The right period at the wrong phase

A beat has two numbers. How far apart the beats are, and where one of them is. The first is the period and the second is the phase, and every figure in the rung that introduced this machinery fixed the second by assumption: the strong positions were laid from step one, and the candidates differed only in how far apart they were.

Take the assumption away and score every phase of one period, and the model’s confidence divides sharply in two.

The period is decided and the phase is not #

Handed the same pattern and asked which period fits, the model answers without hesitation: a beat every two steps scores 1, every four scores −1, every eight scores 0. The two-step reading wins.

Asked which phase, at the four-step period, it returns a three-way tie. At the eight-step period it returns another one — steps 1, 4 and 7 all score 1, which are precisely the three onsets.

So the two unknowns are not equally hard for the same model, and they fail in different ways. A period that is wrong is wrong by a margin the score reports. A phase that is wrong is not wrong at all: it is one of several answers the model regards as identical.

Why a syncopated pattern does this #

The reason is structural rather than accidental, and it follows from what the rules count.

The score rewards a strong position that carries an onset and punishes one that does not, and punishes an onset that falls between strong positions. For a pattern whose onsets are spread evenly across a period — which is what a syncopated pattern is — every phase catches about the same number of onsets, and the counts come out equal.

The tresillo is three onsets in eight steps at 0, 3 and 6. A four-step beat starting at step 1 catches the onset at 0 and misses at 4. Starting at step 4 it catches the onset at 3 and misses at 7. Starting at step 7 it catches the onset at 6 and misses at 2. Three arrangements, one hit and one miss each, identical scores.

The pattern is designed, if a tradition can be said to design, to have its onsets away from where a bar line would put them. A rule set that locates the bar line by looking for onsets is being asked to find the place the pattern is avoiding.

The son clave has the same problem, with two answers #

Sixteen steps and five onsets, and the model still cannot separate two of the four phases. That matters more than the tresillo case, because the two survivors are a semiquaver apart — a displacement small enough that a listener hearing the wrong one would experience the whole pattern as displaced rather than as a different pattern.

There is a well-known thing that happens to people learning this repertoire, which is hearing the clave “backwards” or half a bar out. That is a phase error and not a period error, and it is precisely the error the model cannot rule out.

Sometimes it does not tie, and is wrong #

A tie is the polite failure. The other one is worse.

The cinquillo has onsets at steps 1, 3, 4, 6 and 7 of eight. A four-step beat starting at step 3 lands on the onsets at 3 and 7 — both struck, nothing empty — which is the best possible score, and starting at step 1 lands on 1 and 5, one struck and one empty.

So the model does not merely fail to identify the conventional downbeat; it actively prefers a different one, by six points out of a range of ten. And the conventional downbeat is not a convention in the weak sense — it is where the dancers’ feet go.

That is a different and more serious failure than the tresillo’s tie, because a tie is visible in the output and a confident wrong answer is not.

How often this happens #

Running every phase of a four-step beat over the site’s stock of timelines gives a spread of outcomes rather than a single verdict.

The rumba clave belongs in that table beside the son, and it behaves the same way: two of four phases tied, at −1 each, with the conventional downbeat among them. The two claves differ by one onset and the model’s verdict on them is identical in shape at every period tried — which is itself a small finding, since the difference between the two patterns is the whole of what distinguishes two repertoires.

The pattern in that table is not subtle. The rules find the phase exactly when the phase is already obvious, and go quiet or go wrong on every pattern whose interest lies in its relationship to a bar line. The composite of a polyrhythm is in the same family: its onsets are placed by an arithmetic that has nothing to do with where a downbeat would be.

Phase is rotation #

On a line the phase is where the bar line goes. On a circle it is where the counting starts, and the two are the same operation seen from different angles.

That framing makes the size of the problem visible. A pattern of n steps has n possible phases, and a model that scores them all and reports ties has narrowed the field by some factor and not chosen. Reporting the winner, as the earlier figures on this site do, hides how many runners-up there were.

