Which of the two is the beat

Three against two is written as a bar of two with a triplet laid over it. The bar line, the time signature and the conductor’s arm all say that the two is the beat and the three is the decoration.

The composite is one sequence of onsets, and this site has a rule set for finding the beat in one. Handed the composite of three against two, it does not choose the two.

Why the fit prefers the faster layer #

The reason is structural and it holds for every ratio, not just this one.

By construction the composite contains every onset of both layers. So whichever layer is proposed as the beat, all of its positions are struck and none is empty — the two candidates score identically on the two rules that reward struck strong positions and punish empty ones.

They differ only on the third rule, which punishes onsets that fall between the beats. The composite has a + b − 1 onsets. If the a-layer is the beat, a of them are on the beat and b − 1 are between. If the b-layer is the beat, b are on and a − 1 are between. The faster layer therefore always leaves fewer strays, and it wins by exactly the difference between the two counts.

For three against two that is one stray against two. For seven against five it is four against six. The preference rules prefer the faster layer at every coprime ratio there is, and they do so by an argument that needs no arithmetic at all once it is written down.

That is a result and it is obviously not the whole story, because nobody hears a slow 3-against-2 in three.

The score that cannot be compared #

There is a bookkeeping trap in the middle of this and the site walked into it once before.

The raw preference score of a candidate is a sum over its strong positions, so a candidate with a short period has more strong positions and therefore a larger raw total for the same quality of fit. Reading down the score column, “every step is a beat” ties the three at eight — a reading in which all six steps are beats, four of them struck and two empty, which is not a metre anybody has ever felt.

The site’s own metre machinery carries a normalised score for exactly this reason: the total divided by the number of strong positions. On that measure the three scores 2.67, the two 2.00 and every-step 1.33, and the ordering is the one the figures use from here on. Comparing candidates of different periods on the raw sum is a mistake with a name and this site has made it.

The rule that has a number in it #

The preference rules are about coincidence. There is one other rule in the literature and in this site’s machinery, and it is not about the pattern at all: a beat has a preferred rate.

A series of events can be a beat at all only inside a window of roughly a hundred to two thousand milliseconds, with the preferred rate near five hundred and fifty. At a hundred and eighteen to the minute — the tempo the rest of this essay turns out to be about — a three-against-two offers metrical levels at 254, 339, 508 and 1017 milliseconds, so two of them are close to the preferred rate and neither is at it. That is the whole reason the choice is live: the rate rule has two nearly equal candidates to weigh, and the fit rule has to break the tie.

The window has been measured: a series of events slower than about two seconds apart cannot be felt as a beat, one faster than about a tenth of a second cannot either, and asked to tap comfortably people land near 550 milliseconds. Those three numbers are the whole of the rate rule and this site takes them as published.

Weighting a candidate by its rate needs a shape, and the honest shape here is the one with no free parameter in it: linear in log period, zero at each edge of the window, one at the preferred rate. A smooth bell would need a width fitted to nothing, and a width fitted to nothing is a number that can be moved until the answer comes out right.

The two rules disagree, and the disagreement has a tempo #

At a bar of 1,600 milliseconds the three’s beat is 533 ms and the two’s is 800. The three is nearer the preferred rate and it also wins on fit, so it wins twice over.

At a bar of 700 milliseconds the three’s beat is 233 ms and the two’s is 350. Now the two is nearer 550, and it is nearer by enough to overturn the fit.

Where the crossing is, for every ratio #

The 118 beats a minute this essay turns out to be about is one ratio’s answer, and the same two terms give one for each. The fit is a property of the ratio and the rate is a property of the tempo, so the crossing is where they meet:

Every crossing falls between 118 and 163 beats a minute, which is to say inside the range of ordinary musical tempos and nowhere else. The bar lengths spread by a factor of two and the tempos do not, because a longer bar at a more complex ratio still has to put the slower layer near the preferred rate, and that is a rate rather than a bar.

