Knowing every metre is slower than knowing none

The accent buys two units, however loud it is found a floor under the longest additive metres. A listener inferring a metre from what is heard can prefer a whole bar to its shorter cuts only once the evidence separates them, and the longest cut of any bar — every group but the last — predicts the same onsets, the same long beats and the same downbeats as the whole bar until its last group begins. So no cue carried by the notes can separate them before one step into that group. For a bar of twenty-five that is twenty-three steps, and at the tempo an unequal metre is best played a psychological present holds sixteen.

That essay ended by naming the one escape the floor did not close. It was a statement about induction — about a listener with no metre in mind. 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 listener knows? 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.

That escape is real, and the arithmetic says how much it is worth. Recognition does break the floor, but only for a listener who knows few metres and not the wrong ones, and a listener who knows every possible metre is worse off than one who knows none.

Recognition as contradiction #

The dictionary is the same space the earlier essays enumerated: every metre of twos and threes from nine units to twenty-five that is not a repeat of a shorter one, in every cyclic order, each heard from its written downbeat. There are 1,820 of them.

A listener who knows the dictionary and hears a metre being played can rule an entry out the first time it predicts something the stream does not contain — an onset where there is silence, silence where there is an onset, or, with an accent, a long beat or a downbeat marked where the stream does not mark one. The metre is recognised at the step after the last rival has been ruled out. This is the strictest kind of recognition there is, since it commits to nothing until nothing else is left, and it is the right benchmark for the floor, because the floor was a statement about what the evidence can distinguish at all.

So recognition is a question about which rival lasts longest, and the rivals sort into three kinds. A cut is a shorter metre that the bar begins with. An extension is a longer metre that begins with the heard bar repeated: for as long as it agrees, it sounds exactly like the heard metre played again. Everything else is neither.

For the eighteen, the rival that lasts longest is an extension. The twenty-five plays the eighteen and then starts it again, and it is contradicted only at step 31, where the eighteen plays a short beat and the twenty-five is still inside a long one. The next two to fall are a cut and another extension, together at step 24: the eleven counted 2+2+3+2+2, which the eighteen begins with, and a twenty that begins with the eighteen and its first group again. At the same step falls the first metre of neither kind, a twenty-four that follows the eighteen into its second bar and departs from it at its eleventh group; a twenty-three that departs a group earlier falls two steps sooner.

That order is not peculiar to the eighteen. Across the whole dictionary, a cut is among the rivals that hold out longest for 931 of the 1,820 arrangements and an extension for 879, with 33 arrangements held up by one of each at once. A rival of neither kind is the last to fall for only 43. Recognition among every metre is decided almost entirely by the metres that begin the same way.

A listener who knows every metre never beats the floor #

With the whole dictionary known, recognition on onsets alone takes between 23 and 46 steps, with a median of 28. Not one of the 1,820 arrangements is recognised inside the present, and not one is recognised before its floor.

The reason is the floor’s own reason, one level up. For 1,732 of the 1,820 arrangements the longest cut — every group but the last — is itself a metre of twos and threes in the dictionary, and it agrees with the metre being played on every onset, every long beat and every downbeat until the last group begins. A listener who knows every metre knows that one too, and cannot rule it out any sooner than induction could. The remaining 88 have a longest cut that is not itself an arrangement, and not one of them escapes the floor either: 61 are held up by an extension, 22 by a shorter cut — one by both — and only 6 by a rival of neither kind alone.

The accents do what they did for induction. An accent on the long beats brings the median to 26 steps and recognises none inside the present. An accent on the downbeat collapses most arrangements onto their floor, as it did before, and recognises 85 of them inside the present — all of them bars of fifteen units or fewer, whose floor already lies inside it.

Knowing every metre is slower than knowing none #

The surprising comparison is not with the floor but with induction.

On the sample of 91 arrangements, recognition among every metre is never quicker than induction. For 42 it takes exactly as long; for 49 it takes longer, and sometimes much longer. The Balkan nine of four unequal beats, counted 2+2+2+3, is induced in nine steps — the whole bar is preferred to its cuts as soon as one cycle has been heard — and recognised in twenty-seven.

The difference is which rivals each listener has to dispose of. Induction only has to beat the metre’s own cuts, and it beats them on a score built from preference rules, so it can prefer the whole bar before a cut has been logically ruled out. Recognition has to rule out every metre in the dictionary, and the dictionary contains the extensions, which are not cuts at all. A nine played three times over agrees for twenty-six steps with three of them at once — the twenty, the twenty-two and the twenty-four that begin with it repeated — and all three are first contradicted at step twenty-six, where the nine has come to its long beat and each of them is playing a short one.

