What the onsets left out

Everything this ladder has induced a metre from is a row of ones and zeros. An onset happened, or it did not. Nothing about how long the note lasted, how loud it was, or what chord it belonged to.

That was never argued for. The first rung needed a rhythm a machine could score, took a step grid, and every rung since has scored the same object — including the three that failed. Those failures were recorded carefully and each was read as a limit of the rules: the phase that ties three ways, the downbeat that moves when the downbeats are taken out, and the metre that cannot be offered because it is not in the candidate list.

A listener facing the same music has three things the row of ones and zeros does not carry. This rung gives them to the model, one at a time, with a dial on each — and the three failures turn out to have three completely different fates.

The cue that costs nothing #

The first cue is not extra information at all. It is information the encoding discards.

A tresillo is three, three and two. Its onset row is 1 0 0 1 0 0 1 0, and the inter-onset intervals — 3, 3 and 2 — are sitting in that row, recoverable by counting the zeros. The scoring function treats all three onsets as identical events, because the only thing it asks of a position is whether it is a one.

So the first cue is note length, read off the pattern itself, weighted by a gain. At a gain of zero the scoring is exactly what it always was; as the gain rises, an onset that begins a long note earns more credit than one that begins a short one.

Narrowing three to two and stopping is the right answer rather than a disappointing one. A tresillo genuinely does support two readings, one starting on each of its long notes, and both are used. What the cue removed was the reading that starts on the short note, which nobody hears.

The tie that resolves correctly #

The son clave is the case the ladder has come back to three times, and it ties two ways on onsets alone.

That is a genuine repair of a genuine failure, made with no new data. One of the three recorded failures was a failure of the encoding.

The failure no cue can reach, and the reason is arithmetic #

The second failure is the one that has bothered this ladder most. Take a groove the model reads correctly, remove the onsets on the downbeat, and the model moves the downbeat — where a listener who has heard four bars does not.

The flat line is the result, and it is a proof rather than a measurement. Every cue considered here — length, loudness, harmonic change — enters the score as a multiplier on the credit an onset earns for landing on a strong position. The true phase’s strong positions are steps 3, 7, 11 and 15, and after the stripping there is no onset on any of them. A phase with no hits has no credit; a multiplier on nothing is nothing; the line cannot move.

So the second failure is not a failure of the input. It cannot be, because the evidence a cue would carry has to ride on an event, and the events at issue are the ones that were removed. What repairs it is what the eighth rung found: a model with a state, an oscillator that keeps going through silence, which is a different model rather than better data.

And the free cue makes the stripped case slightly worse: at any gain above zero it picks step 4 outright, where the onsets alone at least left step 1 and step 4 tied. A cue that repairs the clave breaks this.

The failure that is not about evidence at all #

The third recorded failure is the Balkan bar of nine. The induction lays strong positions at a fixed period, so the only readings of nine it can offer are nine, three and one; for a bar of seven, seven being prime, the only readings are seven and one. The right answer is not among the candidates.

No cue touches this, and the reason is different again from the reason no cue touches the silent downbeat. A cue re-weights the candidates. The unequal beat is not a badly-weighted candidate; it is not a candidate. Adding evidence to a model whose hypothesis space does not contain the truth improves the confidence and not the answer.

Three failures, three fates: one of the encoding, one of the model’s memory, one of its hypothesis space. That they are three different things is the finding, and it was invisible while the input was fixed, because a fixed input makes every failure look like the same kind of failure.

When the cue stops informing and starts dictating #

Sweeping the gain answers a question that asking “does the cue help” cannot: at what strength does the cue overrule the onsets?

That bound is the answer to a question the ladder could not previously ask, and sweeping every pattern rather than one says the answer is not an interval:

There are three regimes here and no single working range. On the two ties the cue acts at the smallest gain tested and never does harm at any gain up to six. On the intact groove it does nothing at all until 2.00 and then destroys the right answer in the same step — so between zero and two it is not a weak cue, it is an absent one, and there is no strength at which it is informative. On the stripped groove it is harmful from 0.01 upward.

