A return has to be remembered

Six rungs of this ladder have counted in bars. The matrix is a bar against a bar, the boundary operator has a kernel measured in bars, the coder emits one symbol per bar, and the period that falls out of the diagonals is a number of bars.

A bar is not a duration. It is a unit of notation, and the same eight bars are twelve seconds at one tempo and forty at another. Every figure so far has quietly asserted that a return four bars later and a return twenty-four bars later are the same kind of event differing only in a number — which is the one claim about musical repetition that no listener has ever believed.

The same axis, in the units the listener has #

The first thing to do is the cheap thing: relabel the axis.

The hero figure is the lag profile of four schemes with lag converted to seconds at 108 beats a minute in common time, where a bar lasts 2.22 seconds. The shaded band is the two-to-eight-second window inside which a series of events is heard as one thing rather than as a series, and it does not move when the music does.

The four periods land at 26.7 seconds for the blues, 35.6 for the rondo, 35.6 for the verse-and-chorus plan, and 8.9 for the thirty-two-bar song’s four-bar turnaround. The dashed orange curve falling across each row is not a second data series but an exponential decay with an eight-second constant, drawn to the height of the tallest peak in the figure — how much weight a memory of that length gives to each distance. By the time the blues’, the rondo’s and the verse-and-chorus plan’s peaks arrive it has fallen to four per cent of its height or less; only the thirty-two-bar song’s four-bar peak arrives while it is still at a third.

Only one of the four is anywhere near the window, and it is only near it. Every clock these schemes run on ticks well outside the span in which a listener holds material as present. A return, in this repertoire, is by construction an event in memory rather than in perception — which is obvious once said, and is not something any figure in the previous six rungs could have said at all.

Drawn against the window inside which a series of clicks can be heard as a beat at all — roughly 100 to 2,000 milliseconds, preferred near 550 — a crotchet and a bar at 108 beats a minute fall inside it and four bars and sixteen bars fall far outside. Everything this ladder calls a period lives past the right-hand edge of that figure. And the phrase window above it, the two-to-eight-second one, does not move when the tempo does: which bar count falls inside it is decided entirely by the tempo, so eight bars is inside for a scherzo and a minuet and nothing else, and at 52 beats a minute eight bars is 74 seconds, which is not a phrase in any sense the word normally has.

The discount, and the constant nobody knows #

Relabelling an axis changes nothing about the numbers on it. The change that matters is to weight the comparison.

The measure used below is deliberately the simplest thing that could work. For each bar, find the earlier bar it most resembles, and discount that resemblance by how long ago the earlier bar was:

$$r_i = \max_{j<i} \; M_{i,j} \cdot e^{-\Delta t / \tau}, \qquad \Delta t = (i-j)\cdot t_{\text{bar}}.$$

The piece’s score is the mean of $r_i$ over its bars. At $\tau = \infty$ the discount is absent and the score is the ordinary question — how much does each bar resemble something earlier — which is close to what the redundancy count measures by a different route. At small $\tau$ only recent resemblance counts.

The awkward part is $\tau$. There is no agreed value, and the spread in the literature is more than a factor of ten: estimates of how long an unrehearsed auditory trace survives run from about two seconds for the shortest sensory store to about twenty for the longer one, depending entirely on the paradigm used to measure it. Cowan’s two-store account of auditory memory is the standard reference for that spread; the site’s own two-to-eight-second present is the short end of it, and the pitch of a single note is held to about a semitone across delays of that order.

Picking a number and asserting it would be the least defensible move available. So it is swept.

That figure is the answer to the obvious objection. The argument survives $\tau$ being wrong by any amount at all, provided the error does not carry it across an eight-to-thirteen second band — which is a much weaker requirement than knowing the constant, and is the reason the sweep is drawn rather than a value chosen.

The ranking changes, and the schemes that move are not the ones expected #

With no decay, the six rank: the ostinato at 1.000, then the verse-and-chorus plan at 0.972, the blues at 0.969, the rondo at 0.949, the thirty-two-bar song at 0.930 and the sixteen-bar period at 0.923.

At a decay constant of 8 seconds and 108 beats a minute, they rank: the ostinato at 0.757, the blues at 0.558, the thirty-two-bar song at 0.519, the verse-and-chorus plan at 0.516, the rondo at 0.487 and the period at 0.486.

