Which notes are the chord

A progression is a path through a space of chords, the rate it moves at is a second variable, and the space itself has a geometry. Three rungs, all of which begin with a list of chords already in hand.

Music does not arrive as a list of chords. It arrives as notes, most of the vertical sonorities in an ordinary texture are not chords at all, and something has to decide which notes are structural before there is any progression to analyse.

That decision has a name — segmentation — and the thing that makes it is not harmony.

The passage #

The demonstration needs a passage with nothing in it, so that whatever the analysis turns on is visible. Eight quavers of the scale will do: C D E F G A B C, evenly, over one bar.

The notes on the strong positions are C, E, G and B. The notes on the weak ones are D, F, A and C. Both are perfectly ordinary sets and neither was chosen; they are what the scale run puts there.

Three readings #

Score every triad and every seventh chord at every one of the twelve roots against the passage, weighting each note by the metrical position it falls on.

Three results and each of them says something different.

Barred as written: C major seventh. The strong-position notes are C E G B, which is that chord exactly, and the weak-position notes are passing tones between them. This is the analysis any musician would give.

Barline moved by one quaver: D minor seventh. Now the strong positions carry D F A C, and the notes that were the chord are the passing tones. Nothing about the sound has changed. A recording of this passage, played to somebody who is told the bar starts one quaver later, is a different progression.

Moving it further does not give further readings. Sweeping the barline through all eight positions returns C major seventh at every even shift and D minor seventh at every odd one, and nothing else: the analysis depends on the barline only through its parity. That is a property of a bar of eight, whose weight pattern alternates strong and weak at the quaver, so the set of notes on the stronger positions is one of exactly two sets. A finer subdivision would admit more readings; at this one the choice is binary, and the two candidates are a third apart.

No metre: a four-way tie. With every note weighted equally, C major seventh, D minor seventh, F major seventh and A minor seventh all score 0.625 and there is nothing to choose between them. This is not a near-tie that a better scorer would break — it is exact, and the reason is sharper than the chords contain the same number of notes.

The passage is a scale run from C to C, so its C is sounded twice and every other degree once. A seventh chord scores five of the eight notes if it contains C and four if it does not. Of the seven diatonic seventh chords, exactly four contain C — the ones built on C, D, F and A — and those four are the tie. The tie is not a coincidence of this scorer; it is the set of diatonic sevenths containing the repeated note, and it would have been a three-way tie on a run that did not return to its starting pitch.

A passage with no metre has no harmony. Not an ambiguous harmony: none, in the sense that the question has no answer. And the four candidates that tie are not four arbitrary chords — they are C, D, F and A, which is to say the tonic seventh, the supertonic, the subdominant and the submediant. Any of the four is a defensible analysis of a scale run and a musician asked to pick one would want to know what came before and after, which is exactly the information the metre was standing in for.

Why the scorer needs the parsimony term #

One detail of the method is load-bearing and it is worth exposing, because the obvious scorer gives a wrong answer confidently.

The natural score is coverage: what fraction of the weighted note mass falls inside the chord. Used alone it picks a diminished seventh for very nearly any passage, because a diminished seventh contains four of the twelve pitch classes and so explains a third of anything by accident. The first version of this measurement did exactly that and returned diminished sevenths for every passage it was given, which looked like a finding and was an artefact.

The fix is to multiply coverage by parsimony — how much of the candidate chord was actually sounded. A diminished seventh with two of its four notes present is penalised for the two that are not there. The two terms together select a chord that both explains the notes and is explained by them, which is what an analyst does.

That is a modelling choice rather than a discovery, and the numbers above depend on it — though not equally. The metred readings are decided by coverage, since C major seventh and D minor seventh both have all four of their notes present and therefore a parsimony of one; the two terms only separate when a candidate is partly absent. So the parsimony term is what keeps a diminished seventh out of the answer and is not what chooses between the two answers the essay is about. The finding survives the modelling choice and the field of losers does not, which is the ordinary situation and is worth stating so that the choice is not carrying more weight than it has.

