Two bars of drum hold the barline; a dozen bars of bass do not

A syncopated melody needs its phrases to land set a melody whose long notes all fall on the half-bar over a harmony that knows the bar — one chord a bar, a walking bass — and added up every witness’s opinion about where the bar begins. The syncopation won. Its vote joined the walking bass’s motion, which never votes for the barline, and outvoted the bass’s root two to one in most passages. The barline came back only when the melody’s own phrase ends were held from the downbeat.

That essay’s last section named what its arithmetic lacked. Every reading it used pools the whole passage at once, as though a listener heard eight bars and then decided. The listener musicians describe decides early, from whatever is least ambiguous, and then hears later evidence against that decision: a syncopation is heard as syncopation because the barline has already been settled. The question it left was how early and how firmly a listener must decide for a syncopated tune to survive — and whether the one or two bars of introduction such music so often has are anywhere near enough.

They are enough if the introduction has a drum in it and the listener has a memory. They are nowhere near enough if the introduction is the harmony and bass alone: then a syncopated tune needs about twelve bars of ground to be heard against the bar nine times in ten.

A listener with a running total and a cost for changing its mind #

The listener reads the passage one bar at a time. In each bar every witness casts a vote between two readings, the written barline and the half-bar. The bass’s root votes for whichever of the two positions has the bass sounding the chord’s root. The bass’s motion votes for whichever position has the bass moving. The melody votes for whichever position has the longer note beginning there. Where there is a drum, it votes for the barline. The bar’s net vote is the sum.

The listener keeps two running scores, one per reading, and a cost for switching between them:

$$D_{\text{bar}} \leftarrow \max(D_{\text{bar}},\ D_{\text{half}} - c) + e, \qquad D_{\text{half}} \leftarrow \max(D_{\text{half}},\ D_{\text{bar}} - c) - e$$

and hears whichever is higher. It is the same shape as the key-finder that how much evidence a modulation needs gave a cost for changing key, and it has one interpretation that makes the whole essay easy to read. The difference between the two scores is the listener’s confidence in the barline, and the recursion clips it to $\pm c$ before each bar’s vote is added. So $c$ is how much confidence the listener can bank. At $c = 0$ the listener has no memory and hears each bar by its own votes alone. At $c = \infty$ it never forgets, and its reading is the running total of every vote since the start.

The texture is the earlier essay’s: eight bars of one chord a bar from the diatonic triads of C, a bass walking on every crotchet among its chord’s tones, and a melody of chord tones whose long notes, in half its bars, all fall on the half-bar. In front of it go a stated number of bars of the same harmony and bass with no melody, the ground. Four hundred passages a setting, seeded.

One difference from the earlier readings needs saying before any number is compared with theirs. Those readings let each witness vote once for the whole passage, after averaging its opinion over all eight bars. This listener counts votes bar by bar, and a witness that is right by a little in most bars and wrong by a lot in a few counts differently in the two schemes. With no ground, the earlier reading found the barline in 21 per cent of passages; this listener, never forgetting, hears it in 40 per cent of the melody’s bars. Both say the syncopation wins; they disagree about by how much.

What one bar is worth #

The whole result is in these three distributions.

A bar of walking-bass ground is worth 0.41 of a vote. The root votes for the barline in nearly every bar, because the bass must sound the new root on the downbeat. But the bass’s motion cancels it in about half the bars, for the reason a walking bass marks the bar by the note it is on found: a bass that walks moves less at the barline than half a bar later, since it cannot move onto a root it is already sounding — which is also why the leap at the barline is a fact about register rather than about motion. So a bar of ground usually casts one net vote or none, and the average is a little under half.

A syncopated bar takes back an eighth of a vote. The root still votes for the barline, the motion still votes against or abstains, and in half the bars the melody’s long note on the half-bar adds a vote against. The mean is minus 0.13.

A drum on the downbeat turns 0.41 into 1.41, because it adds a vote for the barline to every bar and nothing against.

