Where a repeat is changed

Almost nothing in music is repeated exactly. A second verse has a different last line; an antecedent phrase is answered by a consequent that agrees with it for six bars and then does something else; a final refrain is the earlier refrain with a real ending on it. Repetition in practice is repetition-with-a-departure, and the interesting question is not whether the departure happens but where.

There is a folk answer, and it is a good one: the change comes at the end. That is why a listener can hear two bars of a familiar tune and settle in, and be right — because whatever is going to be different has not happened yet and is not going to happen for a while.

Both halves of that claim are testable on the encodings this ladder has been using, and only one of them survives.

Cutting a scheme into units without choosing where #

A comparison between repeats needs units to compare, and choosing them by hand would be choosing the answer.

The rule used here is derived instead: the unit is the highest common factor of the scheme’s section lengths and its chorus length. For the blues, whose sections run four bars and whose chorus runs twelve, that gives four. For the ostinato it gives four. For the thirty-two-bar song, the rondo, the verse-and-chorus plan and the sixteen-bar period it gives eight.

Those are the numbers anybody would have named, which is the point: the rule is checkable, it did not have to give them, and having it produce the obvious answer is the evidence that the pairs it goes on to compare are not hand-picked.

Every pair of units is then compared bar by bar. A pair that agrees on every bar is a literal repeat. A pair that agrees on some of its bars and not others is a varied repeat, and the bars it disagrees on are what this essay is about.

The hero figure is the other case. The thirty-two-bar plan’s three A sections are three units of eight; the first two are identical and the third departs at bar 6 and stays departed. Again the changed bars are a run reaching the end: 6, 7 and 8 of 8.

Two cases, and stating that plainly is most of the finding #

With a repeat defined as two units agreeing on more than half their bars, the entire six-scheme corpus contains exactly three varied pairs, and because two of them are the same comparison made twice — the AABA plan’s first and second A sections are identical, so each departs from the third in the same way — that is two independent cases.

Both are terminal runs. Both agree on the first five or six bars and disagree on the last two or three. Neither has a changed bar anywhere in its first half.

Two for two is a perfect record and it is also a sample of two. The probability that three changed bars land as a run reaching the end of an eight-bar unit by chance is one in fifty-six; for two changed bars it is one in twenty-eight; and both together, if they were independent, would be about one in fifteen hundred. That arithmetic is correct and it is worth almost nothing, because it treats two encodings of two conventions as two draws from a population, and they are not.

What the corpus supports is a demonstration, not a statistic. The two cases are the two places where these particular conventions encode a varied repeat, and both do it in the same place. That is a fact about the conventions and it is stated here as one.

The threshold is the finding #

An honest threshold is one that has been moved, and moving this one dissolves the result completely.

The definition above says a repeat is a pair of units agreeing on more than half their bars. Change that to at least half — a change of nothing, in the sense that no pair moves by a single bar of agreement — and eighteen more pairs qualify, all of them in the blues, whose count of qualifying pairs goes from nine to twenty-seven.

Across the whole corpus at the loose threshold there are 44 changed bars, of which 15 fall in the last quarter of their unit — 34 per cent against a chance level of 25 per cent, which is a nudge and not a finding. The terminal-run count falls from three of three to three of twenty-one.

Nothing about the music changed. The pairs that joined are not varied repeats of anything: they are the tonic group and the subdominant group of a blues chorus, which share two chords because a blues is built from three, and calling them a repeat of each other is what the threshold now does.

Which means the threshold should be swept, not moved once #

Moving a parameter to its neighbouring value is the beginning of the argument and not the end of it, so here is the whole continuum. The threshold runs from zero, where every pair of units counts as a repeat, to one, where only identical ones do, and at each setting the corpus is re-counted.

Two things fall out of that table and the second is the one worth having.

The first is that the parameter is real-valued and the answer is a six-step staircase. Agreement can only be a whole number of bars out of four or eight, so almost every move of the threshold changes nothing at all and five particular moves change everything — which is what the dek means by moving it by nothing. A sweep that reported a smooth curve here would be reporting its own interpolation.

