A bass that holds through a change marks the barline

The chords mark the barline by changing there found that a passage’s chords place its barline well when they change once a bar, by comparing how much they differ across a candidate’s barlines with how much they differ across the middle of its bars. The change reading follows the chords, not the bar then showed what that reading really detects. At two chords a bar the chords change at the half-bar as fully as at the barline, and the reading falls to chance. The metre cannot help, because the barline and the half-bar put the same slots under its strong positions.

The bass errs fast where the content errs slow gave the passages a bass and asked it how often the chords change. It closed by naming a second kind of evidence a bass carries, which the rate question had no use for. In much tonal writing a change on the barline is voiced in root position, while a change inside the bar is more often voiced over a bass that is already sounding — a first inversion on a weak beat, a passing six-four, a pedal. So the bass moves more at barlines than between them, even when the chords change at both. That is a convention, and its strength is one number: how often a mid-bar change is voiced over the held bass.

Swept, that number turns out to be very effective at small values and to have a ceiling well below one.

The convention, as a model can hold it #

The passages are the ones the earlier essays built, starting from the chord-and-passing-note model of which notes are the chord: eight quavers a bar, a melody drawn with its onsets weighted toward strong positions and its notes from the current chord, and a chord sequence of diatonic triads of C major, never the same twice running, changing every four slots — twice a bar, on the barline and on the half-bar.

The bass is the earlier essay’s, with one behaviour added. It states each new chord’s root on the slot where the chord begins. Inside a chord it may move to another chord tone on a beat. And when a change falls in the middle of a bar, with a stated probability it does nothing — provided the note it is already sounding belongs to the new chord, so that the result is an inversion of that chord rather than a wrong note.

That proviso matters and it sets a limit. A held bass has to be a tone of the chord that arrives over it, and among diatonic triads that is true of the previous bass note only some of the time. So even when the convention is followed on every occasion it can be, only about a third of mid-bar changes can be held: 33 per cent of them in these passages, with the probability set to one. Every figure here is drawn against that realised share rather than the probability, because the share is the thing a count of real music would report.

Why the ceiling is a third #

The third is not an accident of the seeds, and it can be derived without running anything.

The chords are the six diatonic triads of C major that are major or minor, and each change goes to one of the five that differ from the current chord. A bass holding the current chord’s root can stay only if that root belongs to the chord arriving. Count the triads each root belongs to. C is also in F major and A minor, so a C major chord’s root survives two of its five possible successors. D is in G major only: one of five. E is in C major and A minor: two. F is in D minor: one. G is in C major and E minor: two. A is in D minor and F major: two.

That is ten survivals in thirty ordered changes, exactly one in three. A root-position bass in a diatonic major key can hold through a third of the chord changes that follow it and no more, however firmly a composer or improviser prefers to hold. The passages’ realised 33 per cent is that fraction, measured.

The number is worth having because it bounds the whole question. The convention’s effect, set out below, is steep enough that its strength in real music matters much less above about a sixth than below it, and a third is where the arithmetic of the key stops it. A bass that is already on a chord’s third or fifth — the arrangement an inversion lasts as long as its outer sixth prices acoustically — has different survival counts, and a passing bass outside the chord has none, so the ceiling is a property of root-position writing specifically.

The bass’s evidence #

The bass’s evidence about a candidate barline is then the same shape as the chord-change reading’s: the share of the candidate’s barlines on which the bass moves, less the share of its mid-bars on which it moves. A candidate half a bar later swaps the two and scores the negative, which the figures check on every passage.

Small holds, large returns #

The hero figure’s steepest part is its left end. With no convention at all, the metre plus the bass does exactly what the metre does alone, 50 per cent, because the bass moves on both changes and has nothing to choose between them. With 4 per cent of mid-bar changes held — one in twenty-five — it is 62 per cent. With 9 per cent, 75. With 16 per cent, one change in six, 85 per cent. With 26 per cent, 92, and at the ceiling of 33 per cent, 96.

The bass on its own is weaker at the left end, 34 per cent at one change in twenty-five, because it is being asked to choose among all eight candidates, and six of them score zero whenever the bass moves only on barlines and half-bars. Given the metre’s two candidates to choose between, it only has to say which of two, and it can say that from a single held change.

The chord changes alone stay between 10 and 18 per cent at every strength: at two chords a bar they carry no information about which change is the barline, and holding the bass does not change what the upper parts do.

The evidence itself is simple and small. At the true barline the bass’s score rises in proportion to the held share, 0.83 for each unit of it, and at the half-bar it falls by the same amount. At the ceiling the margin is 0.28 moves per bar. The proportion is slightly under one because a change on the barline is sometimes a root the bass is already sounding: the model’s bass moves only when the pitch changes, and a root that repeats the held note is not a move.

A margin of a quarter of a move per bar is not much in any one bar. It is a consistent quarter, though, always in the same direction, which is the kind of evidence that accumulates.

How many bars it takes #

The figures so far read eight bars. A listener orienting in a passage may have fewer, and the convention’s strength and the passage’s length trade against each other.

In one bar there is at most one mid-bar change to hold, and with so few onsets the metre’s own best candidates are often more than two, so every strength sits near or below one half: 37, 43 and 53 per cent. From there each doubling of the passage buys a roughly constant step. With a third of mid-bar changes held the metre and bass reach 68 per cent in two bars, 82 in four and 97 in eight. With one in six held, 57, 70 and 85. With one in twelve, 50, 59 and 71, and 84 only in sixteen bars.

A weak convention works; it just works slowly. One held change in twelve brings the barline out in eight bars about as reliably as one in three does in two. For a listener, that is the difference between a metre established within a phrase and one established within a period.