Every metre a listener knows is a rival that listener has to rule out. A listener who has never heard of the longer metres needs only to notice that the bar repeats; a listener who knows them all has to wait for each to be wrong. And the evidence that the bar repeats is exactly what an extension imitates, so the better the listener’s knowledge, the less the repetition proves.

Four metres, measured both ways #

The twenty-five counted 7+7+11 — the arrangement of the dance Sedi Donka — is recognised exactly as fast as it is induced, in thirty-nine steps, because its hardest rival for both listeners is the same one: a bar of eighteen, its first eight groups repeated, which agrees with it for thirty-eight steps. That rival is a cut, and a cut is the one kind induction also has to beat. No bar in the dictionary is longer than twenty-five, so the twenty-five has no extensions to wait for. The twenty-five counted 9+9+7 takes twenty-five both ways.

The shorter metres are where knowing more costs more. The eighteen is induced in twenty-five steps and recognised in thirty-two; the eleven counted 2+2+3+2+2, induced in sixteen, is recognised in twenty-nine, nearly twice as long. For a short metre the dictionary is full of longer metres that start the same way, and those are the rivals a listener who knows them must outwait.

The downbeat accent narrows the gap as it narrowed everything: the eighteen to nineteen steps and the eleven to twelve, three and two above their floors. The twenty-five counted 7+7+11 is recognised in twenty-five steps with it — two above its floor of twenty-three, because a twenty-four with the same first ten groups and a short last one agrees with it on everything, downbeats included, until its second bar begins at step twenty-four.

A small repertoire breaks the present #

The escape the earlier essay named is not a listener who knows every metre. It is a listener who knows a tradition, and a tradition is a small subset of the dictionary.

The shape is steep. A listener who knows five metres recognises the one being played within the present 93 per cent of the time, in nine steps on average. A listener who knows thirty recognises 64 per cent of them inside it, in about fifteen and a half steps. A listener who knows a hundred recognises 21 per cent, three hundred 1 per cent, a thousand none.

The downbeat accent barely matters here, which is the reverse of what it did for induction and for the whole dictionary. With a small repertoire the stream is usually unambiguous long before the last group begins, and there is no floor for the accent to collapse anything onto. Only from a hundred metres known upwards, where the repertoire again holds rivals that begin the way the metre does, does the accent recover a few per cent: 21 to 27 per cent within the present at a hundred, none to 5 at a thousand.

The cut decides the floor, and the size decides the rest #

The floor predicts something sharp about repertoires. If the metre’s longest cut is among the metres the listener knows, nothing in the stream can rule it out before one step into the last group, so recognition cannot come before the floor however few other metres are known. If the cut is not known, nothing forbids it. Both halves can be checked by drawing the same repertoires three ways.

With the longest cut forced into the repertoire, not one trial at any size is recognised before its floor, and at most 3 per cent are recognised inside the present — the shortest bars, whose floor already lies inside it. That is the floor holding, as it has to. Drawn at random, thirty known metres beat the floor 85 per cent of the time, because a random thirty seldom contains the one metre that matters.

The third line is the test that could have gone the other way. Keeping out every cut of the metre and every extension of it — all the rivals that held recognition up among the whole dictionary — helps, and much less than the whole-dictionary picture suggests. At thirty known it moves the share below the floor from 85 to 90 per cent; at a thousand, from 5 to 20. Metres of the third kind are the last to fall for only 43 arrangements when the whole dictionary is known, because a cut or an extension nearly always outlasts them. With the cuts and extensions removed, they are what is left, and among a thousand metres there are enough of them agreeing for a while with any stream that one outlasts the floor four times in five.

So the earlier essay’s question has a definite answer, and it comes in two parts. Leaving the metre’s own cut out of the repertoire is necessary to break the floor, and it is not sufficient: the repertoire also has to be small. A listener who knows a few dozen metres, none of which shares the played metre’s opening, will usually recognise even a long bar before its floor; the same listener, taught every metre the arithmetic allows, never does.

