The long note and the strong note

Which notes are the chord established that a progression cannot be read off a texture until something has decided which notes are chord tones, and that the thing which decides is the metre. Eight quavers of a scale, barred as written, are a C major seventh; barred one quaver later, a D minor seventh; barred nowhere at all, four readings that tie.

Every rung since has used that segmentation and every one of them has used it at the same setting. The scorer takes a weightBy and has done since it was written — the metrical weight, the note’s duration, or nothing at all — and the essays have drawn the first and the third. The second has never been drawn.

The second is the rule a theorist actually teaches. A long note is a chord tone and a short one is passing, and no account of harmonic analysis in the world leads with metrical position instead.

What the two cues actually are #

They are worth separating carefully, because they sound like the same idea and are built from different things.

A metrical weight is a number attached to a position in a bar, computed from the subdivision tree: zero on the downbeat and one less at each level down, so in a bar of eight quavers the first is 0, the fifth is −1, the third and seventh are −2 and the four off-beat quavers are −3. It is a property of the grid. Nothing about the notes enters it, and a note gets its weight by falling where it falls.

A duration is a property of the note. It is in the sound, it survives the barline being moved, and a listener has it whether or not a metre has been found — which the beat is inferred says is not a small proviso, since finding the metre is itself an inference over the same notes.

So the two cues are not the same kind of object, and the interesting thing is that the fourth rung’s whole argument depends on which kind is used. Its finding is that harmonic analysis is a function of a variable that is not harmony — the barline, which is imposed. Under a durational cue that finding weakens: a segmentation from durations is a function of the notes alone, and moving the barline does not touch it.

Which means the durational cue is the one that would make harmonic analysis less arbitrary, and it is the one nobody here has drawn.

Putting the two cues on one axis #

The two rules can be run separately and the comparison is then a pair of numbers, which hides where the transition is. So they are put on a single axis instead: each note’s weight is the metrical weight raised to one minus a mixture, times the duration raised to the mixture. At zero the scorer is the fourth rung’s own, unchanged; at one it is pure duration; in between each note’s weight is the geometric mean of what the two cues say it should be.

That construction has a property worth having. At a mixture of zero it reproduces the published readings exactly — same chord, same score, to the last digit the arithmetic carries — so the axis starts where the ladder already stands rather than beside it. And at a mixture of one with every note the same length it reproduces the flat control exactly, because a duration that is constant is no weighting at all. Both are asserted before any figure draws.

The case built to make them disagree #

Take the ladder’s own eight quavers of a scale and hold the even-numbered ones three times as long as the odd ones. That is a perfectly ordinary syncopated line: the long notes are on the weak quavers, which is what an anticipation or a suspension does.

Under the metre the strong quavers are C, E, G and B, which is a C major seventh at a score of 0.850. Under the durations the long notes are D, F, A and C, which is a D minor seventh at 0.813. The notes are identical and the two rules name chords a third apart.

The reading turns over at a mixture of 0.4, which says something modest and useful: the two cues are of roughly comparable strength here, and it takes a bit under half a shift toward duration to overturn a metrical reading. Neither rule dominates the other by construction.

The reading in between is neither reading #

At the crossing the winner is not either cue’s answer. It is A minor seventh, which neither the metre nor the durations select on its own.

That is not an artefact of the interpolation. A minor seventh is A, C, E, G: it shares three notes with the metre’s C major seventh and three with the durations’ D minor seventh, and when the two cues weight the eight notes nearly evenly, the chord that covers most of the weighted mass is the one that overlaps both.

The same thing happens on the arpeggiated line and it happens the other way round. There the metrical reading is D minor seventh and the durational one C major seventh — the reverse of the scale’s answer, because the arpeggio puts different pitch classes on the strong quavers — and the chord that wins at the crossing is F major seventh. Two passages, two different pairs of cues, and in both cases the middle of the axis holds a chord that appears in neither end.

That regularity is not luck either. Any two seventh chords built a third apart in one key share three of their four notes with a third chord lying between them, because the diatonic set stacks in thirds and a chain of thirds overlaps everywhere. So a scorer that rewards covering the weighted mass will always find the intermediate chord attractive when the mass is spread evenly across both readings’ notes.

A compromise between two analyses is an analysis of neither. That is a real objection to blending cues by weight, and it is worth stating as a defect rather than as a curiosity, because the obvious way to combine two segmentation cues is exactly this and it produces readings that no theorist holds and no listener reports.

The alternative form is available and is not what anybody does: combine the two readings rather than the two weightings — score the passage twice and take the better-supported answer, so that the output is always one cue’s chord and never a third. That would produce a discontinuity at the crossing instead of a compromise, which is less elegant and more honest about what is being modelled.

The control, which is the case that ought to be boring #

If the two cues were the same rule in different clothing, they would agree whenever the long notes are on the strong beats — which is most music most of the time, because that is what a metre is for.

Give the same eight notes the opposite durations, so the long ones fall on the strong quavers. The reading is C major seventh at every mixture and never turns over. That is the boring answer and it is the right one.

And they still disagree on one passage in six #

Two constructed passages prove that the cues can disagree, which is much weaker than a rate. So the same comparison is run over four hundred passages at each of six settings, with the durations drawn so that the chance of a long note falling on a weak slot rises from none to certain.

At the left, where every long note is on a strong beat, the two rules still name different chords on sixteen per cent of passages. At the right, where every long note is off one, they differ on forty-three per cent.

The sixteen per cent is the number to keep. It is the case in which the cues are supposed to be saying the same thing, and they are not, because agreeing about which notes are prominent is not the same as agreeing about how much more prominent they are. A metrical weight over a bar of eight spans a factor of eight, from the downbeat to the off-beat quavers. Three-to-one durations span three. Two cues that rank the notes identically still weigh them differently, and the scorer is a weighted sum.

