The passage built to make them disagree

A count and a correlation put two kinds of evidence for a key on one axis: a cadence count, which is an ordered-pair measurement of a few discrete events, and a key-profile correlation, which is a continuous statistic over every note in a window. It made them comparable by scoring each against its own null, and it ended by saying what it had not done.

Every figure here is run on a passage where the two statistics agree, which is the uninteresting case: agreement between a low-information statistic and a high-information one tells nobody anything. The passage that would test the combination is one built to make them disagree — the notes of one key and the cadences of another — and it is constructible in a few bars.

It is constructible in a few bars because two keys a fifth apart share six of their seven notes. A body of chords drawn from C major’s diatonic set, ended by a ii–V–I in G, is a bag of notes that says C and an ordered pair that says G.

The exchange rate #

Up to a body of six bars, the cadence reading says G and the histogram says C. That is the conflict working as designed.

At eight bars the cadence reading changes — and it does not change to C major because the histogram won. Cadence evidence is not a thing the histogram can outvote; the two are different statistics computed from different features. What happens is that the body’s own root motions have accumulated into cadences of their own, and there are enough of them to outweigh the three explicit ones at the end.

So the exchange rate is about six bars per cadence, and the mechanism by which it is spent is not the one the question assumed.

What the agreement case concealed #

That is the finding, and it is the sort of thing only a constructed conflict produces.

The seventh rung treated the two statistics as independent sources of evidence to be combined, which is the natural framing and is what “putting them on one scale” means. On real material they are not independent, and not in a subtle way: every chord added to the body is simultaneously a note added to the histogram and a root motion added to the cadence count. Lengthening one kind of evidence lengthens the other, and there is no way on real material to hold one fixed while the other grows.

Worse, the cadence evidence the body accumulates is not evidence for the key the body’s notes belong to. A body of C major chords full of I→IV motion is full of C→F root movements, and a descending fifth into F is a cadence in F. So the body’s notes say C, its cadences say F, and the written ending says G — three keys from eleven bars, each named by a different reading of the same chords. Running all three readings at every length confirms it and adds something the summary above misses:

The body alone names F on the cadences and C on the notes at almost every length, which is the three-key claim confirmed. And the histogram settles on C by four bars and never moves again, which is the density of a dense statistic.

The column that is new is the first. The cadence reading does not converge. It holds G to six bars, flips to F at eight, to C at twelve, back to F at sixteen and stays. It is not that the ending’s evidence is gradually outweighed and a stable answer emerges; the answer oscillates, because whether the body’s own descending fifths into F or its arrivals on C dominate depends on exactly where the repeating eight-bar pattern is cut.

That is worth more than the exchange rate it was measured to find. A sparse statistic computed over enough material does not become a dense statistic — it becomes an unstable one, sensitive to the last bar or two in a way the histogram is not. The seventh rung’s framing treats the two as evidence of different strengths to be weighed; what the conflict shows is that they have different stabilities, and the one that can be overturned by a bar is the one being asked to carry the decision.

That is not a defect in either statistic. It is what happens when a low-information statistic is computed over enough material: it stops being sparse, and a sparse statistic and a dense one are not two views of one thing.

Why a fifth apart is the hard case and not the easy one #

The construction uses two keys a fifth apart, which is the smallest possible disagreement, and it is worth saying why that is the right choice rather than a convenient one.

Two keys a tritone apart share two notes. A body in C ended by a cadence in F♯ would be a conflict nobody could miss, and every statistic would report the disagreement immediately — the histogram would be bimodal, the cadence count would be isolated, and the passage would sound like two pieces spliced together.

Two keys a fifth apart share six notes out of seven, and the only difference between their scales is one accidental. Keys are neighbours is the essay about that geometry and the circle is a circle and the map is not is about what it does to distance measures. A conflict at that distance is a musical conflict — it is what a modulation to the dominant is, before it is confirmed — and it is the case where a listener genuinely has to weigh evidence rather than notice a seam.

So the exchange rate measured here is measured at the distance where an exchange rate matters. At any larger distance the question does not arise.

The same construction with the body the other way round #

Changing the body changes the answer, and it changes it in a direction that says what the measurement is really about.

A body drawn from C major but built on I–IV–vi motions rather than I–IV–V ones accumulates a different set of incidental cadences, and its histogram settles on A minor rather than C major. The cadence reading is overturned at four bars rather than eight, because the motions in that body are stronger cadential evidence for their own third key.

So the “six bars” is not a property of the two keys or of the cadence. It is a property of how cadential the body happens to be, which is a variable the seventh rung’s framing does not contain at all — its two statistics are “the notes” and “the cadences”, and a body of notes turns out to have cadences in it whether or not anybody wrote one.

What would have to be measured #

The construction gives a number and the number is about a construction. What it also does is say exactly what a corpus measurement would have to be, which is more useful.

The quantity is a rate of accumulation. Cadence evidence arrives in discrete lumps at cadences, and profile evidence arrives continuously with every note. Their relative worth is therefore not a constant — it is a function of how far apart the cadences are, which is a corpus statistic about harmonic rhythm, and how often the chord changes is the ladder’s own essay about the range that takes.

And the confound has a name. A corpus measurement of the two statistics’ relative worth has to deal with the fact that the same chords feed both, which is a collinearity rather than a nuisance. The right measurement is a partial one — how much cadence evidence is worth given the profile evidence — and it needs a corpus of harmonic analyses rather than a corpus of notes.

