The habit worth bringing is to distrust the tune

A listener who knows the style needs no introduction settled a question about where a listener hears the bar begin. The texture is a walking bass under one chord a bar and a melody whose long notes all fall on the half-bar — a syncopated tune — and the listener reads the barline bar by bar from its witnesses: whether the bass sounds the chord’s root at the barline rather than the half-bar, whether the bass moves there, whether the melody’s longer note begins there. The earlier essays had given each witness a record, so that one which kept agreeing with the reading held earned more than a vote and one which kept disagreeing earned less. That listener needed eight bars of walking bass before the syncopated tune was heard against the bar nine times in ten. The listener who arrived with the style’s habits — crediting the bass’s motion and the melody’s long notes at a coin toss from the start — heard the same tune against the bar ninety-three times in a hundred with no introduction at all.

It ended on the obvious objection. A listener arriving at a piece does not know which style it is, and so which habits to bring. The natural listener keeps two sets — one for a style whose tunes mark the bar, one for a style whose tunes syncopate — and weighs them by how well each has predicted the bars heard so far. A metre has to be able to change its mind had found the same trade one level down, for a beat tracker: stickiness holds a beat and stops it following a change. Here the question was whether style can be recognised faster than a barline can be learned.

It can be recognised faster than nothing, and not nearly as fast as it can be assumed. And the reason the choosing listener is slow is also the reason it need not choose. The evidence about style runs one way. A syncopated tune contradicts the habit of trusting tunes; a tune that marks the bar does not contradict the habit of distrusting them. So the habit that serves every style is the syncopating style’s, kept always.

Two sets of habits, weighed by their record #

The texture, the witnesses and the passages are the earlier essays’, drawn with the same seeds, so every difference below is the listener. Each witness keeps its record — how often its vote has agreed with the reading the listener held before that bar — and its weight is read from that record through a starting credit: the credit is how reliable the witness is assumed to be before it has been heard, and the record moves it as bars accumulate.

A style is a set of starting credits. The style whose tunes mark the bar credits every witness at three in four. That is also exactly what the earlier essays’ listener with no habits credits, which is the first small surprise: no habits and the habits of a style that marks the bar are the same listener. The style whose tunes syncopate credits the root at three in four and the bass’s motion and the melody’s long notes at a coin toss, which is the listener that heard the syncopated tune from its first bar.

The choosing listener keeps both. Every bar, before it updates its records, it asks how probable that bar’s votes were under each style — a vote agreeing with the reading held is as probable as that style’s current estimate of the witness’s reliability, a disagreeing vote as probable as its complement — and it moves its belief between the two styles by that ratio, as a Bayesian listener would, with a small leak back towards even so that neither style is ever ruled out for good. Each witness then votes with the belief-weighted mix of the two styles’ weights. Given only one style, this listener is the earlier essay’s to the last digit.

Better than nothing, worse than knowing #

The hero is a clean ordering at every length of introduction. With no ground at all, the listener with no habits hears the syncopated tune against the bar in 39 per cent of its bars, the choosing listener in 67, and the listener who brings the syncopating style’s habits in 93. After two bars of ground the three are at 71, 91 and 99; after eight, at 92, 98 and 100.

So choosing is worth a great deal against having no habits: the choosing listener reaches nine bars in ten after two bars of introduction, where the listener with no habits needs eight. It is worth nothing against simply assuming. The listener who never chooses, and treats every tune as though it syncopates, is ahead of it at every length of introduction, by twenty-six points with none.

The gap has a direct cause in the next figure.

The choosing listener rarely becomes sure #

On the syncopated tune, the choosing listener’s belief in the syncopating style climbs from even to about two in three over the eight bars of the melody. In only nine passages in a hundred does it ever pass nine in ten. With eight bars of ground first it does better — the belief reaches 86 per cent by the end, and passes nine in ten in 37 passages in a hundred — but by then the ground has done the work and the barline is already held.

