The short note is sitting on the floor

The fifth rung of this ladder supplied the frame the first four were missing: a timing deviation is a distance inside a category, categories in time are the simple ratios, and the swing ratio crosses two boundaries as the tempo rises — at 171 beats a minute and at 240 — while the notation says triplet feel throughout.

It left three things on the table, and one of them is a question about the frame rather than about what sits inside it: a category at one tempo may be below the fast limit at another. A ratio divides a beat into a long note and a short one, the short one gets shorter as the tempo rises, and at some point it stops being a note at all.

Working that out takes one line of arithmetic and produces a coincidence.

Every category has a last tempo #

The arithmetic is a division. At a tempo of B beats a minute the beat is 60000/B milliseconds, and a ratio of r to 1 makes the short note 60000/(B(r+1)).

Set that equal to the fast edge of the tempo window — a hundred milliseconds, which is the shortest interval a series of events can be a beat at — and each ratio has a tempo above which its short note is not a note.

4:1 ends at 120 beats a minute. 3:1 at 150. 2:1 at 200. 1:1 at 300.

Below 120 there are four categories available; between 120 and 150 there are three; between 150 and 200 there are two; between 200 and 300 there is one; above 300 there are none, and a beat has no interior at all.

The curve is always leaving a category that has already gone #

Put the two together and the fifth rung’s picture acquires a systematic offset.

The swing ratio leaves the 3:1 category at 171 beats a minute. The 3:1 category ceases to exist at 150. The ratio leaves the 2:1 category at 240; the 2:1 category ceases to exist at 200.

In both cases the category is gone before the ratio leaves it, by twenty-one beats a minute and by forty — fourteen and twenty per cent of the tempo.

So there is a band of tempo, twice over, in which the model says a drummer is playing a 3:1 and the arithmetic says a 3:1 cannot be played: between 150 and 171 the ratio is nearest to three, and the short note of an actual three-to-one would be under a tenth of a second. What the drummer is playing there is a ratio of about 2.8, whose short note is 100 to 110 milliseconds, and the category it is nearest to is one the tempo has already closed.

That is not a contradiction — the ratio is what it is and the category is a label — but it means the categorical reading of the swing curve has to be read with the window beside it. A boundary crossing at 171 is not a change of what is being played; it is the label catching up with a change the tempo forced at 150.

The two hundreds #

Now the coincidence, and it is the rung.

This site’s swing model has one fitted parameter. Friberg and Sundström measured four jazz drummers across tempo and found that the short note of the swung pair holds a roughly constant absolute duration of about a tenth of a second; the falling ratio follows from that. The hundred milliseconds is theirs, measured on drummers.

The tempo window’s fast edge is also a hundred milliseconds. That number is from a different literature entirely — tapping and beat-perception studies, measuring the fastest rate at which a series of events can be felt as a pulse — and this site has been using it since the metre ladder.

They are the same number, and nothing connects them except that both are about how short a musical event can be.

The consequence of their being equal is stronger than a coincidence report, and it is an identity rather than an approximation. The model’s short note is always the constant s, so its ratio is exactly r at the tempo where the beat is s(r + 1) — and a category of ratio r expires at the tempo where its short note, the beat over r + 1, reaches the window’s floor, which is the same equation whenever the two constants are equal. Tabulated:

The swing curve is the locus of expiring categories. Not near it, not correlated with it — the same curve, four for four, because the two hundreds are the same hundred. The offsets the section above reports are real and they are between the midpoint boundaries of the fifth rung’s categorical reading and these exact points; the underlying relation has no offset in it at all.

That also gives the coincidence a test, and one it passes on a number nobody used to fit it. Setting r = 1 predicts that swing vanishes at exactly 300 beats a minute, and the published measurement is that there is no swing left to measure above about three hundred. If the tapping floor were eighty milliseconds instead of a hundred the prediction would be 375 and if it were a hundred and twenty it would be 250, so the agreement is a real constraint and not a range wide enough to hold anything.

Two independent literatures, one constant, and a prediction about where swing ends that neither of them was asked for. That is as much as an arithmetic coincidence can be made to do, and it is enough to make the next paragraph’s reading the one to prefer.