Two readings follow. Three against two has the lowest crossing of the family, so it is the ratio for which the notated beat — the slower layer — holds its claim over the widest range of tempos, and the more complex ratios hand the beat to the faster layer at tempos where three against two would already have given it up. That is the opposite of the usual intuition, which expects the harder figure to be counted in its slow layer for longer.

And seven against two never crosses at all, because the slower layer’s normalised fit is exactly zero: all two of its positions are struck and five onsets fall between them, which the third rule punishes to nothing. So at that ratio the fast layer is the beat at every tempo the window allows, and the notated two is not a candidate on this account rather than a losing one. The disagreement this essay is about exists only where both layers are viable, and the ratio has to be simple enough for the slower one to be viable at all.

Somewhere between those two tempos the answer changes hands, and because both terms are continuous the crossing is a single number.

1,013 milliseconds #

A bar of 1,013 milliseconds is 118 to the minute counted in twos. That is not an exotic tempo. It is a moderate one — somewhere between a slow rock groove and a fast waltz — and it sits in the middle of the range in which most of the repertoire that uses three against two actually lives.

So the prediction is this: the same written figure, played at 100 and at 140, is heard with the beat in different layers, and neither performance has done anything but change the tempo. The notation cannot express the difference because the notation fixes the bar line, and the bar line is precisely what the rules say is up for grabs.

The crossing moves with the ratio #

Four against three crosses at a bar of 1,265 milliseconds. Seven against five crosses at 1,835.

The shape is the one the previous figure drew: the faster layer wins at slow tempos where its own rate is comfortable and loses at fast ones where it has run off the bottom of the window. What the scoring adds is how small the margin is — a fifth of a point on a scale whose other verdicts run to twenty.

The pattern is easy to state. The fit term prefers the faster layer always; the rate term prefers whichever layer is nearer 550 ms, which is the slower one at fast tempos and the faster one at slow tempos. So the crossing sits where the faster layer’s beat has fallen far enough below 550 to lose more on rate than it gains on fit — and since the faster layer’s rate is a/b times the slower’s, a larger a/b pushes the crossing to a longer bar.

The size of the fit advantage moves too, and in the same direction. For three against two the faster layer leaves one stray onset against two, a difference of one; for seven against five it is four against six, a difference of two, spread over a normalisation that divides by seven rather than by five. Working it through, the fit advantage grows more slowly than the rate disadvantage, which is why the crossing for 7:5 sits at nearly twice the bar length of 3:2’s rather than at seven-fifths of it.

Three layers, and a beat that is in none of them #

Ensembles do not stop at two. Three, four and five together produce a composite whose candidate beats include three layers and every common divisor of the grid, and the fit argument above still applies to each pair.

What changes is that the winner need not be a layer at all. With three layers the composite is dense enough that a candidate period nobody is playing can carry more onsets on its strong positions than any of the parts do — and at the rate term’s preferred end, a period that lands near 550 milliseconds beats one that does not, whether or not an instrument is marking it.

That is the formal version of something ensemble musicians describe directly: the beat is a thing the group shares and it is not necessarily anybody’s part. In West African practice the reference is a bell pattern rather than a pulse, and the dancers’ feet mark a division that the bell does not play. The model has no difficulty with that; what it has no term for is why a group would organise itself that way, which is a question about a repertoire rather than about a rule set.

The grid is not a metre #

The third candidate in every figure above is “every step is a beat” — the common subdivision itself, treated as the beat.

It is worth carrying because it is the reading a naive scoring picks. All the onsets are on a strong position, none is off, and nothing is punished except the empty positions. On raw score it ties the winner. On the normalised score it is last, because two of its six strong positions are empty and dividing by six is what makes that count against it.

And it is last for a second reason that has nothing to do with the pattern. At any tempo at which the layers are beats, the grid is a third to a sixth of that, which puts it near or below the fast edge of the window. At the crossing tempo the grid runs at 169 milliseconds, which is inside the window and well away from the preferred rate, and at any faster tempo it is outside it entirely. The subdivision is real and it is not a beat, which is what the window is for.

