written #

Expressed as shares, the categories are at 50, 66.7, 75, 80, 83.3 and 85.7 per cent for ratios 1 through 6. The gaps between adjacent ones run 16.7, 8.3, 5.0, 3.3 and 2.4.

They halve, roughly, at each step. So the room around 1:1 is twice the room around 2:1 and three times the room around 3:1, and past about 4:1 the categories are closer together than the timing precision of a performer.

That is a quantitative account of something notation has always done and never explained. Western notation has symbols for 1:1, 2:1 and 3:1 — a pair, a dot, a triplet — and stops. It does not stop because nobody thought of 5:1; it stops where the categories become too close to be reliably produced or distinguished, and the arithmetic puts that in the same place the notation does.

The polyrhythm ladder’s question, answered from this side #

There is a result here that belongs to another ladder and arrives free.

The composite of a polyrhythm is a sequence of unequal gaps, and that rung argued that what makes 3-against-2 a figure and 7-against-5 weather is the number of distinct gap lengths rather than the length of the cycle. The categorical picture asks a different question of the same object: are the ratios between successive gaps inside categories, or on the boundaries between them?

For 3:2 the gaps are 2-1-1-2 and every successive ratio is 2:1 or 1:1 — category centres, all of them. For 4:3, 3-1-2-2-1-3, every ratio is 3:1, 2:1 or 1:1. For 9:2, ten gaps and every ratio 1:1 or 2:1.

For 7:5 the ratios include 5:2 and 3:2, which are 2.5 and 1.5 — the exact midpoints between adjacent categories, which is to say the boundaries themselves. Four of its eleven successive ratios sit on one.

So two independent criteria — how many distinct durations there are, and whether the ratios between them land in categories — agree that 9-against-2 is easy and 7-against-5 is not, and both disagree with the least common multiple. That is a stronger position than either had alone, and neither was reached for by looking at the other.

The pitch parallel, and where it stops #

The parallel with interval categories is close enough to be useful and it is not exact, and the differences are instructive.

Pitch categories are equally spaced and duration categories are not. Semitones are 100 cents apart all the way up. The duration categories 1:1, 2:1 and 3:1 sit at 50, 66.7 and 75 per cent, so the gaps shrink: 16.7 points, then 8.3, then 5, and beyond 4:1 they are close enough together that the idea of a category stops being useful.

Pitch has a name for every category and time does not. There is a symbol for 2:1 and for 3:1 and nothing at all for 5:2 — which is roughly what a swung pair at 140 beats a minute is.

And a mistuned pitch is corrected upward by the ear and a mistimed note may not be. Categorical perception in pitch is what lets temperament work at all: a third 14 cents from pure is still a major third. Whether a listener performs the same normalisation in time — hearing a 2.4:1 pair as a triplet figure — is exactly the experiment that has been done for intervals and not properly for durations.

The pitch version is the same picture in the other domain: two adjacent interval categories with a crossing between them, where the crossing is 24 cents out of a 100-cent gap. The duration domain’s crossing is proportionally wider and its gaps are unequal, which is the difference the two-part comparison is about — pitch categories are evenly spaced and duration categories are not.

How this sits beside the difference limen #

There is a second threshold in the neighbourhood and confusing the two would be easy.

The smallest pitch difference a listener can detect is a discrimination threshold — the point at which two things stop being distinguishable at all. A category boundary is an identification threshold — the point at which a thing stops being called one name and starts being called another. They are different quantities and in pitch they differ by more than an order of magnitude: a few cents against a hundred.

The same gap exists in time. Listeners discriminate timing differences of a few milliseconds under good conditions, and the category boundary for a 300-millisecond pair sits 25 milliseconds away. So there is a wide band — from about 3 milliseconds up to about 25 — in which a deviation is audible as a deviation and does not change what the rhythm is.

That band is where expressive timing lives, and its width is the reason the thing is possible. It is not that performers deviate by amounts too small to hear; it is that they deviate by amounts that are clearly audible and clearly inside a category. A duo’s residual asynchrony of twenty milliseconds sits in the same band, which is why an ensemble that is audibly not perfectly together is nonetheless playing the written rhythm.