If that is not a coincidence, the swing model changes character completely. It is not “the short note happens to hold constant at a value drummers arrived at”. It is:

The short note is the shortest note there is, and the ratio falls with tempo because the floor does not move.

Which has no free parameter in it. The ratio at a tempo B is (60000/B − F)/F with F the floor, and F is not a property of jazz.

What the floor reading predicts and the constant reading does not #

The two readings agree on every number in the published measurement, because they are the same arithmetic. They disagree about what else should be true, which is what makes the difference testable.

The floor reading says the effect is not about jazz. Any tradition that subdivides a beat unequally at speed should show the same falling ratio, with the same floor, because the floor is a property of listeners. The constant reading has no reason to expect that.

The floor reading says the ratio is a maximum, not a target. A drummer cannot play a higher ratio at a given fast tempo, because the short note would go under the floor; a lower ratio is always available. So the measured ratios should sit at the top of their range with scatter below them and none above, and the constant reading predicts scatter both ways.

And the floor reading explains where the model breaks. At slow tempos the arithmetic gives absurd ratios — at 60 beats a minute the floor would allow nine to one — and the site’s model caps the ratio at 3.5, which is a second fitted parameter and an admitted one. The floor reading says exactly why a cap is needed: below about 170 beats a minute the floor stops binding, and what limits the ratio there is something else entirely, presumably the notation and the style. Above that tempo there is one parameter and it is not free; below it, there are no parameters and no model.

The same floor, one level down #

There is a second place the floor is already doing work in this collection, and noticing it is what makes the coincidence look less like one.

The milliseconds that are the groove found that a fixed physical delay is a different note value at every tempo, so a consistent feel is unwritable in principle. That argument is about a constant in milliseconds against a grid in beats, which is the same shape as this one: a quantity fixed in absolute time, meeting a quantity that scales with tempo, and the ratio between them moving.

The deviations are not noise then split those deviations into an offset and a pattern. An offset is a constant in milliseconds. A pattern is a set of fractions of a bar. The whole ladder has been about the collision between absolute time and proportional time, and the floor is that collision at its hardest edge — the place where absolute time does not merely fail to scale but refuses to go further.

At the slow end the floor is not binding at all: a hundred milliseconds is a small part of a long beat, so the ratio is free to be whatever the player wants and the notation’s three-to-one is briefly right. The floor is a fast-tempo phenomenon, which is why the collision between absolute and proportional time only shows up where the beat gets short.

What it does to the fast end of the repertoire #

The practical consequence is at the top of the tempo range and it is sharp.

Above 300 beats a minute there is no subdivision at all. Not “swing disappears”: every two-note division of the beat has a short note under the floor, so the beat has no interior, and whatever a drummer plays at that tempo is a stream of beats rather than a pattern within them.

That is a prediction about bebop’s upper tempos, which run to 300 and beyond, and it agrees with what those performances sound like, in the way the swing ratio’s own published endpoints do: the ride pattern flattens to even quavers well before the tempo ceiling, and at the extreme the quavers themselves become the beat and the notated beat becomes a hypermetrical level. The music does not stop subdividing; it renumbers, which is a change of metrical level rather than a change of feel — the same renumbering a hemiola forces on a bar, and the metrical levels of a tempo against the window is exactly the figure that shows why.

Where the count of categories goes #

The last thing the arithmetic gives is a number nobody has asked for and which turns out to be the tidiest statement of the whole rung: how many duration categories a beat has, as a function of how fast it is going.

Four below 120 beats a minute. Three from 120 to 150. Two from 150 to 200. One from 200 to 300. None above.

Read against the count of pitch categories an octave holds the parallel is exact in form and opposite in content. An octave’s category count is fixed by a naming capacity and does not depend on anything about the music; a beat’s is fixed by a floor in absolute time and depends on nothing else. Pitch categories are a property of the listener and duration categories are a property of the listener and the tempo, and there is no tempo at which pitch runs out.

