A fourth decision, and two that were never made

The long note and the strong note put the segmentation’s two cues on one axis — a note’s metrical position against how long it is held — and found that the reading turns over partway along it. It ended by naming the obvious next thing to do with the axis: hand it to the search that decides everything else.

Three decisions that constrain each other resolves key, metre and segmentation together rather than one after another, by enumerating a hypothesis space and scoring every point in it. The cue mixture is a free variable inside the segmentation and that search holds it at zero, so a passage whose durational reading would have supported a better barline is never offered as a hypothesis. Adding the axis multiplies the space and changes nothing else.

It multiplies the space from 1,152 hypotheses to 5,760, and the answer does not move: on the passage above the four-axis search picks a mixture of zero and reads C major seventh, which is what the three-axis search picked. That is one of the two outcomes the debt anticipated, and it is the wrong reading of it.

What the search actually enumerates #

Adding an axis means reading the search, and reading it turns up something that was not on the list of things this rung was supposed to find.

The key the joint search reads is a pitch-class histogram correlated against the twenty-four probe-tone profiles, and the histogram counts every event once. So it does not depend on where the barline is, and it does not depend on which notes the segmentation decided were the chord. It is the same key at every one of the search’s forty-eight hypotheses, on every passage, by construction. The key enters the total only through a binary discount on a chord whose root falls outside it.

The metre fares no better on the material the earlier essay draws. The metre score is the onset pattern rotated to each candidate barline and matched against the metrical grid — and both of that essay’s passages have a note on every one of the eight slots, so every rotation is the same pattern and every phase scores identically. The normalised metre factor is 1.000 at all eight, which is a constant that cancels out of a product.

So the search that is described as resolving three decisions together is, on its own material, resolving one: which chord, over eight barline positions that the metre cannot rank and one key it was handed before it started. That is not a small correction. Its headline result — that the pipeline and the joint search disagree — survives, but the mechanism is the out-of-key discount rather than three constraints pulling against each other.

Weighting the histogram by the same cue the segmentation uses repairs the key axis at no cost, and it is what makes the mixture a decision the key can consume in the first place: five different keys appear among the hypotheses for the scale and seven for the arpeggiated line. That is the version searched below.

The score is not comparable along the new axis #

The reason the four-axis search picks zero is not that the metrical cue is doing all the work. It is that the score it ranks by is not the same quantity at different mixtures.

A segmentation score is coverage times parsimony: how much of the weighted note mass a candidate chord accounts for, times how much of the chord was actually sounded. Change the mixture and the weights change, and they do not merely change — they change how concentrated they are. A metrical weighting over a bar of eight runs from 1 on the downbeat to an eighth on the off-beat quavers, a span of eight. Three-to-one durations span three. So the weighted mass is a little over twice as concentrated at the metrical end of the axis as at the durational end, and a concentrated mass is easier for four notes to cover.

The control settles it. Give the eight notes the durations that agree with the metre, and the reading is C major seventh at every mixture on the axis; the earlier essay drew exactly that and called it the boring answer, which it was. The score nevertheless falls from 0.850 to 0.813 across the axis. Nothing about the analysis changed and the number moved anyway.

There is nothing wrong with the score as a score. Within a fixed weighting it does exactly what which notes are the chord built it to do, and the parsimony factor is there precisely so that a diminished seventh cannot win by containing four of the twelve pitch classes. The defect appears only when the weighting itself becomes a hypothesis, which is the moment this rung arrives at.

A search that ranks mixtures by that score is ranking rulers. It will pick the metrical weighting on a passage with no metrical evidence in it, for the same reason a ruler with finer graduations reports a larger number of divisions. And the 14 to 31 per cent of passages on which it does pick a non-zero mixture are not a rate of anything: they are the cases where the durational reading was good enough to overcome a systematic handicap.

A null that holds the weighting fixed #

The repair is one this collection has used one anchor along. A count and a correlation put a cadence count and a key-profile correlation on one scale by scoring each against its own null, and the same device works here: shuffle the passage’s pitch classes among its slots, keeping every duration and every metrical position exactly where it was, and score the best chord again.

