Rhythm and metre

The time signature is a claim

A bar line is not a measurement. It is a claim about where the accents are, made before the sound exists, and there is a metre-induction model here that can be handed the same onsets and asked whether it agrees. On a hemiola it does not: the page says three and the model says six, by a margin of twelve against minus four. And on a bar of seven the signature is not even a candidate — what decides the reading is the beaming, which the signature does not contain.

17 min read 9 figures Notation hides things Time is a circle What the listener supplies

Assumes: The beat is inferred, and sometimes wrongly · The stave is not a ruler

Notation records two things about a note: which pitch and when. The pitch axis turned out to be a coordinate system with a theory in it — the diatonic set, with the accidental as its patch. The time axis has a theory in it too, and the theory is written at the front of the line in two numbers.

The difference is that the vertical axis is a representation, which can be inefficient but cannot be wrong. A time signature can be wrong. It asserts something about the music that the music may not do.

This site has spent five essays building a model of what a listener does with a stream of onsets, and the model produces a number for each candidate reading. So the assertion is testable, and this essay tests it.

What a signature asserts

A time signature does two things at once and they are easy to confuse.

It says how long a bar is: 3/4 is three crotchets. That part is a unit, and units cannot be wrong.

It also says which positions inside that bar are strong. A downbeat is stronger than a third beat, which is stronger than an offbeat, and the whole hierarchy is implied by the signature alone. That is a claim about the sound, and it is the claim the induction model this collection built was written to evaluate.

One pattern, four metres. The same 8-step onset pattern read under 3 candidate metres, each scored by a preference rule set: 3 for a strong position that carries an onset, -2 for one that does not, -1 for an onset that lands off every strong position. The scores are 4/4 -1, 2/4 1, 8/8 as 3+3+2 9, so 8/8 as 3+3+2 wins. The page prints 4/4, which the model does not prefer — it ranks 8/8 as 3+3+2 above it. Nothing about the sound differs between these readings; the bar line is supplied by the listener. — Fig. 1 The hierarchy a signature implies, priced against the onsets it is claimed over. The page prints 4/4 and the rules score it at −1, 2/4 at 1 and 8/8 as 3+3+2 at 9 — so the printed signature is not merely unsupported by the onsets, it is the worst of the three readings offered. Nothing about a sequence of onsets produces a hierarchy; it is supplied by the reading, and a signature is how the page supplies one.

The model scores a candidate reading by asking how much of that hierarchy the onsets actually support: three points for a strong position that carries an onset, two off for one that does not, one off for an onset that lands nowhere strong. Those weights are stated rather than fitted, they are the ones the preference-rule literature uses, and every figure below uses the same three numbers.

A hemiola, which the page and the model disagree about completely

Two bars of 3/4 hold twelve quavers. Four dotted crotchets also hold twelve quavers. A hemiola is a passage that does the second while the page says the first, and it is old enough to be named after a Greek ratio.

A phrase in ordinary notation. A phrase written on a stave. Notation records what a player should do rather than what the air does, so it shows the note names exactly and the pitches only by convention — which is the reason so much of the evidence here is drawn some other way. — Fig. 2 Two bars of 3/4 with the onsets falling every three quavers. The bar lines say the accents are at nought and six; the onsets are at nought, three, six and nine. Every performer reads this correctly and no reader of the page would call the bar line a mistake — which is the interesting part.

One pattern, four metres. The same 12-step onset pattern read under 4 candidate metres, each scored by a preference rule set: 3 for a strong position that carries an onset, -2 for one that does not, -1 for an onset that lands off every strong position. The scores are 3/4 -4, 6/8 12, 12/8 12, 2/4 -4, so 6/8 and 12/8 tie and the model does not choose. The page prints 3/4, which the model does not prefer — it ranks 6/8 above it. Nothing about the sound differs between these readings; the bar line is supplied by the listener. — Fig. 3 The same twelve positions scored under four readings. The printed 3/4 gets minus four: two of its six strong positions carry an onset, four are empty, and two onsets land off the grid entirely. The 6/8 reading gets twelve — every strong position filled, nothing empty, nothing off. The page and the model are not close.

