The change reading follows the chords, not the bar

The essay that found the harmony marking the barline by changing there found that the harmony marks the barline by changing there rather than by being more consonant there, and that a reading scored on how much the pitch-class content changes at each candidate position stands well above the alternatives at the true one.

Every passage it read changes chord exactly once a bar. So does every passage the two essays below it read. That is not a small restriction: one chord a bar is the single harmonic rhythm at which “the chords change here” and “the bar starts here” pick out the same slots, and a reading that scores the first cannot be distinguished from a reading that scores the second until the two are pulled apart.

Pulling them apart is one parameter. A bar of two chords puts a change at the middle; a chord held across two bars puts a hold at every second barline.

The margin collapses #

At one chord a bar the reading stands 1.52 standard units above the average of the other candidates and picks the true barline 48 per cent of the time, against a chance rate of 12.5.

At two chords a bar it stands 0.01 above them and picks the barline 13 per cent of the time. That is chance to within the noise of a hundred and twenty passages. The reading has nothing at all.

At a chord every two bars it stands 0.76 and picks it 35 per cent of the time.

So the reading was measuring the chord rate and calling it the bar. With two chords a bar the changes fall on the barline and the half-bar alike, the two candidates are indistinguishable, and a reading that cannot separate them cannot separate the barline from anything. With a chord every two bars the changes fall on alternate barlines only, so half of the true barlines carry no evidence — and the margin comes out at 0.76 against 1.52, which is half, exactly.

That exactness is worth pausing on. It is not a fitted agreement; it follows from the construction, and the figure leans on it rather than remarking on it. Halving the density of the evidence halves the standardised margin, which is what a linear score in a standardised unit does.

What the earlier essay actually established #

This does not overturn the thirteenth essay; it narrows it, and the narrowing is sharp enough to be worth stating as a sentence.

What the thirteenth essay established is that a harmonic change is evidence about where a boundary is. That stands. What it did not establish, and could not have, is that the boundary in question is the barline — because in every passage it read, the barline and the chord change were the same event.

The metrical half of those essays is untouched. A metre built on a regular weight profile cannot tell a barline from the barline half a bar later, because the two put the same slots under its strong positions, and that ceiling of one half is arithmetic rather than measurement — the same ceiling the chords never move the barline ran into from the other side.

What changes is the other half: the harmony was supposed to break that tie, and it breaks it only when the harmonic rhythm happens to be slower than the half-bar. Two chords a bar is a common harmonic rhythm — it is most of a Bach chorale and a great deal of everything else — and in it the harmony gives a listener no more help with the barline than the metre does.

Why the reading looked so good #

There is a reason a rate detector masquerading as an alignment detector gives a convincing answer, and it is the same reason a pitch detector reports an octave too low.

A chord change every four slots also produces a change at every eighth slot — every second change falls on the slower grid. So the slower grid inherits the faster rate’s evidence, and the bar, being a multiple of any harmonic rhythm that fits inside it, inherits every rate’s evidence.

That means the reading can never be wrong in the direction that would have been informative. If it reported the barline when the chords changed twice as fast, the report would carry information — the reading would have found something the chord rate did not supply. Reporting the barline when the barline is the chord rate is a report with nothing in it.

The one case where the two come apart is the one the figure shows as 0.76: a chord every two bars. There the reading does have evidence at the barline, half of it, and it is genuinely evidence about a two-bar unit rather than about the bar. So even at the rate where the reading looks best behaved, what it has found is the harmonic rhythm.

The rate at which it fails is an ordinary one #

It would be a smaller finding if two chords a bar were an exotic harmonic rhythm. It is not.

A four-part chorale in common time moves at a chord a beat for long stretches and at two a bar almost throughout; a slow movement in four often has two harmonies to the bar; a great deal of nineteenth-century writing changes on every second beat. The harmonic rhythm at which the reading collapses entirely is one of the two commonest there are, and the harmonic rhythm at which it works is the one that produces the least interesting harmony.

That inverts the usual way of putting it. The reading is not a good barline detector with a known failure case; it is a chord-rate detector that coincides with a barline detector at one setting, and the setting is the one where the harmony is doing least.

It is worth being clear that this does not make those earlier readings wrong about the passages they read. It makes them readings about a corpus of one harmonic rhythm, and it means the quantity they reported — how much the harmony helps with the bar — is not a constant of music but a function of a dial nobody had turned.

Two dials that were one dial #

The reason the dial went unturned for three essays is worth naming, because it is the kind of thing that hides in a generative model rather than in an analysis.

The passages are built chord by chord and the chords are laid out one to a bar, so “bar” and “chord” name one and the same unit. Every dial the essays did turn — how regularly the notes fall on strong slots, how strictly each note belongs to its chord, how many notes there are — acts within that structure and leaves it alone. A parameter that is not in the model is not a parameter anybody thinks to sweep, and a chord rate of one a bar was not a setting; it was the shape of the loop.

The separation of those two axes was a real piece of work and it survives. What it did not include was a third axis that the construction had fixed at a value, and the third axis turns out to carry the second one entirely.

What a listener has instead #

Stating the negative leaves the original question open, and the honest position is that the account here has not answered it.

A listener does find barlines in passages with two chords a bar, and they do it somehow. The essays before this one supply two candidates and neither is enough on its own: the metre is a weak witness with a ceiling of one half, and the harmony turns out to be a witness to something else.