The period is easy for a reason #

It is worth asking why one of the two unknowns is so much easier than the other, because the answer says what the rule set is really measuring.

A period is a property of the intervals between onsets. Change the phase and every inter-onset interval stays exactly what it was; change the period and they all have to be re-read. So a period can be found by any method that looks at spacings — autocorrelation, comb filtering, counting — and none of those methods has any way of expressing a phase at all.

The phase, by contrast, is a property of where the pattern sits against an external frame, and the only frame available inside a bare onset list is the list’s own beginning. A pattern that starts on an onset therefore gets a free vote for phase zero, and every transcription starts on an onset.

That is the whole of it: the rules find the period from the pattern and inherit the phase from the page.

What settles it in practice, and none of it is in the pattern #

Every mechanism that actually fixes the phase is outside the onset pattern.

The bass. A low note is a strong candidate for a downbeat and one bass note settles the question instantly. That is not a preference rule about onsets; it is a preference about which instrument.

The harmony. A chord change is a metrical marker of enormous strength, and in most repertoires chord changes fall on downbeats. Harmonic rhythm is a separate periodicity running alongside the onset pattern and it usually agrees with the metre.

The beginning. A piece starts somewhere, and the first event is a strong candidate for step one whatever the rules say. That is an artefact of listening being an ordered process, which a scoring model that reads the whole pattern at once cannot represent.

And what came before. A metre already established carries forward. That is the largest effect of the four and it is the one no preference-rule set has a term for, which is a shortfall this ladder returns to.

That last figure is worth dwelling on. The rate rule is the strongest additional constraint this site has, it settles genuine ambiguities about the period, and it is exactly silent about phase — because two phases of one period have the same rate. The two unknowns need two different kinds of evidence and the literature has supplied one of them.

An upbeat is a phase claim #

The musical device that depends most completely on phase being separately determined is the anacrusis: a phrase that begins before the downbeat.

Heard from cold, an upbeat is indistinguishable from a downbeat. Everything that makes it an upbeat — the sense of leaning towards something, of the real beginning being still to come — is supplied by a phase the listener already holds. Play the same notes to somebody who has heard nothing before them and the gesture evaporates.

It is worth saying what a metre drawn as a tree does and does not fix, because the omission is the whole subject of this essay. A tree fixes a period and a hierarchy: how long a bar is, and which positions inside it are strong. It says nothing whatever about where the tree is anchored in the music. Two performances of the same passage under the same tree, begun a step apart, are described identically by it and are heard as different pieces of music.

That is why every candidate in this essay’s figures carries a strong list rather than only a period. The period is the part a tree can draw; the strong list is the phase, and it has to be written down separately because there is nowhere in a tree to put it.

Notation solves this by fiat. The bar line is drawn, and everybody reads from it. That is why a rewritten bar line is an instruction to performers rather than a claim about listeners, and why so much of the twentieth-century repertoire that renotates metre has a gap between what the players are counting and what an audience hears.

What happens when the model is forced to choose #

A tie has to be broken somehow for a program to produce output, and the usual method is to take the first candidate in the list. That is what the figures above do, and it is worth naming as a decision rather than a result.

Taking the first candidate means taking the phase the pattern was written down at — which for every pattern here is the one the transcriber chose. So a model that reports a winner on a tie is reporting the transcription’s own bar line back, dressed as an inference.

The honest output for the tresillo is three phases, equally scored, and a program that prints one of them is throwing away the most interesting thing the computation found.

There is one exception in the stock and it is the cinquillo, which is why that case is worth keeping in view. A tie-break by list order cannot produce a wrong answer that disagrees with the transcription — it can only produce the transcription’s own reading. The cinquillo’s step-3 verdict is therefore the one result in this essay that could not have come from the convention, and it is the one the model got wrong.

Whose music, and whose bar line #

The rule set is a fit to Western notated music, in which the bar line is a given and the analysis problem is the period. That is why phase is under-theorised in it: for the repertoire it was built on, phase is written on the page.