Two schemes have swapped ends of the field. The thirty-two-bar song rises from fifth to third and the rondo falls from third to fifth — the first of those by three thousandths, which a later section prices and finds is not worth a rank — and the reason is entirely the distance at which each one repeats. The AABA plan’s strongest resemblance is at four bars — 8.9 seconds — and survives the discount nearly intact. The rondo’s is at sixteen bars — 35.6 seconds — and is cut to a fifth of itself.

The verse-and-chorus plan moves too, from second to fourth, for the same reason: its whole repetition is one stripe at sixteen bars, and one distant stripe is exactly what a decay destroys.

A measure with no memory rewards a piece for repeating; a measure with memory rewards it for repeating soon. Those are different properties and the schemes rank differently on them, and only the second has any claim to be about listening.

The constant and the tempo are one number #

There is an identity hiding in the exponent, and it is worth writing out because it collapses two parameters into one.

The weight is $e^{-(i-j)\,t_{\text{bar}}/\tau}$, and $t_{\text{bar}}$ and $\tau$ appear only as the ratio $t_{\text{bar}}/\tau$. So halving the bar duration and halving the memory constant give exactly the same figure. A claim about how much memory a listener has is, in this arithmetic, indistinguishable from a claim about how fast the music is going.

That is the strongest kind of check available to a computed figure: a claim about the arithmetic, a redraw at different numbers, and an agreement that was not arranged. It also has a consequence for reading either figure. A memory constant quoted without a tempo is half a statement, and so is a tempo quoted without one.

Running the same six schemes at 60 and at 180 beats a minute, every one of them scores between 1.40 and 1.71 times higher at the faster tempo. That is not a subtle effect and it is not an artefact: at 180 beats a minute a bar is 1.33 seconds, so a sixteen-bar return is 21 seconds away instead of 64, and the sixteen-bar return is what the whole score is made of.

The ranking moves with tempo too. At 60 beats a minute the thirty-two-bar song is third of six; at 180 it is last, because a fast tempo brings everybody’s returns inside memory and the ranking converges back on the memoryless one.

That has an uncomfortable consequence and it is worth stating plainly. On this measure the same piece played faster is more repetitive, and there is no version of the arithmetic in which it is not. Whether that is a defect depends on whether “repetitive” is meant as a property of a score or of an experience, and the site’s position throughout this ladder has been that the second is the interesting one — which means accepting this.

The one period that is nearly present #

The thirty-two-bar song’s four-bar turnaround lands at 8.9 seconds at 108 beats a minute, which is just outside the eight-second edge of the window and inside it at any tempo above about 120. It is the only period in the collection that gets anywhere near, and it is worth asking what being near buys.

That is a specific and slightly deflating result. The AABA plan’s rise up the ranking under decay is not a discovery that its form is memorable. It is a discovery that its surface repeats fast — a two-bar harmonic figure restated every four bars — and that fast surface repetition is what survives a discount.

The same mechanism explains why cyclic music is where it is. A piece built on a cycle cannot cadence and has no closure to compute, and what it has instead is a pattern whose period is a few seconds, which is to say inside the present. A rhythm drawn as a circle is a claim that the whole object fits in one window; the preferred rate at which a listener counts puts the beat at roughly half a second and the bar at a couple. Groove-based music does its repeating in the region where this figure keeps the ink, and sectional music does its repeating outside it.

The two kinds of music are not repeating different amounts. They are repeating at distances that a decay treats completely differently, and every ranking of one against the other on any memoryless statistic has been comparing a present quantity with a remembered one.

A matrix that forgets #

The decay can be applied to the picture rather than to the number, and doing so produces the only figure in this family that a listener would recognise as asymmetric.

The second picture is the honest version of the first for anyone who wants the matrix to be about hearing rather than about the score. It also makes visible something the undecayed matrix hides: a piece’s structure is not symmetric under time reversal for a listener, and the matrix is, and that symmetry is a property of the drawing rather than of the music.

What survives the discount #

An argument that a measure changes under weighting is only worth having if something also fails to change.

The ostinato remains the ceiling at every constant and every tempo — 1.000 undecayed, 0.933 at 32 seconds, 0.757 at 8, 0.574 at 4, and top of the list every time. Its resemblance is at a distance of one bar, so no plausible decay touches it. That is the same result the earlier rungs gave from three other directions, and its stability across a fourth is worth something.