That destroys the correspondence between the notes and the weights and leaves the weighting untouched, which is precisely the separation the comparison needs. The reading is then reported in standard deviations above what its own ruler gives by chance, and two mixtures become comparable because each is measured against itself.

On the syncopated scale the answer changes. The metrical reading, C major seventh, stands 1.12 standard deviations above its null. The durational reading, D minor seventh, stands 2.15 above its own at a mixture of 0.75 — nearly twice as far — and it is a hypothesis the joint search has never been offered.

The control does something worth stating carefully, because it is the half of the prediction that fails. The debt expected the joint search to choose a non-zero mixture on syncopated passages and zero on the rest. It does not choose zero on the rest. It chooses whatever mixture happens to clear its own null by the widest margin — 0.5 on the control above — and every mixture on that axis names the same chord.

The mixture is unidentified there, not zero, and those are different states of knowledge. A parameter is zero when the evidence says so and unidentified when the evidence cannot tell; reporting the second as the first is how a model comes to look more certain than it is. On material where the two cues agree about which notes are prominent there is nothing in the passage to estimate a mixture from, and the honest output is a range rather than a number.

What the repaired search chooses #

Two constructed passages show that the axis can matter, which is much weaker than a rate. So the comparison runs over five hundred passages, with the chance that a long note falls on a metrically weak slot rising from none to certain — the same construction the previous rung used, so the two numbers can be set side by side.

The middle line is high everywhere and says little: a non-zero mixture wins on 71 to 76 per cent of passages, and most of the time it wins while naming the same chord the metrical reading names. What that measures is that the durational weighting is usually a slightly better-attested ruler, which is a fact about rulers.

The bottom line is the answer. The chord the standardised search names differs from the metrical reading’s chord on 9 per cent of passages where every long note sits on a strong beat, and on 33 per cent where none of them does.

Those two numbers belong beside the previous rung’s 16 and 43 per cent, which is how often the two cues disagree at the ends of the axis. Roughly two disagreements in three are then settled in favour of the metrical reading anyway, and one in three is not. So the fourth axis is a real decision on about a third of syncopated passages and on about one in twelve of the rest, and on the remainder it is a parameter with nothing to estimate it from.

What this does to the rungs above #

The cadence as evidence counts cadences in a passage, and a cadence is a pair of segmented chords. A count and a correlation sets that count against a key-profile correlation and finds the two collinear. Both read a progression that the joint search produced, and on a third of syncopated material that progression would now be a different progression.

There is a second consequence and it is the more interesting one. Weighting the key histogram by the cue does not merely repair a dead axis: it makes the key sensitive to the segmentation’s parameter, which is the constraint the joint search was built to express and did not. A passage read durationally has a different weighted histogram from the same passage read metrically, and the two histograms can correlate best with different keys — five of them appear among the hypotheses for eight notes of a scale.

That is the first place in this collection where the segmentation’s cue actually reaches the key, and it reaches it through the histogram rather than through the chords. It is worth being precise that this is a repair to the search rather than a discovery about music: nothing here shows that a listener’s key-finding is weighted this way, only that a search claiming to resolve three things together cannot do so while one of them is computed from an order-blind, weight-blind count.

Whose analysis this is a claim about #

Every number above is about a segmenter, and it is worth saying plainly which tradition of analysis the segmenter is a model of, because the two cues come from different places and only one of them is a theory.

The durational rule — a long note is a chord tone and a short one is passing — is what a harmony class teaches first, and it has been the stated rule of thumb in figured-bass and species pedagogy since the eighteenth century. It is a claim about the common-practice repertoire from Corelli to Brahms, where a suspension is long, its resolution is short, and a passing note is by definition the quick one between two slower ones.

The metrical rule is not taught anywhere as a rule of analysis. It is a piece of twentieth-century music psychology: Longuet-Higgins and Lee’s weights come from a model of how a listener parses a rhythm, and the beat is inferred is this collection’s account of the inference they describe. Using them to decide which notes are chord tones is a modelling convenience that this collection adopted without arguing for it, and every figure on this anchor since which notes are the chord has rested on it.

