The accent buys two units, however loud it is

A twenty-five is a nine until its last unit pointed the metre-induction model at the Balkan claim that a long additive bar is heard as a group of shorter ones, and found the claim right about the practice and wrong about the mechanism. The model never prefers the group. It prefers the whole bar outright, from the moment there is any evidence at all — but the evidence that separates a bar of twenty-five from its shorter cuts does not exist until at least twenty-five consecutive steps have been heard, and at the subdivision an unequal metre is best played at, a psychological present of three and a half seconds holds sixteen.

That essay was scrupulous about its weakest point. Every figure in it scores a bare list of onsets, and an aksak bar in performance is not a bare list of onsets. Its long beat is louder, longer than written, and in the drum part a different stroke; its first beat is where the dance step lands. The debt it recorded was precise: how strong does an accent on the long beat have to be before the separation window falls inside the present at a bar of twenty-five?

Two things an accent can be #

There are two ways an accent can enter a model that compares readings, and they are different in kind.

The first is the existing one. What the onsets left out added note length, dynamic accent and harmonic change to the same induction, each as a multiplier on the credit a reading earns for an onset on one of its strong positions. An accented onset is simply worth more. That essay found the limit of this itself — a harmonic change and a dynamic accent behave identically once each is reduced to “this onset counts for more”, and on a tie a cue is not induction but transcription.

The second is an accent that a reading has to predict. A reading of a metre is not only a set of strong positions; it says which of its beats are long and where its bar begins. If a heard accent marks the long beats, a reading can agree or disagree with where the long beats fall. If it marks the first beat of the bar, a reading can agree or disagree with where the bar begins. The cue then scores a reading’s claims rather than adding weight to onsets.

Both are run below at every gain from nought to four, on the same compound readings as before. At a gain of nought every version reproduces the onsets-alone scoring exactly, which each figure checks before drawing.

The reading that holds out longest #

On onsets alone the twenty-five counted 7+7+11 needs thirty-nine steps, and the reading that holds out that long is a bar of eighteen: the first eight groups of the metre, repeated. Its onsets coincide with the metre’s for thirty-eight consecutive steps, because eighteen units repeated land on the metre’s group heads for longer than a whole cycle.

One reading holds out against every possible cue, however, and it sets a floor that no accent can move. The longest cut of any metre is every group but the last. For the 7+7+11 that is a bar of twenty-two, and for twenty-two steps it is not merely consistent with the metre, it is the metre: the same onsets, the same beat lengths, the same accents on the same beats, and the same first beat of the bar. The two part only where the metre begins its last group and the cut begins again. Any cue that arrives with a note says nothing about that difference until the note arrives, so no such cue can separate a bar from its shorter cuts before one step into its last beat.

For a twenty-five ending in a beat of three, that floor is twenty-three steps. For one ending in a beat of two it is twenty-four. The present holds sixteen. So the answer to the debt as it was asked is available before any accent is scored: no accent on the notes of a twenty-five, of any kind and any strength, can bring its separation inside the present. What is left to find out is how close each kind comes, and what it costs to be less than perfect.

The weighted accent buys nothing #

With the long beats accented and the accent entered as weight, the 7+7+11 still needs thirty-nine steps, the 9+9+7 twenty-five, the bar of eighteen twenty-five and the bar of eleven sixteen — exactly what the onsets alone need, on all seven metres.

The reason is in what the weight does. Two readings that put their strong positions in the same places earn the same extra credit from an accented onset, because the credit is paid for the onset and not for anything either reading claims about it. The cut that holds out longest agrees with the whole bar on every strong position until it stops agreeing on onsets, so an accent adds the same amount to both and the margin between them does not move.