What is missing from both is everything that is not pitch-class content or metrical position: the bass line’s own contour, the register at which each chord is voiced, the articulation, the dynamic, and the fact that a real passage is played by somebody who knows where the barline is and shows it. How often the chord changes is the essay that made the harmonic rhythm a quantity in the first place, and it is the quantity that turns out to be doing the work here rather than the metre.

That includes what the long note and the strong note separated, which is duration against metrical weight — both of them cues this passage model has and neither of them a cue about alignment once the rate is free. So the practical reading of these four essays together is that the chord rate is recoverable from the notes and the bar’s alignment is not, at least not by anything computed so far. That is a claim with an obvious next test and it is the next essay’s.

What the collapse is not #

Three readings of the collapse are available and two of them are wrong, so it is worth closing them off.

It is not noise. A hundred and twenty passages at each rate is enough to separate 13 per cent from 48, and the margin at two chords a bar is not a small positive number — it is 0.01 standard units, which is zero to two decimal places in a quantity that reaches 1.52 at the next setting. The reading has not become unreliable; it has become empty.

It is not the harmony getting weaker. Every passage at two chords a bar has more harmonic events than one at a chord a bar — twice as many changes, twice as much of exactly the evidence the reading scores. The reading fails with more evidence than it succeeds with, which is the tell that the evidence was never about the barline.

And it is not the standardisation. The score at the half-bar is the score at the barline reversed, exactly, at every rate — an identity of the measure that the figure checks. So the reading is not failing to find a peak; it is finding one and finding an equal trough half a bar away, which is what a period-two ambiguity looks like when it is drawn on an alignment axis.

Put together, those three say the same thing in three ways. A reading that scores a difference across a boundary can only tell a listener where the boundaries are, and “where the boundaries are” is a rate and an alignment together. the earlier essay read the alignment off a passage in which the rate had already supplied it.

Which computation produced the numbers #

The passages are the thirteenth essay’s, with one parameter added. Each is four bars of eight slots, with notes placed at a stated density biased toward the metrically strong slots by a regularity dial, and each note’s pitch class drawn from its current chord with a probability set by a second dial.

The parameter added is how long a chord lasts, in slots. At eight it is one chord a bar, which is what every earlier essay used; at four it is two a bar; at sixteen it is one chord held across two bars. Nothing else changes.

The reading is the thirteenth essay’s change score: for each of the eight candidate alignments, how far the pitch-class content on one side of each boundary differs from the content on the other, averaged over the boundaries that alignment implies. It is standardised against the same passage with its slots shuffled, because a reading’s raw score depends on how many notes there are.

The margin is the standardised score at the true barline minus the average at the six candidates that are neither the barline nor the half-bar. The half-bar is excluded from that average because it is not an ordinary competitor — it is the one candidate the metre cannot distinguish from the truth, and including it would flatter every reading.

Where the model stops #

The chord rate is constant within a passage. Real harmonic rhythm accelerates into a cadence and slows in a prolongation, which is the quantity how often the chord changes measured and left as a constant here, so a real passage has a rate that is a function of position rather than a number. A reading that scored a changing rate would have more to work with and is not what any essay here computes.

And the chords are drawn at random within the key. They avoid repeating and they obey nothing else — no functional progression, no cadential preparation. A progression is a path is The first of these essays, and a listener hearing a real progression has expectations about which chord comes next that these passages give them no basis for.

The chord rate is a power of two. Four, eight and sixteen slots are the rates that fit the metre exactly, and a harmonic rhythm of three slots in a bar of eight — a chord every dotted crotchet — is a case with no clean relationship to the bar at all. That case is where a rate detector and an alignment detector would come apart most usefully and it is not computed here.

Eight slots to a bar is one metre. A bar of three, or a compound metre, changes which candidate alignments are confusable with which.

What the picture cannot show #

It cannot show the bass. Every note here is a pitch class with no register, and the single strongest cue to a harmonic boundary in real music is probably the bass moving. A bass line is not a list of roots is where that becomes a question about register, on a different account.

Nor a performance. A player marks a downbeat with time, weight and articulation, and every one of those is outside the note list. That is not a small omission: it may be the entire answer.

It cannot show a real corpus either. Every passage here is generated, which is what makes the dial available and is also what makes the result a statement about a model. Whether real music at two chords a bar leaves its barline as unmarked as these passages do is a question a corpus would answer and there is none — and there is a reason to think real music does better, since a composer writing two chords a bar has every other cue available and no reason to leave the downbeat unmarked.

And it cannot show a listener who is already entrained. The readings here are cold — each passage is read on its own with no prior. A listener four bars into a piece has a running hypothesis, and the question a real listener faces is whether to keep it rather than which of eight alignments to adopt.

Still open: what the reading finds when it is asked its own question #

The finding above is a failure of the reading as a barline detector and a description of what it is instead. A measure that follows the chord rate is a harmonic-rhythm detector, and a harmonic-rhythm detector can be evaluated properly — by asking it what the rate is and checking the answer against what the passage was built with.

That is a different scoring: instead of eight candidate alignments at a fixed period, a set of candidate periods, each scored at its own best offset and standardised against the same shuffled null. The question is how often it returns the rate it was given, and the interesting part is the shape of its errors rather than the rate of them.

The prediction from the ambiguity above is specific and it is worth writing down before it is run. A slower grid inherits a faster rate’s changes and a faster grid cannot inherit a slower one’s, so the reading should err slow and essentially never fast — which would make it a period detector with the same asymmetry a pitch detector has toward subharmonics, and would explain why it reported the bar so convincingly. The bar is a multiple of every harmonic rhythm that fits inside it.