A melody marks the barline by where it lingers

Three devices, three essays, and the same rate defeated all three. A bass that holds through a change marks the barline, a walking bass marks the bar by the note it is on and the leap at the barline is a fact about register each found a bass convention that places the barline when the metre cannot tell it from the half-bar — and each found it failing, or needing a convention of voicing to rescue it, at two chords a bar. At that rate the chords change on the barline and on the half-bar alike, and everything computed from the chords is symmetric about the half-bar. A texture in which nothing says where the bar begins put the whole account’s witnesses in that texture and found them all at the metre’s coin toss.

The last of those essays ended on the part of the texture nobody had asked. Every passage since the chords never move the barline drew its notes from metrical weight and the current chord, which makes the melody symmetric about the half-bar by construction and guarantees it says nothing. A real melody has habits a chord symbol does not: its long notes arrive on downbeats more often than elsewhere, and its phrases end at barlines. Giving the passages a melody with those habits is the one change that could put evidence where none of the harmony can.

It does. What decides how much is not how many long notes the melody has.

The melody the passages were given #

Each passage is the texture the earlier essays settled on as the hardest: eight bars of eight quavers, two diatonic triads a bar, a bass moving on every crotchet among the tones of the chord sounding, and above it a melody. The melody sounds a crotchet on each beat, a tone of the chord sounding, with two habits laid over it at stated rates.

A long note replaces two crotchets with a minim, either at the start of the bar or at its middle. The share of bars that hold one is one dial, and the share of those long notes that fall on the downbeat rather than the half-bar is the other. A phrase end holds a note through the whole of a phrase’s last bar, from its downbeat. Everything else — the pitches, the chords, the bass — is drawn exactly as before.

The reading is the simplest one a note length allows. For each candidate barline, it takes the length of the note that begins on each of its barlines, less the length of the note beginning on the matching half-bar, a slot where no note begins counting as nothing, averaged over the bars. It is the reading the long note and the strong note used to decide which chord a bar held, turned to a different question: where a bar begins, rather than what it contains. It is asked to break the metre’s tie, exactly as the bass readings were.

The preference is everything #

With the long notes as likely to fall on the half-bar as on the downbeat, the melody’s reading finds the barline in 46 to 48 per cent of passages — the same as the metre alone, whether a quarter of the bars hold a long note, half of them or every one. This is the symmetry argument again, and it holds for note lengths exactly as for chords: a feature equally likely on either side of the half-bar is evidence for neither.

Tilt the preference and the reading moves at once. With a long note in half the bars, 61 per cent of passages at six long notes in ten on the downbeat, 76 at seven, 91 at eight, 97 at all of them. With a long note in every bar the same preferences give 74, 85, 97 and 100. With a long note in only a quarter of the bars they give 53, 61, 71 and 85.

So the quantity a listener’s barline depends on in this texture is a single number about the melody’s style — how strongly its long notes prefer the downbeat — and the number of long notes only sets how quickly that preference is heard. Three quarters of long notes on the downbeat, in a melody with one every other bar, is worth 84 per cent: nearly the 85 per cent that a still bass holding one mid-bar change in six was worth at this same harmonic rhythm, and it needs no bass at all.

An even melody makes the metre worse #

There is a cost hidden in the hero figure’s baseline, and it is worth separating because it runs against intuition.

The metre in these essays scores a candidate barline by whether the strong positions it predicts — the barline and the half-bar — hold onsets. The texture’s own notes already fill those positions on either candidate, which is why the metre ties the barline with the half-bar. A melody that sounds every crotchet fills the second and fourth beats as fully as the first and third, and the metre then ties the barline with every crotchet: 25 per cent, a four-way tie. An even melody has not added evidence; it has removed the one thing the metre still knew, which beats are strong.

Long notes give that back regardless of where they fall. A minim removes the onset on the weak beat it covers, so a melody with a long note in half its bars lets the metre find the strong beats again — 49 per cent, the two-way tie — whether the minims begin on downbeats or half-bars. The metre’s recovery and the melody’s own reading are two separate effects of one habit: the long note tells the metre which beats are strong by what it leaves out, and tells the duration reading which strong beat is the barline by where it begins.

A phrase end held through its bar #

The second habit is rarer and stronger.

A phrase end held through its bar is, by construction here, always on the downbeat, and it covers a whole bar — so it carries a whole bar of duration difference on the barline and none at the half-bar, and it removes the onsets on the other three beats of its bar. Every one is decisive evidence, and they need not be common: one phrase end in four marked, in two-bar phrases, finds 80 per cent of barlines from a starting point of 25, and half of them finds 93.

Four-bar phrases have half as many ends in eight bars and pay for it: one in four marked gives 56 per cent, half of them 79. The strength of a phrase-final cue is in how often it recurs, and that is set by the phrase length. A style of short phrases that holds their ends — a hymn, a dance tune — gives its listener the barline far more cheaply than a style of long ones, which is the direction a phrase is a number of seconds found phrase lengths pulled for other reasons.

The bass is a witness with a vote #

The most surprising number in the figures is what happens when the listener listens to everything.

The earlier essays combined the bass’s witnesses by giving each a vote scaled to its own range, and that was the right thing to do when each witness knew something. At two chords a bar the bass knows nothing: its readings sit at half across the whole range, and scaled to their own spread they cast full-sized votes that are noise. Added to the melody’s reading, they pull the answer toward chance — 76 per cent where the melody alone found 91, 81 where it found 97.

A listener who attends to the bass as well as the melody finds fewer barlines than one who attends to the melody alone. This is not a property of basses; at one chord a bar, the same bass readings are strong, and the earlier essays found them so. It is a property of combining evidence without knowing which witnesses are informative in this texture. A mechanism that weighed each witness by how well it had predicted barlines so far — rather than by its own range — would learn to ignore the bass here within a few bars. Whether listeners do that is a question about attention, and the arithmetic says only that they would need to.