That corrects the interval this section previously reported. The lower bound of 0.05 was the clave’s, the upper bound of 2.0 is the groove’s, and no pattern exhibits both — so “a working range about a factor of forty wide” was an interval assembled from two different pieces of music. The true statement is worse: there is no positive gain that is safe on all four, because the gain that resolves the clave is a gain that has already broken the stripped groove.

And there are two cues here, not three #

The limitations below record that the harmonic-change cue is implemented and untested, and say it “behaves exactly like the dynamic accent”. Running it settles that more sharply than the sentence claims. Scoring all four patterns at every gain from 0 to 3 with an accent on a phase’s onsets, and again with a harmonic change on the same onsets at the same contrast, the two produce identical scores at every gain in every case — not similar rankings, the same numbers.

That is a fact about the scoring function rather than about music, and it is worth stating because it bounds the whole exercise. Both cues enter as an additive contrast on the hit weight, so once a cue has been reduced to this onset is worth more than an average one, its name has been discarded. A dynamic accent, a harmonic change, an agogic lengthening and a timbral emphasis are one parameter in this model wearing four labels.

Which makes the transcription charge above sharper too. The cue that reads the pattern — length — is the only one of the three that carries information the model did not already have to be told, and the other two are one supplied number under two names. The ladder’s input has been varied by exactly one degree of freedom, and the finding that three failures have three fates rests entirely on that one. That is enough for the finding, because the three fates are distinguished by whether a cue can reach the failure at all rather than by which cue it is — but it means the essay’s title promises more variation than the model can express.

The dynamic accent has the same two ends, and sweeping it the same way shows the ends belong to the phase being accented rather than to the cue. Accenting the groove’s own winning phase changes nothing at any gain up to six, because the winner already leads and the cue only adds to its lead; accenting a phase six points behind takes the lead at 2.00, the same crossing the duration cue has on the same pattern. So the price of overturning a preference is a property of how far behind the preference is, not of which cue is paying it. Accent the onsets of a phase the onsets alone already had level, and it wins at the smallest gain tested; accent a phase the onsets had six points behind, and it needs the same gain of 2.00 to take over. So the accent settles ties for free and overturns preferences at a price, and the price is the same one the length cue pays.

Which is a less comfortable result than it looks. The clave’s two-way tie is settled by whichever of the two phases gets the accent — the cue does not adjudicate between them on any evidence of its own, it simply carries whichever answer was put into it. On a tie, a cue is not induction; it is transcription. The only reason the length cue escapes that charge is that its values were not supplied by anybody: they were already in the pattern.

Three failures, three different things #

It is worth putting the three side by side, because a ladder that recorded them one at a time recorded them as one kind of event and they are not.

The phase tie was a failure of the encoding. The evidence needed to resolve the son clave was in the onset row all along, as the spacing between the ones, and the scoring function was reading the row as a set of times rather than as a sequence of notes. The repair is free, it is exact, and it resolves the tie the way the tradition does.

The silent downbeat was a failure of memory. No cue can reach it, because a cue is carried by an event and the events are gone; the true phase’s score is a constant under every cue at every gain, which is a fact about the arithmetic and not a finding about the pattern. Only a model that carries something forward through the silence can hold the beat, and that model exists one rung back.

The unequal beat was a failure of the hypothesis space. Evidence re-ranks candidates and the answer is not among them, so no cue at any gain can produce it. That much is right, and the sentence this essay used to carry next — that nothing short of a different candidate generator would touch it — is wrong, which is worth saying plainly because it changes the size of the defect.