The same fact from the notation side #

Musicians have known the metre decides this for as long as there has been figured bass, and the notational evidence is that every convention for writing harmony assumes a bar.

The passage in notation puts a barline at the front of it, and that barline is not a description of anything a listener was given — it is one of the two things the reading has to infer, printed as though it were an input.

The parity result gives that a sharper edge than move it and the harmony changes. There are two analyses of this passage and the barline chooses between them, so a notation that carries a barline is not merely helpful to an analyst — it is carrying exactly one bit, and that bit is the whole of the difference between the two readings. Everything else on the page is common to both.

A figured bass gives figures per beat. A roman-numeral analysis assigns chords to metrical positions. Chord symbols in a lead sheet sit above bars. Every one of those notations has the metre built into its geometry, so none of them can express a harmony that is independent of one — which is not an oversight but an accurate reflection of the fact that there is no such thing.

How much of a texture is not the chord #

The scale run is a limiting case chosen to make one point. It is worth asking how typical it is, and the answer is that it is more typical than it looks.

A four-part chorale at one chord per beat is the easy case, and it is easy precisely because the harmonic rhythm is as fast as the note rate: every sonority is a chord and there is nothing to segment. Everything faster than its own harmonic rhythm — which is nearly all instrumental music — presents the problem on most of its notes.

What decides the metre, then #

If the harmony depends on the metre, the obvious next question is what the metre depends on, and the answer is not the harmony — or at least, not only.

The same eight events under two metrical readings carry different weights, and it is the weights rather than the notes that decide which pitches count as chord tones — a metrical accent is evidence about harmony, which is why the two cannot be inferred in sequence.

That circle is not vicious in practice, because real music supplies redundant evidence: a bass line, a drum pattern, phrase lengths, a harmonic rhythm that is itself periodic. A listener resolves the two together, and the reason the resolution feels effortless is that the cues almost always agree.

Where they do not agree, the effect is one of the most powerful in tonal music. A passage whose harmonic rhythm implies a downbeat the notated metre puts elsewhere is a syncopation at the harmonic level, and the metre ladder has the machinery for pricing exactly that.

The measurement that is worth making #

There is one number in the three readings that carries more than the others, and it is the tie.

An exact three-way tie is a much stronger statement than a narrow win. It says the metre is not merely the best available evidence for the segmentation — it says that in the absence of metre there is no fact of the matter, at least as far as pitch content is concerned. Any other model that resolves the passage without a metre must be using something the pitch classes do not contain.

That is worth stating as a limit on the headline. The scale run was chosen because its notes are in a neutral order; a passage whose figuration already groups the chord tones gives the metre less to do, and most real music is of the second kind.

The pieces this makes a mess of #

The model is at its most useful where it fails, and the failures are a recognisable list.

A scale run over a static harmony. The passage above is the pure case and the analysis is right for the wrong reason: any competent musician says “C major with passing notes” because the context is C major, not because the odd quavers are weak. Given the same eight notes in the middle of a piece in F, the answer would be F major and the model would still say C major seventh.

Arpeggiated textures, where the chord tones arrive one at a time and the metrical position of each is a consequence of the figuration rather than of the harmony.

Suspensions, which are precisely notes on strong positions that are not chord tones — the one case the metrical rule gets exactly backwards, and the case that most of the interest in eighteenth-century harmony lives in.

That last one is worth stating as a limit rather than a caveat. A suspension is a dissonance on a strong beat resolving to a consonance on a weaker one, so the whole device depends on the metre making a note prominent and the harmony refusing to accommodate it. A model that assigns chords by metrical weight will call a suspension a chord tone every time.

What a moved barline sounds like #

The claim that moving the barline changes the harmony invites an obvious objection: a listener hearing a recording is not told where the barline is, so how can it matter?

It matters because the listener puts one there. Metre is inferred rather than received, and a listener who has inferred a downbeat one quaver later than the notation intends will hear the D minor reading — not as an alternative analysis but as the harmony.