Minus four against twelve is not a marginal disagreement. On these onsets alone the printed signature is the joint worst of the four readings offered, and it is worse than a reading the composer would say is wrong.

Joint worst rather than worst, and the reason is worth a paragraph, because it says something about what a candidate list is. The four readings return two scores between them: 3/4 and 2/4 both get minus four, and 6/8 and 12/8 both get twelve. Neither pairing is a coincidence.

6/8 and 12/8 are the same hypothesis here. Twelve quavers is one bar of 12/8 and two bars of 6/8, and the strong positions 12/8 places at nought, three, six and nine are exactly the ones 6/8 places at nought and three, repeated. Every count in the scoring is identical: four strong positions, four filled, nothing empty, nothing off the grid. The difference between the two signatures is hypermetrical — whether the second group of six is a new bar or the second half of one — and a hypermetrical claim is a claim about the relation between bars. Given one bar there are no bars to relate, so the model has nothing to score and returns the same number twice. Offering both as candidates makes the list look wider than it is.

3/4 and 2/4 agree for a duller reason: with six strong positions each, both put four of them where there is no onset and leave two onsets off the grid, so the arithmetic lands in the same place from different arrangements. That one is close to a coincidence, and it is the kind that a raw score invites — two readings can reach one number by trading a hit against an empty.

The honest reading of this figure is therefore that it offers two distinct hypotheses and a tie-break the onsets cannot supply. That does not weaken the result; the gap between the two hypotheses is sixteen points and the page has picked the losing one. It does mean that “worst of four” was counting names rather than readings.

That is worth sitting with rather than explaining away, because the obvious explanation is available and it is the point. The page is not describing the onsets. It is describing the piece the onsets are part of — a piece in three, from which this passage temporarily departs — and the departure is the musical event. A bar line that agreed with the onsets everywhere could not express a hemiola at all, because there would be nothing for the hemiola to be against.

So the signature’s claim is not “the accents are here”. It is “the accents are here unless the music says otherwise, and a departure is the thing being reported”.

The tresillo, which is the same thing without the excuse

The tresillo is where the metre ladder started, and it is the same shape as the hemiola with one difference: there is no larger piece in four for it to be a departure from. In the repertoires where this pattern lives it is not a syncopation against a metre, it is the metre — a timeline, cyclic, with no downbeat that needs asserting.

Writing it in 4/4 is therefore a translation into a system that has a downbeat, and the syncopation the notation produces is an artefact of the reading rather than a property of the music. This site has a number for how much syncopation a reading imposes, and the number is a function of the reading and not of the sound.

One pattern, four metres. The same 8-step onset pattern read under 4 candidate metres, each scored by a preference rule set: 3 for a strong position that carries an onset, -2 for one that does not, -1 for an onset that lands off every strong position. The scores are phase 1 -1, phase 2 -7, phase 3 -1, phase 4 -1, so phase 1 and phase 3 and phase 4 tie and the model does not choose. Nothing about the sound differs between these readings; the bar line is supplied by the listener. — Fig. 5 The same three onsets scored under the four places a four-step beat could begin. Three of the four tie at −1 and only one is ruled out, so the model does not choose — because a cycle has a shape and no beginning, and that is the thing a bar line cannot represent. Choosing where to put the bar line is choosing which rotation to write down, and the notation forces a choice the pattern does not make.

A bar of seven, where the signature is not a candidate at all

The clearest case is the one where the model cannot score the page’s answer, because the page has not given one.

One pattern, four metres. The same 7-step onset pattern read under 3 candidate metres, each scored by a preference rule set: 3 for a strong position that carries an onset, -2 for one that does not, -1 for an onset that lands off every strong position. The scores are 7/8 as 2+2+3 9, 7/8 as 3+2+2 -3, 7/8 as 2+3+2 3, so 7/8 as 2+2+3 wins. Nothing about the sound differs between these readings; the bar line is supplied by the listener. — Fig. 6 One bar of seven quavers with onsets at nought, two and four, scored under the three ways of grouping seven. The first gets nine, the third gets three and the second gets minus three. The signature “7/8” is identical in all three cases. What distinguishes them is not on the page as a number at all — it is in the beams.