What the earlier rung got wrong, and what it got right #

The first rung of this ladder has a section headed which part wins, and it lists loudness, timbre, register, prior establishment and cultural familiarity. It then says, in as many words, that which one wins “is decided by things that are not in the arithmetic”.

Half of that is wrong. Two of the things that decide it are in the arithmetic — the fit and the rate — and between them they name a specific tempo at which the answer flips. That is a stronger claim than any of the five listed, because it is falsifiable by a tapping experiment with one independent variable.

The other half stands, and stands rather well. Loudness, register and prior establishment are real and they are not in this model at all; the model has no term for which layer is on the bell and which on the hand drum. What the arithmetic supplies is a default, and the extra-musical factors are what a piece uses to override it.

Whose music, and whose experiment #

Every number above rests on a rule set fitted to notated Western music and on a tempo window measured in listeners raised on it. Both are named in the essays they come from and neither is a fact about hearing in general.

The specific worry is that a preference rule set which punishes off-beat onsets encodes a taste for metrical clarity that not every tradition shares. In West African ensemble practice the ability to hold a part that is deliberately off every other part’s beat is a competence, and a rule set that scores such a part badly is measuring conformity to one style rather than perceptual difficulty.

The claim this rung makes is therefore narrower than it looks: given this published rule set and this published window, the beat of a polyrhythm changes layer at a computable tempo. Whether listeners do that is a question for an experiment, and the experiment is cheap — the same recording at two tempos, and count the taps.

What the picture cannot show #

It cannot show which layer is loud. Every figure here treats the two layers as one undifferentiated set of onsets, which is exactly what the composite is and exactly what a real polyrhythm is not. A timekeeper on a bright bell is not one of a set.

It cannot show hysteresis. The model computes a winner from the pattern and the tempo alone, with no memory. A real listener who has settled into one reading is hard to dislodge from it, so a piece that accelerates through 1,013 milliseconds may well not switch, and the same piece decelerating through it may switch at a different point. Metre is a state, and this model has none.

It cannot show the third possibility. Both figures offer the two layers and the grid. A listener might hold neither — hearing the composite as a bar of four unequal beats, 2-1-1-2, which is what an aksak bar is and which the candidate list here never proposes. Adding it is one line and the results of doing so are not in this essay because nobody has measured whether anybody hears it.

It cannot show what happens at the crossing. At exactly 1,013 milliseconds the model reports a tie, and a tie is not a percept. What a listener does at a genuinely ambiguous tempo — hold one reading, alternate, or hold both — is the interesting question and the model’s answer to it is a pair of equal numbers. The same shortfall applies to every scoring account of metre, and it is why the ambiguity itself has to be measured rather than derived.

And the rate weighting is a shape, not a measurement. The three numbers in the window are published; the straight line drawn between them is a choice, made to avoid fitting a width. A different shape moves the crossing. Roughly: a narrower peak moves it later and a broader one earlier, and the qualitative result — that there is a crossing, in the ordinary tempo range — survives every shape tried.

The far end of the same axis #

The tempo axis this rung has been sliding along does not stop at fast. Keep going and the layers stop being beats at all.

So one polyrhythm has three regimes on one continuum, and only the middle one is what anybody would call rhythm: below about 100 milliseconds per event nothing is a beat, between there and two seconds something is and the question is which, and above about 20 events a second the whole thing is an interval. The two boundaries have different mechanisms and different literatures and they are the same axis.

Drawn as trees the two readings have the same six leaves and disagree entirely about the branching — which is exactly why nothing in the sound distinguishes them and why a listener must choose. Every onset is in both trees; only the grouping differs, and grouping is the one thing a signal does not carry.

The ladder from here #

The composite has now been given a beat, and the beat turns out to be a function of tempo. What has not been asked is who is producing the composite: one player with two hands, or two players with one each, and whether that makes any difference to what comes out.

It does, it is measurable, and the measurement that everybody cites for it turns out not to decide the question.

Further on: the cross-rhythm, where one pattern is played against several beats at once rather than against one other pattern — which makes the candidate list above the whole point rather than an intermediate step. And two cycles whose ratio admits no common grid at all, so that no composite exists and the leftover is a comma.