The arithmetic of holding the barline follows directly. Eight syncopated bars spend about one vote on average, and with the spread of the distribution a listener needs to have banked well over that to come through nine times in ten. At 0.41 a bar, that takes the better part of a dozen bars of ground. At 1.41 a bar, it takes two.

The same answer by a shorter route #

For the listener who never forgets, the reading at any bar is the sign of a running total, and a running total of independent votes can be estimated without simulating a single passage. That makes a useful check, because it shows the answer is a property of the two distributions above and nothing else.

A bar of plain ground has mean 0.41 and variance 0.34; a syncopated bar has mean minus 0.13 and variance 0.61. After $k$ bars of ground and all eight bars of melody the total has mean $0.41k - 1.04$ and variance $0.34k + 4.9$, and the chance that it is still positive is, near enough, the normal probability of a value that many standard deviations above zero.

With two bars of ground the mean is minus 0.22 against a spread of 2.4, so the last bar should be heard against the barline in about 46 per cent of passages; the simulation finds 49. With twelve bars the mean is 3.9 against a spread of 3.0, which predicts 90 per cent; the simulation finds 89. With two bars of drummed ground, 1.41 a bar, the mean is 1.8 against 2.3 and the prediction is 78 per cent for the last bar, against the simulation’s 81.

The approximation is not exact: votes come in whole numbers and the first bars are not independent of the passage’s harmony. But it agrees to within a few points everywhere it is checked, and it makes the dozen bars transparent. The ground has to bank more confidence than the syncopation spends, with a margin of about one and a quarter spreads. A drum quadruples what each bar of ground banks, and the spread grows only with the square root of the bars, which is why two drummed bars do the work of twelve plain ones.

How much ground, and how much memory #

Without a drum, the listener who never forgets climbs slowly: 57 per cent after two bars of ground, 71 after four, 83 after eight, 93 after twelve. Two bars of introduction, the length a song most often has, leaves the tune heard against the bar in little more than half its bars.

Memory matters as much as ground, and in a way that sets a hard limit. A listener who can bank only four votes of confidence never passes 75 per cent however long the introduction, because once its bank is full, further ground buys nothing. A listener who can bank two never passes 54. A listener with no memory at all is unaffected by the introduction, at 45 per cent throughout: it hears each bar by its own votes, and the syncopated bars’ own votes are what they are.

With a drum in the introduction, as in the first figure, the listener who never forgets reaches 90 per cent after two bars and 99 after four, and so does one who can bank eight votes. The listener who can bank four reaches 79 per cent after two bars and then stops. The drum does not change what the memory limit is; it fills the bank faster.

So the one or two bars of introduction that such music so often has are enough on two conditions: something in them votes for the barline with every bar, as a drum does and a walking bass does not, and the listener can carry the resulting confidence, around three votes’ worth, through eight bars of syncopation. A dancer who comes in late needs the downbeat marked found how quickly a marked downbeat settles a metre for a listener arriving mid-passage. This is the same finding from the other side: a marked downbeat is also what a listener arriving at the start needs, and for the same reason.

Bar by bar through the syncopation #

Watching the reading bar by bar shows the mechanism directly. The ground builds a lead, and the syncopated bars spend it. After four bars of plain ground, 88 per cent of passages hear the barline as the melody enters, and each syncopated bar lowers that by a few points, to 59 by the eighth. After twelve bars the lead is large enough that the eighth bar is still at 89. After two bars with a drum the reading starts at 100 and ends at 81.

The erosion has no floor inside the passage. A syncopated tune longer than eight bars would keep spending, and in every setting here it would eventually be heard as the bar unless something restored the ground. The listener modelled here never gives the barline up all at once, the way a listener who “hears the syncopation as the beat” is sometimes described as doing. It gives it up one passage at a time, as the running total crosses zero in a larger share of passages.

With a long note in every bar, no ground is enough #

The texture above holds a long note in half of the melody’s bars. A melody that syncopates every bar is a harder case, and the model’s answer is not a matter of degree.