On the tune that marks the bar, the choosing listener never becomes sure of the marking style at all. The belief creeps up to 61 per cent and stops.

The reason is in what each kind of tune can say. A syncopated tune’s long notes vote against the bar, and under the marking style, which trusts the melody, those votes are improbable; under the syncopating style, which gives the melody a coin toss, they are unremarkable. So a syncopated tune testifies — slowly, bar by bar, against the marking style. A tune that marks the bar votes for the bar, and those votes are probable under the marking style and, at a coin toss, only somewhat less probable under the syncopating one. The syncopating style is never contradicted by a tune that agrees with the bar; it merely failed to predict the agreement. Distrusting the melody is a hypothesis a marking tune cannot refute.

What the one-sided evidence implies #

If the syncopating habit cannot be refuted by a marking tune, the question is what it costs on one. The answer is little. On a tune whose long notes all mark the bar, the listener with no habits and the choosing listener hear 98 bars in a hundred against the bar from the first bar; the listener who always distrusts the tune hears 93. Its cost there is five bars in a hundred, and it is paid only when there is no introduction: from two bars of ground on, the always-distrusting listener hears 99 and 100.

The ledger is lopsided. Arriving with the syncopating style’s habits gains 54 bars in a hundred on a syncopated tune over having no habits, and loses 5 on a marking tune. The marking style’s habits — which are also no habits at all — gain nothing on either.

Sweeping the choosing listener’s starting belief makes the same point continuously. The more it starts out believing that the tune syncopates, the more of a syncopated tune it hears against the bar — 51 per cent from a belief of one in ten, 85 from nine in ten, 93 when the belief is certain. On the marking tune the starting belief barely matters, 98 per cent until the belief is certain and 93 then. A listener who starts nine-in-ten convinced that tunes syncopate gives up nothing on a marking tune and gets 85 per cent of a syncopated one. There is no starting belief at which being more sceptical of the tune costs more on one kind of tune than it gains on the other.

Tunes that do both #

Real tunes are rarely all one or the other. Most melodies put some long notes on downbeats and some across them, and a style is a tendency rather than a rule. The last figure sweeps the mixture. The listener with no habits climbs steeply with the share of downbeat long notes, from 39 per cent to 98. The choosing listener climbs from 67 to 98. The listener who distrusts the melody is flat: 93 or 94 per cent at every mixture, since it never takes the melody’s long notes as evidence and reads the bar from the root alone.

So the choice is between a listener whose worst case is 93 per cent and whose best is 94, and listeners whose best cases are 98 and whose worst are 39 or 67. For a listener that has to be ready for any tune, the flat line wins on the worst case at every mixture, and across a repertoire of tunes that are each one kind or the other, the choosing listener beats it on average only if more than about five in six of them mark the bar. Syncopation is a number about the metre measured how much syncopation a rhythm carries against a known metre; this finds that, for a listener still finding the metre, how much syncopation to expect is best set high.

Where the one-sidedness comes from #

The asymmetry is not an accident of the two styles chosen; it was built into the texture three essays before any listener kept a record. A melody marks the barline by where it lingers found that a melody’s long notes are the one thing in this texture that can favour one side of the bar over the other, because the harmony and the metre score the barline and the half-bar alike. A syncopated melody needs its phrases to land found the mirror image: a melody whose long notes all fall on the half-bar makes a listener who adds up the evidence hear the bar begin in the wrong place three times in four. The melody, in other words, is the texture’s loudest witness in both directions, and the root is its only reliable one.

A listener who gives the loudest witness no vote until it has earned one therefore loses little when the loud witness is honest, because the honest witness agrees with the root and the record quickly restores it, and gains a great deal when it is not, because it never lets it steer. Two bars of drum hold the barline, a dozen bars of bass do not found the same shape in time rather than in trust: confidence banked before the tune arrives is what lets a listener ride out a tune that argues against it. Distrust of the melody is confidence of a different kind, banked in the witnesses rather than the reading, and it does the same job without needing an introduction to bank it in.