And the pitch discrimination threshold across the range is around five cents, which is a twentieth of the gap between two pitch categories. The duration limen is about five per cent of an interval, which is a third of the gap between 1:1 and 2:1 and a whole gap between 4:1 and 5:1 — so the two domains have the same structure and completely different resolution.

Whose music, and whose ratios #

The categories 1:1, 2:1 and 3:1 are those of Western notation, and the swing model is a fit to recorded jazz.

Other traditions divide beats differently and notate them not at all. An aksak bar’s beats stand in a 3:2 ratio, which is a category no Western notation has a single symbol for and which Balkan practice treats as basic rather than as a deviation. Traditions with additive metres are, in the terms of this essay, traditions with a different category set — and the fact that they exist is the clearest argument that the categories are conventional rather than perceptual.

The safest claim is the arithmetic one. Given a set of ratios, the boundaries are their midpoints. Which ratios a listener has is a fact about a repertoire, and this essay has used the Western set because the swing model it is testing is a fit to Western performance.

What the picture cannot show #

It cannot show the sharpness. Everything here is boundary positions. Where the transition sits is arithmetic; how quickly identification swings across it is a measurement nobody has published for durations at the standard this site would want, and the figure that draws it says so.

It cannot show context. A pair is drawn in isolation. In music a duration is heard against a running beat, other parts, and an expectation of what the figure is — all of which is exactly the context that makes categorical normalisation possible, and none of it is in a two-note pair.

It cannot show the difference between hearing and reporting. Identification tasks measure what listeners say, and the pitch literature’s central finding is that discrimination and identification come apart: listeners can hear a difference they will not name. The duration equivalent is unstudied here and is likely the same.

It cannot show duration against onset. Every ratio here is a ratio of inter-onset intervals. A pair played legato and a pair played staccato have the same onsets and very different sounding lengths, and which one a category is about is not settled.

It cannot show the fast end. At very fast tempos the pair itself drops below the rate at which anything is a beat, and a category is presumably not available for a duration nobody can count. Where the two limits meet is not drawn, and past about twenty events a second the whole question changes into a question about pitch.

And it treats the swing model as exact. The constant-short-note finding is a fit with real scatter, from a specific corpus, and the crossing tempos of 171 and 240 inherit all of that.

That last caveat used to end by claiming the crossings survive a factor of two in the short note’s length. They do not, and the arithmetic is two lines. A crossing sits where the ratio reaches the boundary, so the tempo at which the figure leaves the 3:1 category is $17{,}143/s$ beats a minute and the tempo at which it leaves 2:1 is $24{,}000/s$, for a short note of $s$ milliseconds. Halve the short note to 50 and those become 343 and 480 beats a minute — nothing plays there, and both crossings have left the room. Double it to 200 and they become 86 and 120, both comfortably inside.

So the survival is one-sided, and how one-sided depends on what “the ordinary tempo range” is taken to be, which the essay never said. Against a metronome’s 40 to 208, both crossings are inside for a short note between 115 and 429 milliseconds and at least one for 82 to 600 — so the published 100 milliseconds has one crossing inside and not two, and losing the second takes a reduction of eighteen per cent rather than a factor of two. Against the axis the figures are actually drawn on, 90 to 320, both crossings are inside from 75 to 190 milliseconds and one from 54 to 267.

The claim’s survival is a fact about the range, and the range was never stated. What is robust is narrower and still worth having: for any short note between about 80 and 270 milliseconds — which spans every published fit and then some — the swing curve crosses at least one category boundary somewhere a musician might play. Two crossings is a claim about the specific fit.

The ladder from here #

Five rungs. The first two established that the swing ratio is not 2:1 and that the deviations that make a groove are unnotatable durations. The third split the deviations into two quantities that share a word. The fourth found that two players hold together by correcting toward each other, and that the standard measurement recovers their total responsiveness and cannot say who is following.

This one supplies the frame the first four were missing: a deviation is a distance inside a category, categories in time are the simple ratios, and the ratio that turns out not to stay inside one is the one this ladder started with.

What the table still has on it: whether listeners normalise durations the way they normalise intervals, which is one experiment; the timing patterns of repertoires whose category set is not the Western one; and the interaction between the categories and the tempo window, since a category at one tempo may be below the fast limit at another.