The second is that the numerator never moves. Across the whole lower two-thirds of the range — every setting at which more than one varied pair exists at all — the number of varied repeats whose changed bars form a run reaching the last bar is three, and it is three at every one of them. What the threshold changes is the denominator: three of thirty-two at zero, three of twenty-one at a quarter, three of three above a half. Above five-eighths the threshold stops admitting coincidences and starts excluding varied repeats, which is a different failure and the one that ends at zero.

So the earlier sentence needs qualifying rather than withdrawing. A result that depends on the line is a result about the line — but the thing that depends on it here is how much else is admitted alongside the finding, and the finding itself is invariant over the entire range in which it can be stated. The threshold decides the noise. It never once decided a terminal run.

A result that depends on where “counts as a repeat” is drawn is a result about that line, and this ladder has now met the same shape at every rung: the key weight decides which structure appears, the kernel width decides which boundaries are found, the alphabet decides which scheme is the most redundant, and the invariance setting decides whether a transposed passage is a return. Every one of them is a choice presented in the output as a discovery unless it is drawn twice.

What the changed bars are #

The two surviving cases have something in common that is sharper than “at the end”, and the site’s own machinery can name it.

The changed bars are the cadential ones, and the site’s own closure vector says why: the first A of an AABA ends ii7–V7 and the last ends IV–I, the antecedent of a period ends ii–V and the consequent V7–I. A repeat is varied where it has to stop differently, and the variation is confined to the last unit of each because that is the only unit whose job differs between the two statements.

In the sixteen-bar period the antecedent’s last chord change is ii–V, which supplies a fifth of root motion and no tonic goal at all — a half cadence. The consequent’s is V7–I, which supplies all three: root motion by a fifth, a tonic goal, and a leading note that resolves. In the thirty-two-bar plan the repeated A sections end ii7–V7, which is a turnaround leaving the harmony open, and the final A ends IV–I, which lands.

So the variation, in both cases, converts an open ending into a closed one, and it occupies exactly the bars that carry the ending. That is a much more specific statement than “in the last quarter”, and unlike the positional claim it does not depend on the threshold at all — it is a statement about what those particular bars do.

It also connects the two ends of this field. What makes an ending an ending is a conjunction of separate signals, and a small ending exists so that a bigger one can: a repeat that is varied at its end is a unit being converted from an interior one to a final one, which is the hierarchy of closure doing its job. The varied repeat and the nested cadence are the same device seen from two directions.

How long a listener actually has to wait #

The second half of the folk claim is the one about commitment: a bar or two of agreement and a listener knows. That is measurable directly, over the corpus, without any threshold at all.

For every scheme, every lag of at least one repeat unit and every bar, take the run of agreement immediately preceding — this passage has matched an earlier one for $k$ bars — and ask whether the next bar matches too.

One bar of agreement is worth nothing and two are worth nothing. In an alphabet of a dozen chords with a strong tonic bias, a single matching bar is mostly coincidence: any two tonic bars in a piece full of tonic chords agree, and the agreement predicts nothing whatever about the next bar.

The threshold is four bars, and it is sharp — 48 per cent, 84, 95 across two steps. Four bars is one hypermeasure, half a standard phrase, and at 108 beats a minute in common time it is 8.9 seconds, which is just outside the window inside which a series is heard as one thing.

That is a straightforward refusal of the slated claim, and it makes the folk answer more interesting rather than less. A listener really does commit early — but the commitment is not evidence-based at one or two bars, because the evidence is not there. Whatever is doing the work at bar 2 is a schema learned from other pieces rather than a measurement of this one.

Removing the restriction on lag lets the same computation run over repeats at every distance rather than at the unit length alone, and the answer does not move: agreement predicts agreement at four bars whatever the offset, because what the count is measuring is a property of the material rather than of the plan.

And over the only two schemes in the collection whose repeats are literal for most of their length, the same threshold appears — four bars, in both, arrived at independently.

Reading that figure requires reading the numbers under the points, and it is drawn to make the case for the one above it rather than against it. Eighteen cases and four cases are not a rate anybody should quote; what they show is which schemes the full-corpus dip belongs to, and the answer is the two that were expected to own it.

Where the broken runs are #

Twelve runs of more than three bars still broke, and where they broke is the two halves of this essay meeting.