A signature of 7/8 says the bar is seven quavers long and stops. Whether those seven are 2+2+3, 3+2+2 or 2+3+2 is a different question with a different answer, and the model scores the three at nine, three and minus three — as wide a spread as any comparison in this essay.

The information exists on the page. It is in the beaming: quavers beamed in twos and threes tell a player where the beats are. But the beaming is a typographic convention rather than part of the signature, it is routinely regularised by engravers, and it is discarded entirely when a piece is described as being “in 7/8”.

Two of those are regular — one number of beats, one division, all the way down — and 7/8 as 3+2+2 is not, because its beats are of two different lengths. This collection has no word for that beat any more than the notation has a number for it. A signature can say how many units a bar has; it cannot say how they are grouped unless the grouping is uniform, which is the sharpest limit on what the claim is able to claim.

So the notation is under-specified where it is usually said to be over-specified. A time signature is at best two thirds of a metre: it fixes the bar and the unit and leaves the grouping to a convention that is not part of it. In additive metres that missing third is the whole of what makes one piece different from another.

The backbeat, which cannot be written in onsets

One more case, and it goes the other way: here the page is right and the model has nothing to score.

One pattern, four metres. The same 16-step onset pattern read under 2 candidate metres, each scored by a preference rule set: 3 for a strong position that carries an onset, -2 for one that does not, -1 for an onset that lands off every strong position. The scores are 4/4 12, 2/4 4. At a bar of 2000 ms each candidate's beat also has a rate — 4/4 at 1000 ms, 2/4 at 500 ms — and weighting the fit by how near that rate is to the preferred 550 ms gives 4/4 1.61, 2/4 0.47, so 4/4 wins on the two rules together. The page prints 4/4, which the model also prefers. Nothing about the sound differs between these readings; the bar line is supplied by the listener. — Fig. 7 Four onsets on four beats, which is the least informative rhythm there is. The printed 4/4 fits perfectly and so does everything else that divides it; adding the rate rule breaks the tie in the page’s favour, because a bar of two seconds puts 4/4’s beat at five hundred milliseconds and that is where the preferred beat rate is.

Now put a backbeat on it — accent the second and fourth beats, as every drummer in a century of popular music has. Nothing in the onsets changes. The pattern is identical, the model’s score is identical, and the thing that makes the music what it is has not been written down at all: it is in which drum plays, how hard, and with what attack.

The page can say snare and it can write an accent mark, and both are outside the metrical system. A signature describes a hierarchy of positions; an accent that contradicts that hierarchy without moving any onset is not expressible in it. The model cannot see it either, for the same reason — it takes onsets and nothing else.

Hand a phase-corrected oscillator a passage whose beats stop being sounded and it coasts, and it coasts for a while, because the beat is a prediction rather than an observation. A backbeat is the same situation in reverse: the strongest events are in the places the hierarchy says are weak, and the hierarchy survives anyway. In both cases what survives is the claim — which is the thing a signature writes down and a sequence of onsets does not contain.

Where the page and the model do agree

It would be easy to read the last four figures as an indictment, and that is not what they are. Three of the five notated cases this collection carries are cases the model settles the same way the page does, and the two that disagree are both cases where the disagreement is the musical point.

That is the more interesting result. A notation that agreed with an onset-counting model everywhere would be describing the onsets, and a notation that describes the onsets has thrown away the difference between a hemiola and a bar of six. The page’s claim is not about this passage’s onsets; it is about the metre the piece is in, of which this passage’s onsets are an instance and sometimes a deliberate exception. Every disagreement above is an exception being flagged.