How many metres a tradition holds #

The dance repertoires the long metres come from name a small number of them. A Bulgarian or Macedonian dance tradition uses a handful of bar lengths and, for each, one or two arrangements — the seven counted 3+2+2 or 2+2+3, the nine counted 2+2+2+3, the eleven counted 2+2+3+2+2, the long composite bars of the named dances. That is a repertoire in the tens, not the hundreds, and it is the size at which recognition works.

That puts a different interpretation on the tradition’s rule that long bars are groups of shorter ones. A twenty-five is a nine until its last unit found the rule right about the practice and wrong about the mechanism: a listener inferring the metre never prefers the group, but cannot reach the evidence that separates the bar from it. With recognition in the picture, the rule can be read as a statement about which repertoire the listener has. Of the shorter metres the twenty-five counted 7+7+11 begins with, two matter most: the eighteen that holds it up for thirty-eight steps, and the twenty-two that is its longest cut. A listener who knows the twenty-two cannot recognise the twenty-five before step twenty-three, and one who knows the eighteen cannot recognise it before step thirty-nine. A listener who knows neither is limited only by whatever else the repertoire holds.

There is a connection here that reaches outside rhythm. How many boxes an octave holds found that a listener can tell between 151 and 356 pitches apart in a direct comparison and can name only six or seven without one, which is the number of degrees scales actually have. A repertoire of metres is recognisable for the same kind of reason a scale is nameable: identification is a comparison against a stored set, and a stored set is useful only while it is small.

The arithmetic of ruling out #

The arrangements are generated by the same enumeration the earlier sweep used — every metre of twos and threes from nine to twenty-five units containing both, not a repetition of a shorter metre, in every distinct rotation — and the count of 1,820 is checked against it. Each arrangement is tiled from its written downbeat to 120 steps, marking its onsets, which of them begin a long group, and which begin the bar. For a stream and a rival, the first contradiction is the first step at which their onsets differ, or, under a cue, at which an onset is marked long or marked as a downbeat in one and not the other. The recognition window is one step past the latest first contradiction over every rival in the dictionary, and the rivals figure recomputes each contradiction from the tiled onsets it draws.

Induction is the earlier essays’ preference-rule scoring unchanged. The repertoires are seeded random subsets of the dictionary of the stated size, always containing the metre being played, drawn three times for every seventh arrangement. The cut test draws each repertoire three ways from the same seeded sequence of random choices — at random, with the longest cut added, and with every cut and extension barred — so the three lines differ only in the rule.

What recognition by contradiction assumes #

That the listener knows where the downbeat is. Every stream here is heard from its written first beat, which a dancer joining a dance at its start has and a listener tuning in halfway does not. A listener who must also find the downbeat has every rotation of every metre as a rival, and recognition would be slower still.

That recognition waits for certainty. Ruling out every rival is the strictest criterion, and a listener who commits once the rivals are improbable rather than impossible would recognise sooner — in a small repertoire the difference is small, and in a large one it is the whole question.

That the repertoire is random. A real tradition’s metres are not a random draw; they may be chosen to be distinct from one another, which would make recognition quicker than the random repertoires suggest, or they may share their openings, which by the cut test would make it slower.

And that memory is perfect. The stream is compared step by step against the dictionary for as long as it takes. A cycle that outruns the memory holding it is a different problem, and a listener who holds only the last few seconds is back under the present the floor was measured against.

What a table of rivals cannot establish #

That dancers recognise metres this way. A dancer knows the dance, not only the metre, and the steps, the melody and the words all say which dance it is long before the rhythm alone could. The arithmetic here isolates the rhythm and says what it can do by itself.

That a larger repertoire is a worse one. A musician who knows more metres can play and name more of them, and the cost measured here is only the time to identify one from its onsets. What the arithmetic says is that the cost is real and grows fast, and that it is paid in exactly the currency — steps of evidence — that the present rations.

Still open: a listener who has to find the downbeat too #

Every recognition window here starts at the written downbeat. A listener who joins a dance already in progress hears a stream starting at an unknown point in the bar, so every rotation of every metre in the repertoire is a rival, including the metre’s own rotations. Finding the beat is two problems, how far apart the beats are and where the first one falls, and a timeline that says where it is answers the second for a single cycle by its asymmetry. A metre has to be able to change its mind found induction revising its reading as a stream arrives; the recognition version of that question is how many steps a listener entering at a random point of a bar needs to know both the metre and where its bar begins, against a repertoire of a stated size — which would say whether a small repertoire still recognises a long bar inside the present when the listener has not been told where the bar starts.