So the mixture is not a dial between two theories of harmony. It is a dial between a theory of harmony and a theory of rhythm perception, and the interesting thing about a third of syncopated passages preferring the durational end is that they prefer the older and more explicitly musical of the two. The repertoire that would test it hardest is the one where long notes routinely fall off the beat — the anticipations and pushed chords of mid-century popular song and of Brazilian and Cuban dance forms, where the harmony arrives an eighth early as a matter of style. Syncopation is a number about the metre prices that displacement on the rhythm ladder, and the note that has a length is the collection’s account of duration doing structural work a pitch-only model cannot see. Neither of them has been pointed at a harmonic segmentation, and both could be.

Which computation produced the numbers #

The segmentation is cueFit, which is the fourth rung’s scorer with its weighting on a mixture: each note’s weight is its metrical weight raised to one minus the mixture, times its duration raised to the mixture. At a mixture of zero it reproduces the published scorer exactly, which every figure here asserts before drawing. The metrical weights are Longuet-Higgins and Lee’s, computed from the subdivision tree.

The joint search is the ninth rung’s, unchanged in its three factors — a normalised metre fit, the segmentation score, and the key correlation with its out-of-key discount — with the mixture as a fourth loop and the key histogram weighted by the same cue. Every candidate chord in the collection’s table is tried at every one of the twelve roots, at each of eight barline phases and five mixtures.

The null is a set of shuffles of the passage’s own pitch classes among its own slots — forty on the single-passage figures and twenty-four on the sweep, seeded in both, so the figures redraw identically. Those are few, and they are enough because the quantity being estimated is a mean and a spread over a bounded score rather than a tail probability.

The rate sweep draws pitch classes uniformly from the major scale and gives each slot a long or a short duration with a probability that depends on whether the slot is metrically weak — the previous rung’s construction exactly, so that the two rates are measured on the same population.

Where the account stops #

A null is not a likelihood. Standardising against shuffled orderings makes two mixtures comparable; it does not turn either into a probability, and the standard deviations here are not a test with a size. What they license is a ranking, which is what a search needs and all it needs.

The passages are eight notes long. A bar of eight quavers is the smallest object on which the mixture can do anything at all, and it is also the object on which the metre score has least to say — one bar, one onset pattern, eight rotations. Over four bars the barline phase would be a real decision and the interaction between it and the mixture would be visible; here it cannot be.

And the durations are two values. Real durations are continuous and are correlated with metrical position, which is exactly the correlation the sweep destroys. The 33 per cent is therefore an upper bound on a difference and not an estimate of one, in the same way and for the same reason the previous rung’s 43 per cent is.

Where this ladder goes next #

Ten rungs: a progression as a path through a space, a progression that drifts a comma flat, how fast harmony can move, which notes are the chord, three decisions resolved together, a cadence as evidence, a count set against a correlation, a passage built to make the two disagree, the segmentation’s own untried cue, and now that cue handed to the search — which found two of the search’s three decisions taking one value each on the way.

What is owed after this is a passage long enough for the barline to be a decision. Every hypothesis space on this anchor is one bar of eight slots, and on one bar with an onset everywhere the metre score is a constant. Four bars of the same construction give it eight candidate phases with genuinely different onset patterns under them, and the question the ninth rung meant to ask — whether the barline the metre prefers is the barline the harmony prefers — becomes askable for the first time. It is arithmetic and it needs nothing this collection does not have: the same scorer, the same null, a longer event list. The prediction is that the interaction is asymmetric — that the segmentation moves the barline more often than the barline moves the segmentation — because the metre score has four values to work with and the segmentation has a hundred and forty-four.

The second debt is a null for the metre, and it is the same argument one axis along. The metre score is normalised by its own range across those eight candidate phases, which is a device for putting it on a scale and not for saying whether any phase is well attested; when every phase scores the same, that normalisation reports 1.000 rather than reporting that it has nothing to say. Scoring each candidate phase against shuffled onsets would give the metre a comparable quantity and would let the search weigh a confident barline against a confident chord instead of multiplying two numbers on unrelated scales.

The third is not arithmetic and should be named as such. Everything here is measured on passages whose durations are drawn independently of their pitches, and the one quantity that would move all of these numbers is the correlation between the two in real music — how often a composer gives a chord tone both length and a strong position. That is a count over analysed scores, it is the corpus this anchor has been recording as owed since the cadence as evidence, and no amount of shuffling substitutes for it.