The same onsets, read from four places, four syncopation counts. A 12-step pattern with 4 onsets, against 4 readings of its bar. A position's weight is zero on the downbeat and one lower at each level down the subdivision tree, drawn here as the depth of the bar hanging beneath it. A note on a weak position followed by a rest on a stronger one costs the difference. from step 1 scores 2 — the note at step 4 against the rest at step 5, costing 1; the note at step 10 against the rest at step 11, costing 1. from step 2 scores 2 — the note at step 6 against the rest at step 7, costing 2. from step 3 scores 3 — the note at step 2 against the rest at step 3, costing 1; the note at step 5 against the rest at step 7, costing 1; the note at step 8 against the rest at step 9, costing 1. from step 4 scores 2 — the note at step 4 against the rest at step 5, costing 1; the note at step 10 against the rest at step 11, costing 1. — Fig. 8 The same twelve onsets read from four different starting points under one metre. Rotating where the bar begins changes the syncopation score without moving a single onset, which is the sharpest form of the claim: the number the page’s reading produces is a property of the reading. A listener who has not been told where the bar starts is choosing among these rows, and the page has chosen for them.

Read that way, the two numbers at the front of the line are doing something notation is generally rather bad at: they are stating a prior. Everything else on the page is a record of events, and this is an assertion about the frame the events are to be heard in. It is the only place in the system where that happens, and it is why the disagreements are informative rather than embarrassing.

What the bar line was for before it was a claim

The metrical reading of a bar line is not as old as the bar line. Mensural notation, which carried European music through the fourteenth and fifteenth centuries, had no bar lines at all; the length of a note was written into its shape and the relations between shapes were governed by rules of imperfection and alteration that a modern reader finds baroque in the wrong sense.

Bar lines arrive in the sixteenth century in keyboard and lute sources, and they arrive as an alignment device: several voices on one system have to be readable together, and a vertical line every so often keeps the eye in the right place. Scores of vocal music kept the parts unbarred for another century.

The metrical meaning accumulated afterwards, and it accumulated because the music being written happened to be strongly metrical. That is worth saying plainly: the bar line became a claim about accent because for two hundred years the claim was nearly always true, and the notation was then exported to music where it is not.

One pattern, four metres. The same 12-step onset pattern read under 3 candidate metres, each scored by a preference rule set: 3 for a strong position that carries an onset, -2 for one that does not, -1 for an onset that lands off every strong position. The scores are 6/8 5, 3/4 7, 12/8 5, so 3/4 wins. The page prints 6/8, which the model does not prefer — it ranks 3/4 above it. Nothing about the sound differs between these readings; the bar line is supplied by the listener. — Fig. 9 Twelve quavers whose first half groups in threes and whose second half groups in twos — the pattern that gets two time signatures printed one after the other, because no single one accounts for it. On the raw score 3/4 wins with seven against 6/8’s five; divide each by the number of strong positions it has to fill and the order reverses, 1.25 against 1.17. The model is as undecided as the page is, and for the same reason. This is the honest case: the notation prints two signatures because there are two, and it has no way to print one thing that is both.

How much of this depends on the three weights

Every number above rests on three constants — three for a filled strong position, minus two for an empty one, minus one for an onset off the grid — and they were stated rather than fitted. That is the right way round, but it leaves an obvious question unanswered: which of these results are about the music and which are about the constants?

It is a cheap question to answer, because the scoring is linear in the weights and comparing two candidates only ever needs their difference. Scaling all three by a positive constant scales every score and cannot reorder anything, so one weight can be fixed at one and the other two swept. Holding the hit at 1 and running the empty penalty and the off-grid penalty independently from 0 to 2 covers every ratio anyone would defend, including the degenerate corner where neither miss costs anything at all.

The hemiola result is invariant. Across the whole grid, 6/8 beats the printed 3/4 in every single cell. That is not a close call surviving: 6/8 fills four of four strong positions and leaves nothing off the grid, so its score is four hits and nothing else whatever the penalties are, while 3/4 can only lose as the penalties grow. There is no assignment of the three weights under which the page’s reading wins, and the essay’s central disagreement is therefore a fact about the onsets rather than about the preference-rule literature.