Change the grid and the contrast and the floor moves with them: in a bar divided into two and then four, with long notes only twice the length of short ones, the metrical span is four and the durational span two, and the floor falls to eleven per cent with a ceiling of thirty-three.

The rate is a rate about durations, not about music #

One more caution about the sixteen per cent, because it is the number most likely to be quoted out of the essay.

It is measured over passages whose durations are drawn independently of their pitches. Real music does not do that: a composer who wants a note heard as a chord tone tends to give it both length and a strong position, which is exactly the correlation the sweep destroys. So the rate on real material would sit toward the left of the figure and probably below its floor.

What the floor does say is that the two rules cannot be substituted for each other silently, which is what has been happening: the collection teaches the durational rule in its prose and computes the metrical one in its figures, and the two have been treated as one rule with two descriptions.

The right reading of the sweep is therefore as an upper bound on a difference rather than an estimate of one. Sixteen per cent is what the cues disagree by when nothing correlates them; the correlation in real music is a fact nobody here has counted; the true rate is somewhere below.

That is the same shape as the short note is sitting on the floor, where a measured quantity turned out to be pressed against a limit set outside the music rather than free to vary — and the honest report in both cases is the bound and the reason.

What this does to the rungs above it #

The fourth rung’s finding was that harmonic analysis is a function of a variable that is not harmony. This adds a second such variable, and the two interact.

Everything from three decisions that constrain each other onward reads a progression out of a texture and then does something with it — counts cadences, correlates against a key profile, prices a surprise. On roughly one passage in six, and on nearly one in two where the durations are syncopated, that progression would be a different progression under the other cue, and everything downstream of it would be a different number.

The ladder has not been reporting an error bar on that because it has not had one. It now has a first estimate: the segmentation’s own cue choice moves the reading about as often as moving the barline does.

That is not a reason to distrust the numbers above. It is a reason to say what they are measurements of, which is a progression as read by a metrical segmenter, and to notice that the alternative reading is not exotic — it is the rule taught first in every classroom.

The rungs it bears on hardest are the ones that count events rather than describe them. The cadence as evidence counts cadences in a passage, and a cadence is a pair of segmented chords: change the segmentation and the count changes. A count and a correlation sets that count against a key-profile correlation and finds the two collinear, and only one of the two is sensitive to the cue — a pitch-class histogram does not care which notes are chord tones, and a cadence count cares about nothing else.

So the cue choice moves one of that rung’s two statistics and not the other, which means it changes the very quantity that rung reports: how far apart two collinear predictors are. That is a consequence worth chasing and it is not chased here, because doing it properly means recomputing both statistics under both cues over the whole scheme table, and the constructed passages above are not the material to do it on.

The note that has a length is the collection’s account of duration as a melodic variable, and it says duration is doing structural work a pitch-only model cannot see. This is the same claim about harmony, arrived at from the segmentation rather than from the tune.

Which computation produced the numbers #

The scorer is the fourth rung’s, and it is unchanged in everything except the weight it gives a note. A candidate chord’s score is its coverage — how much of the weighted note mass it accounts for — times its parsimony, how much of the chord was actually sounded, which is what stops a diminished seventh winning every passage by containing four of the twelve pitch classes.

Every triad and seventh in the collection’s chord table is tried at every one of the twelve roots, so the answer is which chord the passage is rather than which chord built on C fits it best.

The metrical weights are Longuet-Higgins and Lee’s, computed from the subdivision tree, and are the same weights the metre ladder uses for syncopation. The durational weight is the note’s length in slots.

The constructed passages take pitch classes uniformly from the major scale and assign each slot a long or a short duration with a stated probability that depends on whether the slot is metrically weak. Four hundred passages at each setting, with a fixed generator, so the figures redraw identically.

Where the account stops #

Two cues are not the cues. A theorist deciding which notes are chord tones uses at least four things: metrical position, duration, whether the note is in the bass, and whether it is approached and left by step. This essay adds one of the missing three and the collection has none of the others in its segmenter.

There is no register in any of it. A passage here is a sequence of pitch classes in slots, so the bass — which is the strongest single cue a theorist has — cannot be expressed. A chord is a register is the collection’s account of what that discards, and it discards a great deal.

The durations are two values. Real durations are continuous and correlated with position, and a passage in which the long notes are three times the short ones is a caricature of a rhythm. The sweep over the contrast says the numbers move with it and does not say how they move on real material.

And the disagreement rate is over constructed passages. Random pitch classes from a scale, uniformly drawn, are not music: they have no voice leading, no repetition and no cadence, and a rate measured over them is a rate for a population nobody listens to. The direction and the ordering survive that; the sixteen per cent does not travel.

Where this ladder goes next #

Nine 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, and now the segmentation’s own untried cue.

What is owed after this is the fourth axis of the joint search. Three decisions that constrain each other already resolves key, metre and segmentation together rather than in sequence, and it does so by enumerating a hypothesis space and scoring every point in it. The cue mixture is a fourth free variable inside the segmentation and that search holds it fixed at zero — so a passage whose durational reading would have supported a better key, or a better barline, is never offered as a hypothesis. Adding the axis multiplies the space by however many mixtures are swept and changes nothing else, which is arithmetic this collection can do today. The result it would produce is stateable in advance: if the cue mixture is genuinely a fourth decision, the joint search should choose a non-zero one on the syncopated passages and zero on the rest — and if it chooses zero everywhere, the metrical cue is doing all the work and the parameter swept here turns out not to matter, which is also an answer.

The larger debt this anchor keeps recording is a corpus of harmonic analyses, and it is a different thing from this. That corpus would settle a coefficient. This would settle a structure, and the collection can settle it alone.