A barline read from the bass alone #

What the always-distrusting listener actually does is read the bar from one witness: the bass’s root. It gives the bass’s motion and the melody no vote to start with, learns their reliability from the record, and trusts the root as the style does. A walking bass marks the bar by the note it is on found that the root is the witness a walking bass reliably supplies; a listener who weights its witnesses needs eight bars of bass found it earning a second vote once the record was kept. So the habit is less a belief about styles than a belief about witnesses: that the root is the reliable one and the others are to be proved.

That makes the result a statement about which kind of error is cheap to make. Distrusting a witness that turns out reliable costs little, because the record restores its weight within a few bars and the root carries the reading meanwhile. Trusting a witness that turns out unreliable costs a great deal, because it votes against the bar from the first bar and the record has to be built against it. In a texture in which nothing says where the bar begins that asymmetry had nowhere to act, since no witness was reliable; here, with one reliable witness and two doubtful ones, it decides everything.

The model the numbers rest on #

Passages of one chord a bar, a walking bass on every crotchet with the chord’s root favoured at the barline, and a melody of crotchets on chord tones with a long note in half its bars — on the half-bar for a syncopated tune, on the downbeat for one that marks the bar, and a stated mixture between. Four hundred passages for each length of introduction, seeded as in the earlier essays. The listener’s two running scores for the barline and the half-bar, with no cost to switch between readings since that is the setting the earlier essays reported; its per-witness records with a starting credit worth four bars; and, for the choosing listener, a belief over two styles updated each bar by the likelihood of that bar’s votes under each, flattened by two per cent a bar towards even. A reading is scored against the bar as it is in the score.

What the model leaves out #

More than two styles. The choosing listener knows exactly two, and one of them is correct. A listener with a repertoire of styles, most of them wrong for the piece, would be slower still; the always-distrusting listener would be unaffected.

Other evidence of style. Real listeners recognise a style from its instruments, its tempo, its bass pattern and its harmony long before a tune has syncopated or not. None of that is here, and it is exactly the evidence that could make choosing fast.

The cost of distrust elsewhere. A listener who distrusts the melody reads the bar from the bass. In textures without a bass that marks the root — solo melody, a cappella singing — the habit leaves nothing to read, and the choosing listener, which would come to trust the melody, would do better. The arithmetic here has a walking bass in every passage.

Learning across pieces. Each passage starts from the same beliefs. A listener who carried its belief from one piece to the next, as a listener immersed in a style does, would be the choosing listener with a better start — and the fourth figure shows how much that start is worth.

Whose music #

The texture is the one this sequence has used throughout: a walking bass under one chord a bar and a melody above it, the texture of much jazz, popular song and dance music, with its tunes syncopated or not. The claim is about listeners reading the bar from that texture: that a listener who does not know whether the tune will syncopate is best served by assuming it will, because a tune that marks the bar cannot show the assumption wrong and costs it little, while a syncopated tune punishes the opposite assumption heavily. Whether listeners unfamiliar with a style do in fact read its barlines from the bass and hold its tunes at arm’s length is a question for listening experiments with ambiguous passages, and the prediction here is specific: newcomers to a syncopated style should find its barline faster than a model trusting the melody predicts, and slightly slower on a style that marks the bar.

Still open: when the root lies #

Every passage here has a bass that sounds the chord’s root at the barline more often than at the half-bar, so the witness the always-distrusting listener relies on is reliable. Some styles move the root off the barline — a bass that anticipates the next chord by a quaver, as much Latin American dance music does, or a pedal under changing harmony. There the habit of trusting the root and distrusting the rest is the wrong way round, and the choosing listener might recover the advantage the always-distrusting one has had here. The computation adds a third style, whose bass anticipates, and asks whether any single set of habits still serves all three, or whether style has to be recognised after all once the witnesses can each be the unreliable one.