A syncopated melody needs its phrases to land

A melody marks the barline by where it lingers found that at the fastest harmonic rhythm, where every harmonic witness is at a coin toss about where the bar begins, a melody with the habit of putting its long notes on downbeats finds the barline — 91 per cent of eight-bar passages when a long note falls in half the bars and four in five of them on the downbeat. It found the preference was everything: at even odds no number of long notes helped. And it ended on the far end of that dial. A melody that prefers the half-bar marks the wrong barline with exactly the confidence the other marks the right one.

In real music such a melody is rarely alone. A syncopated tune sits over something that knows where the bar is — a harmony changing once a bar, a bass stating each root on the downbeat. The question left was whether syncopation is then a figure against a ground the harmony supplies, or a second opinion that competes with it.

Added up with the harmony, it competes, and it wins. What brings the barline back is the melody itself.

A texture in which the harmony knows the bar #

The passages are the earlier essay’s, which grew out of the chords never move the barline — eight bars of eight quavers, diatonic triads in C, a melody of chord tones in crotchets with minims and held phrase ends laid over it at stated rates — with the harmony slowed to one chord a bar. At that rate the chords change only at barlines, and the witnesses the earlier essays built out of the bass know it: a walking bass marks the bar by the note it is on found the root reading placing the barline over a bass that walks on every crotchet.

The melody is set to the far end of the earlier essay’s dial: every long note on the half-bar, the syncopated counterpart of a melody that lingers on downbeats. The readings are combined as they were before, each scaled by its own spread over the candidates the metre leaves tied, and added.

With no phrase end held, the combined reading finds the barline in 21 per cent of passages and the half-bar in 77. The bass by itself had found the barline in 69 per cent. One syncopated voice has not merely weakened the harmony’s answer; it has reversed it.

Why one voice outvotes the harmony #

The reason is visible the moment the witnesses are counted one at a time.

The harmony’s evidence about the barline reaches the combined reading through two of the bass’s readings, and they do not agree. The root reading votes for the barline in every passage: at one chord a bar the bass states each new root on the downbeat and is anywhere in the chord at the half-bar. The motion reading — whether the bass moves — never votes for the barline, and in more than half the passages votes for the half-bar.

That is the walking bass’s own habit. It moves on every crotchet among the tones of the chord sounding, never repeating the note before; but on the downbeat it must sound the new root, and whenever the note before the barline already was that root, the bass does not move there. Inside the bar there is no such constraint and it always moves. So a bass that walks moves less at the barline than half a bar later, which is the opposite of the “bass leaps at the barline” convention and the reason the leap at the barline is a fact about register had to put the leap into the register rather than the motion.

The tally follows. In 57 per cent of passages the root votes barline, the motion votes half-bar and the melody votes half-bar: two against one. In the other 43, the root and the melody tie and the reading splits them. Twenty-one per cent of passages reach the barline, which is what that arithmetic gives.

The harmony knows where the bar is, and says so through one witness out of two. A bass that holds through a change marks the barline found a convention that gives the bass a second barline witness, a held note under a mid-bar change; at one chord a bar there is no mid-bar change to hold, and the convention has nothing to act on. A syncopated melody needs only one vote to cancel it, and the bass’s other witness has already sided with the melody.

How little a vote the melody needs #

The melody’s vote was set equal to each bass reading’s. The combination can be asked what happens if the melody counts for less.

The curve has three flat steps and they are the vote-counting above. Below one, the melody decides only passages the bass leaves tied — which at one chord a bar is every passage in which the root and the motion disagree, more than half of them — and it decides them all for the half-bar. At one, the melody ties the root, and the motion decides. Above one, the melody outvotes the root alone and, since the motion never votes the other way, wins nearly every passage.

So any weight at all takes the barline from the harmony in the passages where the harmony’s own witnesses disagree. A listener who gives a syncopated melody a tenth of the credit of a bass note loses a quarter of the barlines the bass alone would have given; a listener who gives it full credit hears the bar begin in the middle.

With a chord every two bars the bass is a little stronger — the root reading still votes for the barline, the motion is silent more often — and the same steps occur at slightly higher values: 59 per cent at a tenth of a vote, 29 at a full one. The shape does not depend on the harmonic rhythm; the harmony is simply a weaker ground than it looks, because half of what the bass does is not about the barline.

What the melody itself gives back #

If the harmony cannot hold the barline against a syncopated melody, something in the melody must, and the earlier essay had already found the strongest single thing a melody can do: hold the end of a phrase through its bar, from the downbeat.

A syncopated melody usually does that too. Its long notes fall off the beat inside the phrase, and its phrases still end on a downbeat — the syncopation is a tension that the arrival resolves. Laid over the hero figure’s melody at stated rates, the held phrase ends bring the barline back quickly. With a syncopated long note in half the bars, a quarter of the phrase ends held gives 67 per cent together and 58 from the melody alone; half of them held gives 86 and 84; three quarters, 97.

The melody’s own duration reading shows why. Each syncopated minim puts four quavers of duration on the half-bar; each held phrase end puts eight on the barline and removes the notes that would otherwise follow it. At two-bar phrases, the barline’s surplus overtakes the half-bar’s somewhere between one phrase end in ten held and one in four, and from there the melody points at the barline by itself, with the harmony adding to it rather than being outvoted.

With a syncopated long note in every bar the melody’s pull toward the half-bar is twice as strong, and it takes half the phrase ends held to reach 77 per cent. The two dials trade against each other in a way a composer would recognise: the more a tune syncopates inside its phrases, the more firmly it must land at their ends to be heard in its metre.

The same held phrase ends at two chords a bar tell the other half of the story.