This collection has now recorded that corpus debt four separate times and has found twice that the counts overlap. This is the third overlap: the key-relations ladder needs a prior on how often keys change, the tonal-expectation ladder needs root-motion frequencies, and this needs cadence spacing. All three come out of one pass over a corpus of harmonic analyses, which makes the smallest useful corpus smaller again.

What a listener does with this passage #

The arithmetic says three keys and a musician says one thing, which is worth reconciling because the reconciliation is the useful part.

A musician hearing eight bars of C major ended by a ii–V–I in G says: it modulated to the dominant. That is not a third reading; it is the two readings put in order. The body established C, the cadence established G, and the sequence of the two is the modulation. Neither statistic can say that, because a bag of notes has no order and a cadence count is a count.

Which is exactly what the eighth rung of the key ladder found from the other side, and it is the reason a passage like this one is the standard test case for an ordered model. The disagreement measured here is not a puzzle about how to weigh two numbers; it is the signature of a process that neither number represents.

So the honest reading of the exchange rate is narrower than it looked. Six bars is how long the profile evidence takes to accumulate past one cadence’s worth when both are read as static summaries of a whole passage — and reading a modulation as a static summary is the mistake, not the weighting.

Which computation produced the numbers #

The body is a loop of chords in one key — I, IV, I, V, IV, I, ii, IV — repeated to the required length, followed by ii–V–I in the other. Every chord is a diatonic triad and the passage is expressed as roman numerals with a key, which is the representation both statistics already take.

The histogram reading is keyCorrelations over the pitch-class content, which is the Krumhansl–Schmuckler apparatus this collection has used since keys are neighbours. The cadence reading is cadenceEvidence, the seventh rung’s own: for each of the twelve keys, a count of how many components of the closure vector are satisfied by each ordered pair in the passage.

The reported number is the body length at which the cadence reading’s top key changes, found by stepping the length rather than solved for. That is a coarse measurement — the steps are two bars — and a finer grid would put it between six and eight rather than at eight.

Nothing here is fitted and the construction is stated in full, which matters more than usual: a constructed conflict is a thing whose answer depends entirely on how it was built, so the loop, the two keys and the cadence are all arguments to the figure and can be changed.

The ordered model, for comparison #

There is a third reading available and it is worth putting beside the two, because it is the one that does not have this problem.

A key-finder that keeps the order is a hidden Markov chain over key-and-degree states, and it does not have two statistics to combine: the root motions are its transition weights and the pitch content is its emissions, so the two kinds of evidence enter one model at different places and are combined by the arithmetic rather than by a decision. It has no exchange rate because it never needed one.

What it has instead is a key-change cost, which is a single parameter deciding how readily it modulates — and the key-relations ladder’s ninth rung is entirely about sweeping it. So the ordered model has moved the free parameter rather than removed it, which is the same verdict the seventh rung reached about its own combination and is worth stating as a pattern: every way of joining these two kinds of evidence ends with one number nobody has measured.

Where the model stops #

The body is a loop and no music is. Repeating an eight-chord pattern to twenty-four bars is a way of adding evidence at a controlled rate, and it adds it in a pattern the statistics can both see. A real passage’s evidence arrives unevenly.

Voicing is absent. Both statistics take pitch classes: the histogram counts them and the cadence vector reads root motions from them. A real passage has a bass line, and a bass line is most of what tells a listener which chord it is — so both readings here are working with less than a listener has.

And the cadence count is a count of components, not a weighted score. The seventh rung chose that deliberately, because what makes an ending an ending found that no weighting of the five components is derivable. The exchange rate measured here inherits that choice, and a weighted cadence score would give a different number of bars.

What the picture cannot show #

It cannot show a listener changing their mind. The figure reports what each statistic says about a whole passage of a given length, which is a reading taken once at the end. A listener’s reading develops through the passage and has a moment where it flips, and how much evidence a modulation needs is the ladder that measures that in chords.

Nor can it show which reading is right. The passage was constructed and has no correct answer: a body in C with a cadence in G is a passage that has modulated, or a passage in C that ends on its dominant, and those are two descriptions of one object. That ambiguity is the point of the construction and it is also its limit.

And it cannot show the third key. The figure names what each reading says and the F major that the body’s own motions accumulate is visible only as a change in the label under the axis. It is the most interesting thing in the figure and it is one letter.

Whose harmony, and when #

The construction is a common-practice one and it is deliberately a cliché: a passage that sits in a key and ends with a cadence in the dominant is what the first half of nearly every binary form does. That is the point — the conflict being measured is not exotic, it is the standard ambiguity of tonal form, and a key-finder that cannot say what it thinks about it is a key-finder that cannot read a minuet.

The exchange rate is therefore a claim about a repertoire in which cadences are sparse and explicit. In a repertoire where they are dense — a hymn, where nearly every phrase ends with one — the cadence evidence accumulates as fast as the profile evidence and the trade does not arise. In one where they are absent, the profile is all there is.

Where this ladder goes next #

Eight rungs. A progression is a path across a space with a geometry; it never comes home in just intonation; its rate has limits set outside music; the objects it connects are produced by a segmentation that depends on the metre; three decisions change two readings in five; the ordered pairs carry evidence a bag cannot; the two kinds of evidence on one scale; and now the case built to make them disagree.

What is owed after this is the partial measurement, and the construction has said precisely what it is. The two statistics are collinear on real material, so their relative worth is not one number and is not measurable by comparing them on a passage — it is a coefficient in a model that has both, fitted over a corpus in which the cadence spacing varies. That is a corpus this collection now needs for three separate ladders, and each of the three has recorded it independently.