Eight of the twelve break in the second half of an eight-bar unit, which is the varied-repeat finding arriving from the listener’s side: a prediction made on five or six bars of agreement fails, and it fails near the end, because that is where the alteration is.

The other four break at a unit’s first bar, and those are a different event entirely. They are alignments at some arbitrary lag — nine bars, eleven, twenty-five — that happened to run for several bars and then ran out because the piece arrived at a new section. Nothing was varied; a coincidence stopped.

Distinguishing the two took no extra machinery, only counting where the break fell, and the first version of this figure asserted that all of them were at the end. They are not, and eight of twelve is a good result that would have been reported as twelve of twelve by anybody who did not check.

In a matrix the broken runs are the ends of the stripes: the bright diagonal at an offset of eight runs for six cells and stops, and where it stops is the bar the two statements differ in.

Whose music varies its repeats, and where the encoding already said so #

The two cases belong to two repertoires and the practice in each has a name and a date.

The sixteen-bar period is a classical object: an antecedent closing on the dominant answered by a consequent closing on the tonic, the standard shape of the theme of a great many movements between roughly 1770 and 1830. Its variation is not optional decoration. A period whose two phrases ended identically would not be a period; the whole device is a question and an answer, and the answer differs from the question in exactly the bars where the cadence is. The thirty-two-bar AABA plan belongs to American popular song of roughly 1925 to 1955, and the same logic applies to its final A: the first two leave the harmony open on a dominant so the piece continues, and the last one closes so it can stop.

In both cases the placement of the change is not a stylistic habit that happens to fall at the end. It is forced, because what the change does is convert a unit that continues into a unit that finishes, and finishing happens at the finish.

There is a third piece of evidence sitting inside the encoding, and it was written down before any of this was computed. The twelve-bar blues in this collection carries the note “the commonest of several variants; bar 12 is the one that varies most” — a description of practice, recorded when the scheme was encoded and used by every figure on the site since. Bar 12 is the last bar of the chorus.

So the corpus’s own annotation, the classical theory of the period, and the computation over the encoded bars all put the variation in the same place, and only the third of those is a measurement. Three weak agreements are worth more than one of them alone and are not worth a corpus, and saying which is which is the whole of the honesty available here.

What this cannot show #

The corpus is six schemes and it contains two varied repeats. That is the binding limitation and no amount of care about thresholds fixes it. What is demonstrated here is that a computation can find the altered bars of a varied repeat without being told where to look, and that when it does so on these encodings the bars are at the end. What is not demonstrated is anything about how often, in any repertoire, that is where they are.

The encodings are also plans rather than pieces. A convention written down as a chord succession has already had most of its variation removed — the whole point of writing a plan is that it says “A” twice — so the corpus is biased towards literal repetition by construction, and the two varied repeats survive in it only because the variation is structural enough to be part of the convention.

And a bar is still the unit. A varied repeat in practice is very often varied in its melody over identical harmony, or in its scoring, or by an added bar, and none of those is visible here. The single commonest kind of varied repeat in the classical style — the same phrase with an ornamented melodic line — would register in these figures as a literal repeat.

Where the ladder stands #

Eight rungs have now measured repetition eight ways: as a picture, as a set of boundaries, as a number of bits, as a period, as what a first hearing can have of it, as something a transposition may or may not preserve, as an event in memory, and here as a thing with a place where it stops being exact.

Every one of them ran on a bar-level chord scheme, and every one of them ended at the same wall. A chord scheme cannot see a melody, and melodic repetition is most of the repetition there is. It cannot see rhythm, and a returning rhythmic figure over new harmony is one of the commonest ways music refers to itself. It cannot see who is playing.

What the eight rungs do establish is that none of that is a limitation of the method. The matrix, the checkerboard, the coder, the lag profile and the decay are all indifferent to what goes in the vector; they were run on chord symbols because chord symbols are a matter of public record and a transcription would not be. Feed them a spectral description of a recording and every figure above is computable and none of the arguments changes shape. The arguments about thresholds, about which parameter is a question in disguise, and about what a first hearing cannot have are properties of the measurements rather than of the encoding — which is the reason for doing it on encodings first.