The beat that is never sounded

Take an ordinary sixteen-step groove and ask the rules where the beat is. They answer without hesitation.

Now take the onsets away from the beat it chose — remove every onset on steps 3, 7, 11 and 15, and leave everything else exactly where it was — and ask again.

The beat has moved. Not degraded, not become uncertain: moved, to a different phase, with the old winner now last.

The slate for this rung said the opposite #

The rung that introduced this machinery lists, among the things a scoring model gets wrong, that “the beat survives silence — remove events for a bar and the metre continues; a listener taps through a gap without hesitation”. It states that as a known limitation and does not measure it.

The plan for this essay was to measure it, and the prediction was mild: that the score would still peak at the right phase, more weakly. It does not. On four patterns tried, the winning phase moved in all four.

That is a worse failure than a weakening, and it is worth being precise about why. A model that became less confident would be a model that had noticed something was missing. This one becomes more confident: the charleston with its beat onsets removed scores its new winner at 2 against −10 for the others, where the intact pattern had a tie.

How much it takes #

Removing all four beat onsets is a strong intervention. Removing fewer says how much the model’s commitment is worth, and the answer is: one beat.

With one of the four removed, step 3 still wins — 3 against −3, weakened but intact. With two, the margin is gone entirely and three phases tie at −2. With three or four, step 3 is beaten outright.

So the model holds a beat through one silent beat in four and not through two. A listener holds one through eight, which is the length of the two-bar break in the figures above, and through considerably more than that in a piece they know.

That ratio — one beat against eight or more — is the size of the gap, and it is a factor rather than a margin.

Why it moves, and why it must #

The mechanism is not subtle and that is the point.

The score rewards a strong position carrying an onset and punishes an empty one. Removing the onsets from a phase converts every one of its hits into an empty — the largest single penalty in the scheme — while every other phase is untouched. The chosen phase does not merely lose its evidence; it acquires positive evidence against itself.

Against the weights of a bar of sixteen, read from step 1, the stripped groove puts its onsets two and three levels down and leaves every one of the strongest positions empty. That is what a syncopated bar looks like from inside a metrical hierarchy, and it is exactly what this pattern is — which is the difficulty the rest of the essay is about, because a scoring function is being asked to find a beat on positions that carry nothing.

So this is not a bug. A model that scores a pattern all at once, with no memory and no prior, must prefer the phase whose positions carry onsets, because that is the only thing it is made of. Given a bar whose beats are silent, it will always conclude that the beats are somewhere else.

Which is what a listener would also conclude, if the stripped bar were the first thing they heard.

The whole difference is what came before #

That last sentence is the substance of the rung. Heard cold, the stripped pattern really is a pattern whose downbeat is elsewhere — a listener with no context would hear it that way, and the rules are right about it.

Heard after four bars of the intact groove, it is not. It is the same groove with its downbeats knocked out, and a listener hears it that way effortlessly, keeps tapping, and finds the effect exciting rather than confusing.

The difference between the two situations is entirely in the listener’s history, and the model has no representation of a history at all. Every score on this page is a function of one pattern. There is no argument to the function that could carry “and the last four bars were in this phase”.

That is a structural limit, not a missing feature. A preference-rule set is a scoring function over readings, and a scoring function cannot hold a commitment.

What a prior would have to be worth, which is a number #

“A scoring function cannot hold a commitment” can be made quantitative rather than left as an argument from form. Add the missing term in the cheapest possible way — a bonus given to whichever phase won last time — and ask how large it has to be.

The answer is exact and it is the same in every pattern. Each silenced beat costs the chosen phase six points relative to its rivals, so the bonus needed to survive k of them is 6k, less whatever margin the intact pattern gave it:

Six is not a fitted number. Removing an onset from a strong position of the chosen phase turns a filled position into an empty one, which is three points gained becoming two lost — a swing of five. And the same onset was off the grid for every rival phase, costing each of them one, so removing it hands them a point back. Five plus one is six, in every scheme with these weights, on every pattern.