An arpeggiating bass, and a slower harmony #

The earlier essay found that a bass reading of the rate was undone by arpeggiation — the same fast-side error that asked for the rate, it answers a multiple found mirrored on the slow side in the content reading, because a bass that moves inside a chord looks like a faster rate. The barline reading is exposed to the same thing in a different way.

At two chords a bar nothing happens. Both beats of the bar are already chord changes, so an arpeggiating bass has nowhere to move that a change has not claimed, and the held changes keep their whole value at every setting: 95 to 97 per cent with a third held, and exactly the metre’s 50 with none held.

At one chord a bar the barline reading needs no convention: the only change is on the barline and the bass states it there, so a bass that never arpeggiates places the barline every time. But as the arpeggiation rises the reading turns over — 98 per cent at a move on half the beats, 79 at three quarters, and 16 per cent when the bass moves on every beat. The reason is the same asymmetry the hold exploits, running backwards. An arpeggiating bass always moves at the half-bar, since it moves to a tone different from the one it is on. On the barline it states the new root, which is sometimes the very note the arpeggio has just reached, and then it does not move. So the barline becomes the only beat on which the bass sometimes stands still, and a reading that looks for where the bass moves points half a bar late.

A bass marks the barline by what it does less often there, and which thing that is depends on the texture. In a slow harmony with a busy bass, the barline is where the bass is most likely to arrive and stay; in a fast harmony with a held inversion, it is where the bass is most likely to move. The reading as defined sees only the second.

Which evidence to ask first #

Three readings are now available on the same passages, and there is more than one way to put them together.

At one held change in six, the metre alone places the barline in 50 per cent of passages, the chord changes alone 13, the bass alone 73, and the metre with the bass breaking its tie 85. At one in three: 50, 15, 95 and 97.

Two orderings that bring in the chord changes are worse. Using the changes to choose among the metre’s candidates before the bass is asked gives 74 and 82 per cent, because at two chords a bar the change reading is at chance, and a filter at chance discards the right candidate some of the time and leaves the bass nothing to choose. Asking the changes first and the bass only to break their ties gives 13 and 15, no better than the changes alone.

This is the same lesson the chords are a weak witness to the barline arrived at from the other side. A reading at chance is not neutral when it is used to prune: it has to be weighed, or left out. At a fast harmonic rhythm the upper parts’ changes are exactly that kind of reading, and the bass is not.

One passage #

The averages hide what the evidence looks like in a single case, so the last figure draws one.

The chords are A minor and D minor in the first bar, E minor and A minor in the second, E minor and A minor in the third, C and A minor in the fourth. Every mid-bar change could be held, because in each case the bass note already sounding belongs to the arriving chord: A under D minor makes a second inversion, E under A minor a second inversion, and C under A minor a first. The bass moves on the first, second and fourth barlines. On the third it states E minor’s root, which is the E it was already holding, and does not move.

The melody’s onsets tie the barline with the half-bar, as they always do in these passages, and at two chords a bar the chord changes carry no preference that could be trusted. The bass alone separates them, 0.75 to −0.75, from three moves and four held notes — and a listener hearing these four bars has the same three moves to go on.

Which computation produced the numbers #

The passages, the metre reading and the chord-change reading are the earlier essays’ own, unchanged. The metre’s candidate scores come from the melody’s onsets against its strong positions, and the chord changes from the dissimilarity of the best triad on each half-bar across candidate barlines and mid-bars. The bass is the one the bass errs fast where the content errs slow added, with the hold applied only at mid-bar changes and only when the sounding note belongs to the new triad.

A reading’s credit for a passage is one over the number of candidates it ties for best if the true barline is among them, and zero otherwise. “Metre, then bass” takes the metre’s best candidates and keeps those the bass scores highest. The rates are over 160 to 240 passages each, from fixed seeds.

Where the model stops #

The chords are drawn at random. Real progressions are not, and a real cadence is itself a barline cue; nothing about closure enters here. The effect is that these passages give the chords less barline evidence than music does.

A hold is the only convention modelled. A bass that leaps to a root on the barline and moves by step inside the bar marks the barline by the size of its motion, not by whether it moves, and a walking bass that moves on every beat marks it by landing on the root. Neither is visible to a reading of moves.

The bass is heard perfectly. A bass line is not a list of roots found that inversions confuse a key-finder’s bass cue, and a listener’s bass is also partly masked by the texture above it; the moves here are counted as if every bass note arrived.

What the numbers cannot say about a listener #

How fast real harmonic rhythms are where this matters. How often the chord changes put the range across styles at a factor of thirty, and the two-chords-a-bar case is only one point in it.

How strong the convention actually is. One in six held mid-bar changes is enough for most passages and one in three is the most a diatonic bass can manage. Where real repertoire sits between nought and a third is a count over analysed scores with their bass lines, which is data not to hand here.

Whether listeners use it. A barline found by a model from a bass’s holds is a prediction about what a listener could use. That listeners find the barline in fast harmonic rhythms at all, and sooner in some styles than others, is compatible with it and does not test it.

Still open: the busy bass that lands #

The arpeggiation figure left a reading that fails in exactly the texture a great deal of music has — one chord a bar over a bass that moves on every beat — and it failed because it asked where the bass moves rather than where it arrives. A bass that moves on every beat marks its barline by landing on the chord’s root there, and between barlines by passing through other chord tones and notes outside the chord. Scoring a candidate barline by how often the bass sounds the current chord’s root on it, against how often it does at the half-bar, is the same computation with the root in place of the move. It would say whether the two conventions together — hold through a mid-bar change, land on the root at a barline — cover both textures, or whether the texture in between, two chords a bar over a walking bass, is one in which neither the chords nor the bass can place the barline.