For the repertoires whose patterns produce these ties — Cuban, West African, and everything descended from them — the phase is the interesting question and the notation is the imported thing. A cyclic pattern has no privileged starting point, and which point a player enters on distinguishes one part from another.

So the model’s failure here is not a general failure. It is a model built where the answer was free, applied where it is not.

What the picture cannot show #

It cannot show the tie-break. The figures print a winner because a figure must. Every one of them where the model ties is reporting a decision made by list order.

It cannot show a graded answer. The scores are integers from a small set, and near-ties and exact ties look the same to a reader. A model whose output was a distribution over phases would be more informative and it would need a different scoring scheme. The integer grain is coarse enough to matter: the son clave’s four phases score −1, −13, −1 and −7, so the two survivors are not merely close, they are the same integer, and no amount of reading the numbers more carefully will separate them. That is a property of a rule set whose rewards and penalties are small whole numbers, and it means a tie here is never a near miss reported as a tie — it is an exact equality of two counts.

It cannot show onset strength. Every onset is one dot. A pattern in which the downbeat is played on a bass drum and the offbeats on a hi-hat has its phase settled by the timbre, and none of these figures has a timbre in it.

The figures cannot show half-a-bar errors and the model can be asked, which is worth doing because that is the error people learning the repertoire actually make — hearing the two-side as the three-side, eight steps out. Scoring the son clave at a period of eight rather than four puts the question directly, and the answer is the tie the tresillo gave at the beat level: phase 1 and phase 5 both score −3, and the model has no preference between the clave and its own half-bar rotation. The rumba clave gives the identical pair of scores. So the most consequential phase error in this repertoire is one the rule set cannot see at all, and it is invisible in the figures above only because they ask at the wrong period.

Asked at a period of sixteen — the whole bar rather than the half — it separates them, and by a wide margin: phase 1 scores −1 and phase 9 scores −7, six points apart, on both claves. That is a real result and it comes with a warning attached, because the same score is available to any rotation the pattern is written down at. At period sixteen there is exactly one bar, so a strong position at phase p is a single step, and what the model is rewarding is that step 1 carries an onset and step 9 does not. It has not recognised the clave; it has recognised that transcriptions begin on onsets, which is the tie-break this essay is about, arriving as a score instead of as a list order — a decision by the transcriber, dressed as an inference from the pattern.

So the half-bar error divides the model against itself. At the level a listener works at — a four-step beat inside an eight-step half-bar — the two sides of the clave are indistinguishable. At the level at which they separate, the separation is the transcription’s convention scored back. Neither reading is the mechanism a musician uses, which is that the three-side and the two-side sound different because of what the rest of the ensemble is doing.

Hypermetre is the same computation one level up and it has the same ambiguity, larger.

And it cannot show the listener’s history. Every score here is computed from the pattern alone, with no prior. That is the largest single omission in this rung and it is not repairable inside a scoring model, because a scoring model has nowhere to put a prior. Two rungs further on this ladder there is a model that does.

The ladder from here #

Once a metre is fixed — by the rules, by the bass, by the notation or by habit — a second quantity becomes computable that was meaningless before it: how much the pattern contradicts the metre it is being heard in. That number is syncopation, it has a published definition, and it turns out to be a number about the pair rather than about the rhythm.

Then two failures worth measuring rather than asserting: what the rules do when the beat has no onsets on it, and what they do with a bar whose beats are not the same length.

And after those, the model that has what all of this is missing — a state, so that a phase once chosen is kept rather than recomputed from scratch every bar. That model also settles a question this rung raised and could not answer, which is what happens when the tempo will not stay still.

One thing this rung has quietly assumed throughout deserves saying out loud, because the microtiming ladder exists to deny it: every onset here is exactly on a step of an even grid. Real performances are not, the deviations are systematic, and a performer marks a downbeat partly by playing it slightly early or slightly long. That is evidence about phase, it is thrown away by every figure on this page, and it may be most of what settles the question in practice.