The bar of seven is invariant too. 2+2+3 wins in every cell of the grid, which matters more than it looks, because that comparison is between three candidates of identical period and identical strong-position count — the sort of comparison a raw score is most likely to get wrong. Here it cannot: 2+2+3 puts a strong position under all three onsets and the other two groupings do not.

The tresillo is invariant in its winner and not in its loser. 3+3+2 wins everywhere. Whether the printed 4/4 comes last is weight-dependent — it does in 86 per cent of the grid, and in the remaining corner, where an empty strong position costs almost nothing, 4/4’s two strong positions cost it less than 2/4’s four and it climbs past it. The claim that the page’s reading is the worst of the three is a claim about the penalty ratio; the claim that it is not the best is not.

And the America bar is undecided at exactly the point the essay said it was. The raw winner is 3/4 in 86 per cent of the grid and 6/8 in the other 14, so the two readings trade places under a change of weights as well as under a change of normalisation. That is a stronger version of what the figure already reported. A comparison whose answer moves under both a reweighting and a renormalisation is not a comparison that has an answer, and the notation prints two signatures for the same reason.

The pattern across the four is worth naming. The comparisons that survive are the ones where a candidate fills every strong position it claims, because then the penalties have nothing to act on and only the hit term is live. The comparisons that move are the ones settled by the relative price of two kinds of miss, and that price is the part of the model with no evidence behind it. So the robustness of a result here is legible in advance from the counts in the figure, without any sweeping: a reading with no empties and nothing off the grid is winning for a reason that no weighting can take away.

Which computation produced the numbers

Every score in this essay comes from the same function, with the same three weights: three for a filled strong position, minus two for an empty one, minus one for an onset off the grid. Nothing was tuned per figure.

Candidates whose period does not divide the pattern length are excluded rather than scored badly, because a reading that does not fit the span is not a reading. That is why the tresillo figure offers three candidates and the seven-quaver figure offers three of a different kind.

The normalised score divides by the number of strong positions, which is what makes readings with different period lengths comparable; the raw score is what the figures print, and where the two disagree — the America bar above — both are shown, because the disagreement is a real feature of the comparison and not an artefact to be hidden.

The patterns are entered as grids of equal subdivisions. That is itself an idealisation: a performance has no grid, and the deviations from one are not noise but the substance of a groove. Everything here is about the page and the model, both of which quantise.

What the picture cannot show

It cannot show tempo. A signature says nothing about speed, and the same onsets at two speeds are two different metres — the preferred beat rate is a fact about the listener and the page addresses it only through a word in Italian. The one figure here that uses a rate has it passed in.

It cannot show the other things a bar line does. Repeat marks, rehearsal numbers, the convention that an accidental lasts to the end of the bar: the bar is a unit of housekeeping as much as of metre, and none of that is in the model.

It has no dynamics and no timbre, which is exactly why the backbeat defeated it. Accent in real music is carried by loudness, attack and instrument at least as much as by position, and a model that reads onsets is deaf to all three.

And the weights are stated, not measured. Three, minus two, minus one are a plausible ratio from the preference-rule literature and every conclusion here inherits them. Sweeping them, as the section above does, shows which conclusions that actually costs — the two large disagreements survive every weighting and the two narrow ones do not — but sweeping a parameter is not the same as measuring it. A ratio that no experiment fixed is still a ratio somebody chose, and the fact that this essay’s headline results do not depend on it is a fortunate property of those particular patterns rather than a defence of the numbers.

The ladder from here

This rung asked what a time signature claims and found it claims a hierarchy the onsets sometimes contradict and a grouping it does not contain. The next moves from when to how loud, and finds that the page’s other ordinal ladder — pp to ff — is not an instruction about level at all: on a struck string a harder blow shortens the hammer’s contact and moves the first spectral null from the third partial to the sixth, so a fortissimo is a different sound rather than a louder one.