The direction of every scheme’s response is the same: all six fall as $\tau$ falls, monotonically, and none crosses zero. The measure does not invent repetition anywhere.

And the size of the field is stable. Undecayed, the six occupy the range 0.923 to 1.000 — a spread of 8 per cent, which is uselessly narrow. At an eight-second constant they occupy 0.486 to 0.757, a spread of 56 per cent. The discount does not merely reorder the schemes, it separates them, and that is a better argument for weighting than the reordering is: a statistic that puts five very different plans within 8 per cent of each other is not measuring anything a listener would call a difference.

Whose music, and at what tempo #

Because the tempo is half of the parameter, the repertoire has to be named before any of these numbers means anything.

The figures above are drawn at 108 beats a minute in common time, which is a plausible tempo for a mid-century popular song and for a blues, and an implausible one for most of the other four. A classical rondo finale runs faster and is usually felt in a shorter unit than a crotchet; a sixteen-bar period in a slow movement runs a great deal slower. Redrawn at each scheme’s own conventional tempo, the field would spread further than it does here, and in the direction that separates them more: the fast sectional forms would climb and the slow ones would fall.

That is a limitation of the drawing and it is also the point. The scheme is a plan and the tempo is not part of it. A twelve-bar blues is a chord succession, and the same succession is a slow blues at 60 and a jump number at 200, and this measurement says those are different pieces by a factor of 1.6 while every other measurement in the ladder says they are the same object. Which of those is right is not a question arithmetic settles, but it is a question the earlier rungs could not even pose.

There is a second period-and-place claim worth stating. The two-to-eight-second window is a finding about human listeners from twentieth-century laboratory work, chiefly on tone sequences and on tapping, and it is not a fact about music. Composers of these plans did not have it. What they had was a practice, and the practice happens to put phrase lengths in that window at conventional tempos — which is the argument the phrase essay makes at length and which this rung takes as given rather than re-deriving.

What this cannot show #

The decay is exponential because exponential is the simplest thing with one parameter, and there is no evidence that auditory memory decays exponentially. Recognition data are usually better fitted by a power law, and both are fitted to aggregate performance rather than to anything happening in a listener.

That last sentence used to end by guessing that a different functional form with the same half-life would give the same ordering. The guess can be checked and it is five-sixths right. An eight-second exponential has a half-life of 5.5 seconds, so here are the six schemes rescored under power-law decays of that same half-life:

Everything the essay above makes an argument out of survives. The ostinato is the ceiling under every form; the blues rises to second under every form; the rondo falls to fifth under every form; the sixteen-bar period is last under every form. And the separation survives too — at k = 1 the six occupy 0.469 to 0.714, a spread of 52 per cent against the exponential’s 56, where the memoryless statistic gave 8.

What does not survive is the one place the essay’s ranking was decided by three thousandths. The thirty-two-bar song is third under the exponential at 0.519 against the verse-and-chorus plan’s 0.516, and fourth under two of the three power laws. So the claim that the AABA plan rises is a claim about its direction, which is robust, and not about the place it rises to, which is not. The rondo’s fall, which is the same paragraph’s other half, is a full two places under every form tried.

A gap of 0.003 was never worth a rank, and the sweep is what says so rather than the caution.

The measure has no rehearsal, no chunking and no learning. A listener hearing a refrain for the third time is not making a fresh comparison against a two-minute-old trace; they are matching against a schema built on the first two hearings, which is a far more robust object than anything with a twelve-second constant. That mechanism — a hierarchy built by counting what has been heard — is exactly what makes long-range form audible at all, and none of it is in this arithmetic. The figures here therefore understate what a listener can do with a distant return, systematically and by an unknown amount.

And the whole apparatus still runs on chord symbols. A returning refrain in a recording is announced by its melody, its orchestration and its words long before its harmony is diagnostic, and none of that is in a roman numeral. The measure has memory now and still has no ears.

Where the ladder goes #

One question is left, and it is the one this rung’s arithmetic makes urgent rather than answers.

If a return has to be recognised while it is happening, then how much of it must go by before it is recognised — and what happens when a repeat turns out not to be one? The last rung of this ladder measures both: how many bars of agreement are needed before the next bar can be relied on to agree, and where in a repeat the disagreement falls when it comes. The first number is larger than anybody would guess and the second depends on a threshold in a way that changes the answer completely.