Which is why no fixed prior works #

That linearity is the whole finding, and it is worse for the model than the essay’s structural argument was.

A bonus of six survives one silenced beat. Twelve survives two. Surviving the eight a listener manages through a two-bar break needs forty-eight — and the entire score range across all four phases of these patterns is about twenty-four points, from −16 to +8. The prior would have to be twice the size of every piece of evidence the model is capable of weighing, and it would then never lose, which is not a beat-finder but a constant.

So the missing term is not a prior at all, in the sense of a fixed bonus. It has to be something whose weight grows as the contrary evidence accumulates, or something that stops re-scoring altogether. A quantity that must be unbounded to do its job is a quantity of the wrong kind, and that is a sharper statement of the same conclusion: the thing that keeps a beat is not a heavier thumb on the scale but a mechanism that is not weighing.

It also says why the oscillator two sections below works. It does not out-argue the evidence; it ignores it. Nothing arrives near the expected beat, so nothing corrects the phase, and the phase persists because persisting is its default rather than its verdict. A scoring function’s default is to answer the question again.

The groove’s row is worth one more note, because it is the case that looks like a counterexample and is not. It survives two silenced beats for free, which is exactly the essay’s own finding above — and it survives them because its intact margin was twelve, which is two silenced beats’ worth. A pattern’s tolerance for silence is its margin divided by six, and the margin is a property of how unsyncopated the pattern was to begin with. The grooves that most need the beat held through a break are the syncopated ones, which are precisely the ones with no margin.

The devices this makes possible #

Music is full of passages that exist because a listener keeps a beat that is not being played, and each of them is a passage this model cannot analyse.

The break. A rhythm section stops for two bars and everybody in the room keeps time. The re-entry is precisely on the beat, and it is only satisfying because the beat was still running in the audience.

The dropped downbeat. A pattern that omits the first beat of a bar depends on the listener supplying it. Played to somebody with no context it is simply a pattern that starts late.

The tie-over. A note held across the bar line means nothing without a bar line to be held across, and the bar line at that instant has no onset marking it.

And the whole apparatus of syncopation. Syncopation is a note followed by a rest on a stronger position, which is to say: it is defined by something that does not happen at a place the listener is nonetheless attending to. Every syncopation is a small version of this experiment.

The charleston, which is the cleanest case #

The clearest demonstration is the smallest pattern.

Take away the two onsets on step 1 and its cycle, and the remaining two onsets are both at step 7 of their bars. The model then reports step 3 at 2 against −10 for everything else: a confident answer, from a pattern that has been reduced to two events.

That is the shape of the failure at its most exposed. The less evidence there is, the more decisively the model commits, because what it is measuring is agreement between the surviving onsets and a grid, and two surviving onsets agree perfectly with the grid they happen to sit on.

A listener given the same two events after four bars of the charleston hears two late notes. The model hears a new metre.

An expectation is a thing that can be violated #

The framing that makes this coherent is not rhythmic at all, and this site has it elsewhere.

A metre is a set of expectations about when events will happen. An expectation is a state — it persists, it is carried forward, and it is what makes a violation feel like something rather than simply being a different stimulus. The same argument runs in the pitch domain, where a chord that does not arrive is an event, and there is nothing in the signal at the moment it fails to arrive.

The parallel is exact and it is worth stating because it explains why the fix has to be structural. A model of expectation needs somewhere to keep the expectation. A scoring function has nowhere.

At a hundred and seven to the minute the groove’s beat sits at 560 milliseconds, which is as near the preferred rate as a beat gets — and that is a fact about the rate rather than about the pattern. It is most of why this particular beat is so easy to keep through the silent bars: the level a listener holds on to is the one whose period is already where the tempo window is widest, so the pattern is not being asked to supply anything the rate is not already supplying.

What a model with a state does instead #

There is such a model, it is standard, and this ladder gets to it properly one rung further on. The short version is worth having here because it completes the argument.