A candidate here is a period and a list of strong positions, and nothing has ever required the strong positions to be evenly spaced. An additive metre is period: 9 with the strong list [0, 2, 4, 6] — the cumulative sums of 2, 2, 2 and 3 — and offered that candidate the scoring picks it out of the aksak bar at 12 points against 2 for the best isochronous reading. The essay that priced it does so figure by figure. The hypothesis space was never closed; it was merely never enumerated, and the difference between a model that cannot say a thing and a model that was never asked to is the whole of what this rung was trying to measure.

A model can be wrong because it was told too little, because it forgets, or because it cannot say the right thing. This ladder found all three and could not tell them apart, and the reason it could not is that telling them apart requires varying the input — which is the one thing eight rungs held fixed.

Which computation produced the numbers #

The scoring is metreFit generalised: an onset on a strong position earns the hit reward times a weight, an empty strong position costs the empty penalty, an off-grid onset costs the offbeat penalty, and the weight is one plus the gain times the cue’s contrast. Every cue is normalised to its own mean, so a uniform rhythm is neutral at every gain and the cue only ever expresses a difference. At gain zero every score in this essay is identical to the number the earlier rungs printed, which is the check that the generalisation is one.

Note length is the inter-onset interval, taken cyclically so the last note runs to the first of the next repetition. The three failure cases are the patterns the earlier rungs used, unchanged: the tresillo, the son clave, the sixteen-step groove and that groove with two of its four beat onsets removed.

The crossing gains are found by stepping the gain in hundredths and watching for the leader to change, so they are accurate to 0.05 rather than solved for.

Whose music, and when #

The three test patterns are Afro-Cuban and its descendants, and the claim about them is about induction rather than about practice. A clave player is not running a scoring function; the phase of the clave is known because the tune is known, and the ambiguity this essay measures is an ambiguity for somebody hearing eight steps in isolation.

The Balkan nine belongs to a tradition where the unequal beat is the norm rather than a difficulty, and the failure is entirely on the model’s side.

What generalises is a methodological claim with nothing musical in it: a model’s recorded failures cannot be diagnosed while its input is fixed. Three failures that looked alike for eight rungs — the model returns the wrong metre — turned out to be a discarded feature, an absent state and an inexpressible hypothesis, and the only way to tell them apart was to vary the thing that had never been varied.

What the picture cannot show #

The harmonic-change cue is exercised above and turns out to be the accent under another name, which is a statement about this scoring function and not about harmony. Testing whether a real harmonic change acts like a dynamic accent on a listener would take a corpus with chords and onsets aligned, which this site does not have — and there is a circularity waiting in it, because the site’s own harmonic segmentation weights chord fit by metrical strength. Harmony deciding metre and metre deciding harmony cannot both be run at once without saying which is prior.

Note length is notated length. Real performances stretch and shorten notes, and the deviations are not noise: a swung quaver’s length is a function of tempo, and an agogic accent in performance may be several times the difference this cue is reading.

The gain is a free parameter with no measurement behind it. Published boundary models weight duration against pitch at fixed ratios and this site has already found that no fixed weighting serves two tunes. The same warning applies here with more force, because the working range found above is wide and nothing locates a listener inside it.

And the credit model is multiplicative by construction. A cue that could create credit where there is no onset — an expectation, a remembered downbeat — would break the flat-line argument entirely, and that is exactly what the oscillator rung’s model is. The proof above is a proof about this family of scoring functions, not about induction.

The ladder ends here #

metre-induction closes at nine rungs. The model is a periodic pattern of strong positions scored against events in time, and the ladder now bounds it in every variable it has: whether the metre is in the signal (one), what rate the listener’s own clock prefers (two), what happens one level up (three), whether the phase is found as well as the period (four), what the mismatch between grid and onsets is worth (five), what survives silence (six), which metres the candidate set can express (seven), what a model with a state does differently (eight), and now what the model is given to work with.

Name a variable the model has, and it is on that list. What is not on it is not a further rung of it: how far from the grid a real performance sits is microtiming, what the page asserts about all of this is notation, and how long the whole apparatus can hold anything is phrase.