At two chords a bar the melody alone finds the barline as readily as at one: half the phrase ends held gives 83 per cent, every one gives all of them. Added to the bass, it now does worse than alone — 69 per cent at half, never above 82 — which is the dilution a melody marks the barline by where it lingers found: a bass at chance casts full-sized votes that are noise. So the two harmonic rhythms fail the syncopated melody in opposite ways. At one chord a bar the bass’s motion sides with the syncopation; at two it sides with nothing, and costs the phrase ends a fifth of what they would have bought.

A drum restores the ground, if the harmony helps it #

The one witness most syncopated music has and these passages lack is a drum, and its evidence is the simplest of all: it strikes every downbeat.

A drum is a witness of the same kind as the root: it votes for the barline in every passage. At one chord a bar, with a drum worth one bass reading, the tally in the passages where the bass’s motion leans toward the half-bar becomes two against two, and the reading splits them; in the rest, the root and the drum outvote the melody. The barline comes back in 72 per cent of passages, and from a drum worth one and a half, in all of them.

At two chords a bar the root no longer votes, and the drum has to outvote the melody and the bass’s noise on its own: 54 per cent at a drum of one, 82 at two, and three before it holds every passage. So the kick drum on the downbeat that ragtime, swing and most popular music keep under their syncopated melodies is doing most of the work of holding the metre, and it does it more cheaply the slower the harmony moves. A syncopated melody over a drum and a slow harmony is heard against the bar; over a fast harmony and no drum it is heard as the bar — unless its own phrases land.

Figure and ground, or two opinions #

The question the earlier essay posed had two answers in mind. If syncopation is a figure against a ground, the harmony fixes the metre and the melody is heard as displaced against it; if it is a second opinion, the two are weighed and the stronger wins.

A listener who adds evidence, as every reading in these essays has, is of the second kind by construction, and the arithmetic says what that listener hears: the half-bar, three times in four, whenever the melody’s syncopation is not answered by its own phrase ends. That is not how syncopation is described by musicians, who hear an off-beat long note as pulling against a beat that stays where it is. So either listeners weigh the witnesses very differently from this — giving the harmony’s root a vote several times the melody’s — or they are not adding at all.

Nothing in the arithmetic can decide between those from the outside, because both listeners would report the same thing about a syncopated tune with a drum under it; they differ only where the ground is weak, which is exactly where the figures above live. The second possibility has a precise form. A listener who settles the barline first, from whatever is least ambiguous, and then holds it, reads later evidence against a fixed grid rather than as a vote on the grid. A dancer who comes in late needs the downbeat marked found how quickly a marked downbeat settles a metre; a listener who settled it from a phrase end or from the first few roots would then hear every later syncopation as syncopation. That listener is the figure-and-ground account, and it is a listener with memory — which no reading here has.

How the passages are built and read #

As in the earlier essay: eight bars, one chord a bar (or one every two bars where stated) drawn from the diatonic triads of C with no repetition, the texture’s notes at every harmonic setting of one, a bass walking on every crotchet among its chord’s tones and never repeating a note, and above it a melody of chord-tone crotchets with minims and held phrase ends at stated rates. The metre scores each candidate by its strong positions’ onsets; the root reading by whether the bass sounds its chord’s root at each candidate’s barlines and half-bars; the motion reading by whether it moves there; the melody’s reading by the length of the note beginning there. Each is scaled by its own spread over the candidates the metre ties and summed with the stated weights. Three hundred passages a setting, seeded.

What the arithmetic leaves out #

Pitch, again. A syncopated melody’s pitches carry metre too — a phrase that climbs to its peak on a downbeat, an appoggiatura resolving onto one — and the melody here is pitch-blind.

The drums. Most syncopated music is accompanied by a percussion part whose whole job is the ground: a kick drum on the downbeat is a witness that votes for the barline in every passage, as the root does, and would change the tally from two against one to two against two. The harmony alone is a weak ground; the combination of harmony and drums may not be.

Memory. Every reading pools eight bars at once. The figure-and-ground listener needs a state that remembers where it put the barline and charges for moving it, and the arithmetic here is exactly the part of the account that listener would replace.

Whose syncopation #

The textures this is about are the ones where syncopation is the style rather than an event in it: ragtime’s right hand over a stride left hand, swing and bossa nova melodies over a walking or ostinato bass, the anticipated downbeats of much popular song, where a phrase’s long notes arrive a quaver or a beat early and its ends are held across the barline. The long note and the strong note found that duration and metrical weight disagree about which chord a bar holds on a large share of passages; here they disagree about where the bar is, and in the styles above the disagreement is written in on purpose. The fastest of them move their harmony twice a bar, which is the texture a texture in which nothing says where the bar begins found every harmonic witness blind in; a syncopated melody there has nothing underneath it to be heard against except what the drummer supplies. The arithmetic says those styles rely on something other than the harmony to keep the metre — a drummer, a stride bass that leaps, or phrase ends that land — and that when none of those is present, a listener adding evidence would hear the syncopation as the beat.

Still open: a listener who settles the barline and then holds it #

Every reading here weighs all of the evidence at once. A listener who decides where the bar begins from the first bars and then charges for changing that decision is the listener musicians describe, and the one question the arithmetic can ask of it is how early and how firmly it must decide for syncopation to survive.

The computation reads the passage bar by bar with a state that carries the current barline and a cost for moving it, as the key-finder carries a key and a cost for changing it, and feeds it the syncopated melody after a stated number of bars in which only the harmony and the bass are heard. It would say how many bars of unambiguous ground a syncopated tune needs before it is heard against the bar rather than as the bar — and whether that number is anywhere near the one or two bars of introduction that such music so often has.