That is why rhythmic notation has always needed a tempo mark and pitch notation has never needed one. A crotchet is not a duration; it is a ratio, and the set of ratios available is not the same set at every speed.

The floor has a consequence for what a swung pair is, and it is sharper than the ratio numbers suggest.

The same notated figure is a different rhythm at each end of an ordinary tempo range, by the categorical criterion, and the notation says triplet feel throughout. That is the floor’s real cost: it does not merely bend the ratio, it walks the ratio across a category boundary while the page stays the same.

Which computation produced the numbers #

Two published constants, one division, and this site’s own swing model.

TEMPO_WINDOW is the site’s existing three numbers — 100, 550 and 2000 milliseconds — from the beat-perception literature, and it has been used since the metre ladder for exactly one thing: whether a candidate metrical level can be a beat.

swingRatio is the existing model, unchanged: the ratio is the beat minus the short note, over the short note, capped at 3.5.

categoryWindow divides: a ratio r at a tempo B has a short note of 60000/(B(r+1)) milliseconds and a long note of r times that, and the category is available when the short note clears the fast edge and the long note is inside the slow one.

The one stretch is using the fast edge for a note rather than for a beat, and it is stated in the machinery as a stretch. The published hundred milliseconds is the shortest inter-onset interval at which a pulse can be felt; whether it is also the shortest at which two notes can be told apart as long and short is a different question with a different literature and a smaller answer — order judgements survive down to about twenty milliseconds. The direction of the error is known: a smaller floor pushes every category end upward in tempo, and the coincidence with the swing constant then stops being a coincidence and becomes a puzzle in the other direction.

Whose music, and when #

The swing measurement is jazz drumming from recordings across the twentieth century, and the four drummers Friberg and Sundström measured are named in the source. The tempo window is not about any repertoire.

The floor reading, if it holds, makes a claim about traditions this collection has not entered. Unequal subdivision at speed exists in Viennese waltz playing, in Norwegian and Swedish dance forms, in Turkish and Balkan practice, and in Brazilian and West African music — and the prediction is the same falling ratio with the same floor in all of them, because the floor is a listener’s rather than a style’s.

That is the fourth time in a row that a rhythm rung here has ended at a measurement this collection cannot make, and it is worth saying plainly rather than as an aside: the test is a corpus of performance timings across tempo from several traditions, it exists in the world, and none of it is here.

What the picture cannot show #

The floor is used for two different jobs and it may not do both. A hundred milliseconds is the fast edge of a beat, and the essay applies it to a note inside a beat. If the second limit is lower, every category end moves up, the coincidence weakens, and the whole argument is a rearrangement of one number.

The coincidence is between two round figures. Friberg’s short note is reported as about a tenth of a second; the tempo window’s fast edge is variously 100 or 120 milliseconds in different sources. Two quantities agreeing to within their own precision is weaker evidence than two quantities agreeing to three figures, and this is the first kind.

Categories are treated as existing or not. A real boundary is soft, and a short note ten milliseconds under the floor does not vanish — the pair becomes harder to hear as two unequal notes, gradually. Nothing here has a gradient in it.

And no drummer’s short note was measured by this collection. Every number about swing here descends from one published study of four players, which the fifth rung said and which is worth saying again.

Where this ladder goes next #

Six rungs. The swing ratio is not two to one and falls with tempo; the deviations that make a groove are unnotatable; they split into offset and pattern; two players correct toward each other and the measurement cannot say who is following; the deviations sit inside categories whose boundaries are the simple ratios; and now the categories themselves have tempo ranges, and the constant the whole ladder rests on may be the edge of the tempo window wearing a different name.

The rung after it is the one the floor reading makes available, and it needs no corpus. If the short note is at the floor, then a player’s control over it is control at a boundary, and everything this ladder has measured about deviations — the thirty milliseconds a soloist sits behind the ride, the fifty a Viennese second beat arrives early — is a deviation of a note that has no room to move in one direction. The distribution of a quantity resting against a wall is not the distribution of a free one, and every statistic this ladder has computed on those deviations assumed it was. Two players correcting toward each other is the case where it would show first, because a correction that cannot be made in one direction is not a correction.