Over all 1,820 arrangements the weighted accent changes the number of steps on thirty, and on every one of the thirty it adds a step. A seventeen counted 3+2+2+3+3+2+2 goes from twenty-three to twenty-four, a twenty counted 3+2+2+3+3+3+2+2 from twenty-six to twenty-seven. Where a cut happens to put its strong positions on more of the accented long beats than the whole bar’s early steps do, it gains more credit from them, and the whole bar has to wait one step longer to pull ahead. On the seventeen, at twenty-three steps, the whole bar scores 3.00 per strong position on onsets against 2.89 for its best cut, a bar of ten counted 3+2+2+3; with the accent as weight the whole bar scores 4.20 and the bar of ten 4.22, because two of the ten’s four beats are long and only three of the seventeen’s seven are. A cue that pays for loudness wherever it falls rewards a reading for how many loud notes it happens to call strong, and on these metres a shorter wrong reading happens to call more of them strong.

An accent on the long beat buys a short beat #

When each reading has to say which of its beats are long, the accent does separate readings, and it separates them by the same amount everywhere.

On the 7+7+11 the prediction brings thirty-nine steps down to thirty-seven. On the 9+9+7 it brings twenty-five to twenty-three, on the bar of eighteen twenty-five to twenty-three, on the bar of nine nine to seven. Over all 1,820 arrangements the long-beat accent needs exactly two steps fewer than the onsets alone, without a single exception.

Two is the length of a short beat, and that is the mechanism. On onsets alone a listener learns that a beat is long only when the onset two steps after its head fails to arrive: until then a beat of three and a beat of two look the same. An accent on the long beat says so at the head. It moves every piece of evidence about beat lengths forward by exactly the time a short beat takes, and it can move nothing else, because the positions of the onsets were already known. So wherever the onsets alone would have found the first disagreement in beat lengths, the accent finds it two steps sooner — and on the 7+7+11, whose longest-agreeing reading is a bar of eighteen, the first disagreement arrives at step thirty-six rather than thirty-eight.

An accent on the downbeat reaches the floor #

An accent on the bar’s first beat does something the long-beat accent cannot. A bar of eighteen repeated predicts that its bar begins again at step eighteen. The metre’s first beat is at nought and at twenty-five, so a heard downbeat accent at nought and none at eighteen refutes the bar of eighteen at step nineteen — twenty steps before the onsets would have. Every shorter cut predicts a downbeat somewhere inside the first cycle where the metre has none, so every one of them is refuted as soon as its own bar ends.

Every one except the longest, which predicts its downbeat at twenty-two. So the 7+7+11 needs twenty-three steps with a downbeat accent, which is the floor, and over all 1,820 arrangements the downbeat accent meets the floor on every one. It collapses every arrangement of a bar length onto one or two numbers — the bar less its last beat, plus one — however far above that floor the onsets alone had put it. The 7+7+11 falls from thirty-nine to twenty-three; the 3+2+2+3+2+2+2+2+3+2+2, from thirty-five to twenty-four.

Strength is not the variable #

The question as it was recorded asked how strong the accent had to be, and the model’s answer is that strength does not enter. At a gain of 0.01 the long-beat accent already needs thirty-seven steps and the downbeat accent twenty-three, and at a gain of four they need the same. For the downbeat accent a gain of 0.001 does as well.

That is not a flaw in the scoring; it is what a prediction is. The whole bar is the true reading, so every agreement term favours it and none favours a cut. Until the first disagreement the two readings earn identical scores however large the term is, and after it the whole bar is ahead by an amount proportional to the gain but greater than nought at any gain. A cue that a reading must predict wins at the first place it is contradicted, and the loudness only sets the margin it wins by. The strength of a real accent matters to whether a listener hears it at all — an accent below the ear’s threshold for a change in loudness is not an accent — and above that threshold this model says it matters for nothing.

The present the tradition was measuring #

The sweep settles the longest bar that can be heard as one. On onsets alone the fewest steps any arrangement of a bar needs is the bar’s own length, so sixteen units is the longest that fits sixteen steps. With either accent predicted, the fewest is two less — the bar less a last beat of three, plus one — so eighteen units fits and nineteen does not. With the downbeat accent every arrangement of seventeen units separates inside the present, and twenty-six of the fifty-nine arrangements of eighteen do. An accent moves the longest single bar from sixteen units to eighteen, and nothing carried by the notes can move it further.