A progression drawn as a path is what the previous rungs of this ladder produced, and every one of them took the chord sequence as given. This rung is about where that sequence comes from.

Which makes the well-known experience of “hearing a piece wrong” — coming in on the offbeat and being unable to shake it — a harmonic phenomenon as well as a rhythmic one. The whole analysis moves with the grid, and it moves consistently, which is why the wrong hearing is self-supporting rather than incoherent.

What an analyst does that the model does not #

It is worth setting the model beside the practice it is a model of, because the gap is instructive rather than embarrassing.

An analyst deciding what the chord is uses the metre, and also uses: what key the passage is in, what the previous chord was, what the bass note is, how long each note lasts, whether a note is approached and left by step, whether it is a suspension prepared in the previous bar, and what the composer’s habits are. The model uses one of those.

That it gets the right answer on the scale run is therefore not much of a recommendation — it gets the right answer because the scale run was built so that the metre is the only evidence there is. What the model demonstrates is not that metre is sufficient but that metre alone is enough to produce an answer, and that without it there is none. Those are different claims and only the second is surprising.

Which computation produced the numbers #

The weights are Longuet-Higgins and Lee’s, generated from a subdivision tree: zero on the downbeat and one less at each level down, so a bar of eight is 0, −3, −2, −3, −1, −3, −2, −3. Each note’s weight in the score is two to the power of its metrical weight, which makes the downbeat eight times the weakest position.

The candidates are the nine chord types this site’s table carries, at all twelve roots — 108 candidates — scored as coverage times parsimony and sorted. Sweeping the roots is essential: a scorer that only tests chords built on C answers a question nobody asked, and an earlier version did precisely that and reported a “result” that was an artefact of the root being fixed.

Moving the barline is implemented by rotating the notes’ metrical slots rather than the notes, so the sequence of pitches is untouched and only the grid moves.

And the weights for a bar of eight with onsets on the strong half of it are the input the whole reading turns on — change which slots are strong and the chord changes, which is the dependency the pipeline breaks and the joint search keeps.

Whose music, and when #

The metrical rule for segmentation is a description of European tonal practice, roughly 1650 to 1900, and of the popular idioms descended from it. It is the rule every harmony textbook in that tradition states and the one every analysis program implements.

It is not general. Music with a free or unmeasured rhythm — plainchant, alap, taqsim, a great deal of twentieth-century writing — has no metrical grid to weight by, and the harmonic analysis of such music is either done by other means or not done. That is consistent with the finding above rather than a counterexample to it: no metre, no determinate progression.

And it is worth noting which way the historical dependence runs. The bar line arrives in European notation in the sixteenth and seventeenth centuries, and functional harmony arrives immediately after. Whether the second required the first is not a question this site can settle, but the sequence is suggestive and the mechanism above is at least a candidate for why.

What the picture cannot show #

It cannot show duration. Every note in the passage is one quaver. In real music a note’s length is at least as strong a cue to structural status as its position, and the model has a duration weighting available and unused — because using it would have muddied the demonstration, which needed everything except the metre held equal.

It cannot show register. A note in the bass is a much stronger candidate for a chord tone than the same pitch class in an inner voice, and the bass’s own partials are why. The model has one voice.

It cannot show what came before. Harmonic analysis is enormously constrained by context — a progression is a path, and the previous chord narrows the next one substantially. The model analyses one bar in isolation, which is the hardest possible version of the problem and not the one an analyst faces.

And it has no voice leading. Two readings that fit the notes equally well are not equally good if one produces a smooth bass line and the other does not, and the machinery for scoring that is elsewhere in this collection and unconnected to this.

The ladder from here #

Four rungs, and this one has put the other three on a footing: a progression is a path, its rate is a variable, its space has a geometry — and the objects being connected are produced by a segmentation that depends on the metre.

What is owed is the joint problem. Every model here decides one thing at a time: the key from the pitch classes, the metre from the onsets, the chords from the metre. A listener does all three at once and the three constrain each other, and a model that resolved them jointly would be a considerably better model and a considerably harder one to draw.