The same melody at three harmonic rhythms #

The dilution is a claim about one texture, and the way to test that is to change the texture and keep the melody.

The melody’s reading hardly moves across a fourfold change of harmonic rhythm — 86 per cent with a chord every two bars, 89 at one a bar, 91 at two — because nothing in it comes from the chords. The small rise is the metre’s, not the melody’s: faster chords leave more of the texture’s own notes on strong beats, and the tie the melody has to break is a little cleaner.

The bass moves a great deal. With a chord every two bars it finds the barline 76 per cent of the time and with one a bar 70, which is the strength the earlier essays measured; at two a bar it collapses to 51. And the combination tracks the bass’s usefulness exactly. At one chord a bar, adding the bass to the melody lifts 89 per cent to 92 — two informative witnesses, partly independent, doing what independent evidence should. At a chord every two bars it adds nothing and costs nothing. Only at two chords a bar, where the bass knows nothing, does the sum fall below the melody alone.

So the rule a listener would need is not “ignore the bass”. It is “ignore the bass when the chords change twice a bar” — which is a rule about the texture, learnable from the first few bars of any passage, and exactly the kind of rule the essays on the chord rate found listeners must already apply to read harmonic rhythm at all.

Why a melody can do what the harmony cannot #

The chords are symmetric about the half-bar at two chords a bar because the chord sequence has no memory of the bar: it changes at both points, by the same rules. Everything downstream of the chord sequence inherits that — which octave a root goes in, whether the bass moves, whether it holds. The hold convention and the anchored root broke the symmetry only because they were conventions of voicing applied at one point and not the other.

A melody’s habits are the same kind of thing, and more of them. A long note on a downbeat is an agogic accent, one of the oldest markers of metre in the theory of rhythm; a phrase end at a barline is the reason phrases are counted in bars at all. Neither is in a chord symbol, and neither depends on the harmonic rhythm. That is why they survive the rate that defeated the bass: they were never downstream of the chords.

It also explains why the melody’s evidence is so much stronger than the bass’s ever was. The bass devices marked a barline by an absence at the other point — a mid-bar change voiced over a held note, a mid-bar root played in the nearest octave — and each one needed the chord to change at the half-bar before it could be absent there. A long note marks the barline by a presence at the barline and needs nothing from the harmony.

How the passages are built and read #

Eight bars of eight quaver slots, the metre two levels of two over the bar, two chords a bar drawn from the six diatonic triads of C major with no chord repeated, the texture’s notes drawn as in the earlier essays at every harmonic setting of one, a bass moving on every crotchet among its chord’s tones. The melody sounds a crotchet on each beat, a chord tone; a long note is a minim at slot nought or four of the bar; a phrase end is a semibreve at slot nought of a phrase’s last bar. The metre scores each of the eight candidates by its strong positions’ onsets, texture and melody together; the duration reading is the mean over bars of the note length beginning on the candidate’s barline less that beginning on its half-bar. A candidate is credited in proportion when readings tie. Three hundred passages per setting, seeded.

What the model leaves out #

Pitch. A melody’s pitches carry evidence too — a leap to a phrase’s first note, a high point in the middle of a bar, a cadential descent onto a downbeat — and the melody here is pitch-blind: its notes are random chord tones. Everything above is what duration alone can do, and a melody that shapes its pitches to its phrases would add to it.

Where the preference comes from. Every figure takes the share of long notes on downbeats as a dial. In real repertoires it is a property of the style — high in hymns and marches, lower in music that syncopates its long notes onto weak beats — and no corpus in this collection counts it. The figures say what each value is worth, not which value a style has.

The listener’s memory. The readings pool eight bars at once. A listener hears them in order, and whether three or four marked downbeats are enough to lock a barline that later bars then need only confirm is a question about updating, which is what the change reading follows the chords, not the bar began to ask of the harmony.

Whose melodies #

The textures that put two chords in every bar are the ones where the question arises in practice: the chorale and the hymn, where the harmony often changes on every beat; fast dance movements whose accompaniment turns over twice a bar; the chord-per-beat vamps of much popular music. In the first of those the melody’s habits are unusually strong. A chorale melody moves mostly in equal notes and holds its phrase ends, marked in print with a pause, and a hymn tune holds the last note of each line through its bar. On the arithmetic here those phrase ends alone are enough to find the barline, and it is the harmony — changing on every beat — that has least to contribute.

The counter-example is a style that puts its long notes on the weak strong beat on purpose. A sarabande is the textbook case in triple time: its characteristic long note falls on the second beat, not the first, and the dance is recognised by exactly that displacement. The duple equivalent — a melody whose minims sit on the half-bar — would, by everything above, be heard with its barline half a bar late, unless something else in the texture overruled it. That is where the next question begins.

Still open: the melody that syncopates #

The preference that decides everything here has a far end. A melody whose long notes fall on the half-bar more often than on the downbeat — a syncopated melody, which much popular and dance music is — does not merely fail to mark the barline. By the arithmetic above it marks the wrong one, half a bar late, with exactly the confidence a downbeat-preferring melody marks the right one.

In real music that melody sits over something that knows where the bar is: a drum pattern, a bass, a harmonic rhythm of one chord a bar. The question is what happens when the two disagree — a syncopated melody over a harmony that marks the barline — and whether the combination finds the bar, finds the half-bar, or, as the bass’s vote did here, drifts toward chance. It would say whether syncopation is a figure against a ground the harmony supplies, or a second opinion that competes with it, and at what strength the melody would win.