The tracker has two numbers of its own — a period and a next beat time — and when nothing arrives near the beat it simply carries on. That is the whole mechanism. It is not clever, it does not know anything about music, and it does the thing the scoring model cannot do at all, because it has somewhere to put a commitment.

The same failure, without any deletion at all #

A cleaner version of the experiment needs no surgery. Some patterns simply have no onset on any beat, and the tradition still has a beat for them.

Handed that pattern cold, the rules put the beat on the onsets, and there is nothing else they could do. Handed it in context — as an offbeat guitar part, a ska upstroke, a rhythm-section comp — every listener puts the beat in the gaps, because something else in the texture put it there and the part is defined by its relationship to it.

That case is more informative than the deletion, because nothing has been taken away. The pattern is complete and unaltered, and the correct reading of it is still not recoverable from it. A part can be written whose entire content is its relationship to a beat it never marks, and an analysis that takes the part alone has thrown that content away before it starts.

What the picture cannot show #

It cannot show how long the beat survives. A listener keeps a beat through two silent bars and does not keep it through twenty. The tracker above coasts indefinitely, with its error growing only from its own period estimate, which is certainly wrong — continuation tapping degrades measurably after a handful of beats.

It cannot show what the silenced bars actually sound like. Every figure treats the removal as a clean deletion. In practice a break is not silence — there is usually something, a held chord, a voice, a cymbal — and whatever is there is doing metrical work that no onset list records.

It cannot show the strength of the prior as a listener’s quantity. The section above prices what the model would need — six points a silenced beat, growing without bound — but that is a price in the scoring scheme’s own units, not a measurement of how strongly a listener commits. Where a listener sits, and how it depends on how long the groove ran first, is not on this page and would need an experiment rather than an arithmetic.

It cannot show the pleasure. The reason music does this is that a beat kept against contrary evidence is enjoyable, and there is no term for enjoyment in any of it.

It cannot show the level above. Everything here is a four-step beat inside a sixteen-step bar. A break that removes a whole bar is a failure at the level above the beat, where the same argument runs with the same conclusion and larger numbers.

And the deletion is artificial. Removing exactly the onsets on one phase is a laboratory operation. Real music that silences the downbeat usually does other things at the same time — changes the density, the register, the harmony — and each of those carries metrical information of its own.

Whose music, and how general the failure is #

The rule set is a fit to notated Western music and the grooves used here are from popular music of the last seventy years and the Cuban repertoire behind it. That narrowness is real and it is not the load-bearing part.

The load-bearing part is that the failure is structural rather than parametric. It is not that the weights are wrong for this repertoire, or that the pattern set is unrepresentative. No choice of weights makes a memoryless function keep a commitment, so no repertoire would produce a different result. That is what makes this a bound on the model rather than a bug in it.

Which is the strongest form of a negative result this site knows how to produce: not the model got this wrong, but the model has no term the answer could live in.

The ladder from here #

The next rung takes the same rule set to a different kind of edge — a bar whose beats are not all the same length, which the machinery cannot express because it lays strong positions at a fixed period.

Then the model with a state, properly: what it does with a tempo that will not stay still, how far behind it settles, and the one number that says when it gives up.

There is a question this rung has raised and cannot answer, and it belongs to a different anchor on this site. How long an expectation persists is a memory question, and memory has been measured for musical returns rather than for metrical ones. Whether the two decay at the same rate is not known here and would be worth knowing: a beat held for eight bars and a theme recognised after eighty are both a listener carrying something forward, and it would be surprising if they were the same mechanism.

One more thing is worth recording as unfinished. Everything on this page treats the onsets as exact. Performers mark a beat partly by timing rather than by presence — a note slightly early, a note slightly long — and a break in real music is very often preceded by a ritardando that tells the listener what is coming. None of that is in an onset list, and it is plausible that a good deal of what keeps a beat alive through a silence is put there before the silence starts.