That has a consequence for the essay it continues, and it is not a comfortable one. That essay ran its argument backwards: a tradition that treats bars up to sixteen units as single bars is reporting how many subdivisions its listeners can hold, and at 218.35 milliseconds a subdivision sixteen units is 3.49 seconds — the published typical value of the psychological present to two decimal places. That agreement was computed on onsets alone. If the listeners the tradition reports on hear the downbeat marked, as a dancer does, a bar of sixteen separates in fourteen or fifteen steps, and the same threshold implies a present of 3.06 or 3.28 seconds. Both are inside the published band of two to eight. The sharp agreement is not; it belonged to the least realistic input the question admits, and the more a listener is given, the shorter the present the threshold measures.

The arithmetic #

The readings are the same as before: the whole metre as one period, and every proper prefix of its group list as a shorter period repeated, each scored by the preference rules — three for a strong position holding an onset, minus two for an empty one, minus one for an onset off every strong position — and divided by the number of strong positions. The weighted accent adds the gain to the credit of every onset at the head of a long beat. The long-beat prediction adds the gain to the credit of an onset where the reading’s beat there and the heard beat are both long or both short, and subtracts it where they differ; the downbeat prediction does the same for whether each onset is the first of a bar. Credit is never allowed below nought. The separation window is the fewest consecutive steps, from the written downbeat, after which the whole bar scores strictly above every cut. The arrangements are every cyclic order of every metre of twos and threes from nine units to twenty-five that is not a repeat of a shorter bar, 1,820 in all, and the subdivision is the one at which the two beat lengths are jointly at their preferred rate.

What a cue in the notes cannot show #

It cannot show a cue that arrives before its note. The floor holds for anything carried by the notes themselves. A dancer’s weight shifting before the step, a caller, a singer’s line that starts a verse, or a listener who already knows the dance — any of those can say where the bar begins before the note does, and none of them is in this model.

It cannot show an accent that is inconsistent. Every accent above is on every long beat or every downbeat, every time. A performance that marks the bar sometimes and not others is a noisy cue, and a noisy cue costs evidence rather than giving it for free; the floor would then be where a perfectly consistent accent puts it, and real listeners would need more.

It cannot show the threshold of hearing an accent. Any gain above nought wins here, and a real accent has to be large enough to be heard as louder at all. That makes the model’s answer an upper bound on what a real accent buys, not an estimate of it.

It cannot show a longer note as distinct from a louder one. An agogic accent — the long beat played longer than written — is also a cue that arrives with the note, and it is bounded by the same floor. It differs in timing: its information is complete only when the next onset comes, which is exactly the delay the long-beat accent removes.

And it cannot show learning. A dancer who has heard a twenty-five ten thousand times is recognising a stored pattern rather than inferring a period, and recognition is a different computation with a different bound — as different as a metre that has to be able to change its mind is from one that already knows which metre it is in.

Whose accents #

The long metres are Bulgarian and Macedonian dance metres, and they are played marked. The tapan, the double-headed bass drum of that music, is struck with a heavy beater on one head and a thin stick on the other, so the drum itself distinguishes heavy strokes from light ones; the dance puts its weight on particular beats; and the teacher who counts a twenty-five as seven, seven and eleven is marking where three short bars begin. A bar of nine is four beats of unequal length, and the ratio between them is fixed at three to two by arithmetic; the length of the bar is bounded by the present. None of those results was about accents, and this one says the accents cannot rescue the longest bars from that bound. What they can do is make the bound two units kinder than the onsets alone suggest — and make the counting of a twenty-five as three shorter bars, which marks downbeats the metre does not have, a practice that actively hides the bar it is teaching.

Still open: how soon a listener who knows the repertoire can tell #

The floor is a statement about induction: a listener with no metre in mind, choosing between the whole bar and its cuts. A listener who knows the tradition holds a repertoire instead, and the question becomes recognition — after how many steps is the heard stream consistent with only one of the metres the tradition actually uses? That is arithmetic on the same 1,820 arrangements treated as a dictionary, with and without the accents above. What it would settle is whether knowing the repertoire breaks the floor, since a listener who knows that no dance uses a bar of twenty-two does not need to hear where the twenty-two would begin again — and so whether the rule that long bars are groups is a limit of the present or a limit of a first hearing.