With a long note in every bar, a syncopated bar’s mean vote falls to minus 0.61, and eight such bars spend about five votes. With no ground the listener who never forgets hears 6 per cent of the melody’s bars against the barline. Sixteen bars of plain ground bring that to 85 per cent, and only for a listener who never forgets: one who can bank eight votes reaches 66. For a tune that syncopates in every bar, the bass alone is not a ground at any length a song would use. Something that votes harder is needed — a drum, or the melody’s own phrase ends.

The two remedies add #

The earlier essay’s remedy was the melody’s own: phrase ends held through their last bar from its downbeat, each of which votes for the barline where the long notes vote against it. It also works for this listener, and it adds to the ground rather than replacing it.

With half the phrase ends held, two bars of plain ground bring the tune to 81 per cent, and four bars to 87. With every phrase end held, two bars reach 93. So a syncopated melody over a walking bass can be heard against the bar after a short introduction with no drum at all, provided its phrases land. A melody marks the barline by where it lingers, and the listener counts that lingering bar by bar in the same total as the ground. The result is the earlier essay’s figure-and-ground account, stated as numbers: a tune is heard against the bar when the evidence before it and inside it for the bar outweighs the evidence of its syncopation, and a listener with a memory can add the two.

What memory buys, and what it cannot #

This listener’s single parameter, how much confidence it can bank, turns out to do most of the work, and it cuts both ways.

A listener with a large bank holds the barline through syncopation after enough ground. The same listener, given no ground, is captured by the syncopation for just as long: a running total that starts negative stays negative. A listener with a small bank is never captured for long, but also never holds the barline through more than a few bars of syncopation, whatever came before. The two failures are the price of each other. A listener who carries a decision across eight bars is a listener who will carry the wrong decision across eight bars.

The chords never move the barline and the essays after it looked for a single witness that settles the bar. This one finds that the answer depends on the listener as much as on the witnesses. The ground, the memory and the evidence are three separate quantities, and a performance and a listener between them decide which barline is heard.

What the listener leaves out #

Two readings only. Every passage here is read as either the written barline or the half-bar, the two readings the metre leaves tied in this texture. A real listener considers other offsets and other metres, and a syncopation that shifts by a quaver rather than half a bar is a different contest.

Votes of equal size. Every witness casts one vote a bar. The earlier essays found that a witness’s weight could be varied and that the answer depends on it; here the drum is simply one more vote, and a drum that a listener trusts more than a bass would shorten every introduction in this essay.

A fixed bank. The listener’s cost for changing its mind is constant. A listener whose confidence grows with how long it has held a reading, or who remembers the first bars better than the middle, would fall between the curves drawn here.

No pitch, no dynamics. As before, the melody is pitch-blind, and a syncopated note’s loudness — often an accent in performance — is not modelled. An accented syncopation would vote harder, and so would an accented downbeat in the bass.

Whose introductions #

The music this is about is the music the earlier essay named: ragtime, swing, bossa nova and the popular song whose phrases arrive a quaver or a beat early. In most of it the introduction is short and the drummer plays in it: a count-in, a two-bar fill, a bar of hi-hat before the band enters. A texture in which nothing says where the bar begins found that at two chords a bar every harmonic witness is blind. This essay finds that at one chord a bar the harmony is not blind but weak, a little under half a vote a bar. On this arithmetic, the drummer’s first two bars are what make a short introduction long enough. Without them the band would need a dozen bars of vamp before the tune could syncopate against anything.

Still open: a listener who trusts its witnesses differently #

Everything here gives every witness one vote. A listener learns, over a piece, which of its witnesses are reliable — that in this style the bass walks and so its motion says nothing, that the drum is always right about the downbeat. The next computation lets the listener weight each witness by how often it has agreed with the reading it currently holds. It would say whether a listener who has heard a few bars of a walking bass stops counting the bass’s motion at all, which would roughly double what a bar of plain ground is worth. It would also say whether a syncopated melody, which disagrees with the held reading bar after bar, ends up discounted as a witness — which is one precise sense in which a syncopation is heard as syncopation.