A texture in which nothing says where the bar begins

Nine essays of this account have asked where a bar begins, and the last of them arrived at a pair of readings that between them cover every bass a listener is likely to be given. A walking bass marks the bar by the note it is on showed that a reading of where the bass moves and a reading of where it is sounding the chord’s root are exact complements — one is certain wherever the other is worthless — and that a rule asking whichever of the two is the more decided picks correctly without being told which texture it has. At one chord a bar the barline comes out of eight candidates in ninety-four passages in a hundred or better, at every busyness of bass from a held minim to a line that moves on every crotchet.

That is one harmonic rhythm. How often the chord changes put the range across styles at a factor of thirty, and the rate immediately faster than the one all of this was measured at is the rate these essays have been failing on since the change reading follows the chords, not the bar first drew it.

Why everything fails at once, and it is not four failures #

The flat lines in that figure are a single fact seen four times, and it is worth setting out as the symmetry it is rather than as a list of disappointments.

A bar of eight quavers at two chords a bar has a chord change on its first slot and another on its fifth. The candidate that puts the barline at the fifth slot is therefore not a mistake a reading has to work to avoid; it is a reading of a passage that is, in every respect any of these instruments can measure, identical to the true one.

• The metre scores a candidate by how much of the passage’s onset weight falls on its strong slots, and a bar and its own half put the same slots under the same weights. This is the symmetry the right period at the wrong phase is about, and it is exact: the metre is not weak here, it is silent.
• The chord changes are the reading the chords mark the barline by changing there built, and it scores a candidate by how much the harmony differs across its barlines against across its mid-bars. At two chords a bar the harmony differs across both, equally, by construction.
• The bass’s moves are the hold reading’s. The bass states a root at both changes, so it moves at both.
• The bass’s root is the walking-bass reading’s. The bass is on a root at both changes, and walks away from both.

So four instruments, four independent principles, one answer: fifty per cent, which is the metre’s two surviving candidates split by a coin. The sum of the two bass readings is fifty per cent as well, because nothing plus nothing is nothing.

The one thing worth taking from the hero beyond the flatness is the shape of the curves at their right-hand ends, which wander by a few points in both directions. Those points are the tie-breaking of a few passages in two hundred, and their size — two or three points — is the scatter these passages give at that sample. Getting them to sit there took a repair, which is the subject of a later section.

Where the hole is, exactly #

The failure is not general, and drawing it against the neighbouring rates says so.

At one chord a bar the still bass gives the move reading 100 per cent and the moving bass gives the root reading 100. At a chord every two bars the numbers are 100 and 93, slightly softer because a chord held across a barline means half the barlines carry no change at all — the same halving the change reading follows the chords, not the bar measured from the other side. At two chords a bar every cell is between 49 and 55.

So the hole is a property of the harmonic rhythm and not of the bass. It appears at exactly the rate at which the chord change stops being a barline event and becomes a beat event, and it takes every reading built on the chord change down with it — including the two that were built on the bass precisely because the bass was supposed to be independent of the harmony. It is not: the bass in this model does one thing, which is state roots, and it does it wherever the roots are.

That is the sharpest version of a complaint that has been implicit in these essays from the chords never move the barline onward. Every witness here is downstream of the chord sequence. A listener has cues that are not — the melody’s own shape, its phrase ends, the dynamics, the drummer — and none of those is in this model.

The convention, and the reason it is the only remedy #

One thing in the model is not downstream of the chord sequence, and it is the convention a bass that holds through a change marks the barline was built to price: in tonal writing a change on the barline is voiced over its root and a change inside the bar is often voiced over the bass already sounding, as an inversion. That is a fact about voicing, and voicing is a decision the chord sequence does not make.

The two passages are the same chords — A minor, F, D minor, C, G, C, E minor, C — the same melody and the same walking bass, and they differ in two notes out of sixteen. At the half-bar of the first bar the chord turns to F major and the first passage’s bass states F; the second holds the C it had reached a crotchet earlier, because C is the fifth of F major and a held C under it is a second inversion. At the half-bar of the fourth bar the chord turns to C and the first states C; the second holds its G, which is C major’s fifth. The other two mid-bar changes cannot be held, because the note the bass is on does not belong to the chord arriving, and that is the one-in-three ceiling showing up in a passage of four bars as two.

Two held notes take the root reading from 0.00 at every candidate to 0.50 at the written barline and −0.50 half a bar later, and that is the whole of the difference between a passage that can be read and a passage that cannot.

Two details of the pair are worth more than the headline. The first is that the root reading, even repaired, still ties four candidates at 0.50 — the barline and the quaver after it, and the two at the far end of the bar — so on its own it takes a quarter credit and needs the metre to finish, exactly as it does at the slower rate. The second is that the move reading in the unheld passage does not score zero; it scores −0.33, which is a vote for the half-bar. A reading with no information is not a reading that returns nothing. It returns noise, and in any single passage the noise has a sign.

What the convention supplies that nothing else does is an asymmetry that is not a consequence of where the chords are. The chords are symmetric about the half-bar; the voicing is not, and that is a choice a composer makes bar by bar.

What the busy bass costs, and what it does not #

Whether the two devices a bass has — hold through a mid-bar change, walk inside the chord — get in each other’s way is the question the walking-bass reading ended on, and it has two possible answers with different mechanisms.

The first mechanism is not the one. A change can be held when the note the bass is already sounding belongs to the arriving chord, and the essay that priced the convention derived from the diatonic triads alone that this is true of exactly one change in three. A busy bass might have been expected to lower that — it is on a third or a fifth as often as on a root, and a third has different survival counts from a root. Measured, it does not: the realised share runs 34, 31, 34, 34 and 34 per cent across the whole range of busyness. The ceiling is a fact about which triads follow which, and where the bass has wandered inside the bar does not touch it.

The second mechanism is. At the convention’s ceiling the root reading finds the barline in 98 per cent of eight-bar passages over a bass that holds its note, 96 at a quarter busy, 96 at a half, 92 at three quarters and 90 at a bass that moves on every crotchet. The move reading falls further and faster: 96, 94, 90, 88, 82. At half the convention’s strength both fall by about the same nine points.

The reason is arithmetic rather than musical. The held change is a difference between two shares, and a walking bass adds motion — and root-sounding — to slots on both sides of the comparison. Every extra note the bass plays inside the chord raises the numerator and the denominator of both candidates alike, so the signal stays where it was and the noise around it grows. A busy bass does not hold fewer changes; it makes each held change a smaller share of what there is to count.

Two further things fall out of that figure and both are new.

The first is that at two chords a bar the root reading beats the move reading at every strength and every busyness, and the gap widens with the busyness: two points over a still bass, eight over a walking one. The hold convention was priced with the move reading throughout, because it was the only reading there was. On its own subject — the barline carried by a voicing convention — the root reading is the better of the two, and by more the more like real music the bass is.

The second is the direction the whole result runs. The convention is strongest exactly where the bass is least interesting, and a continuo line that moves on every crotchet under two chords a bar is a common enough object. Ninety per cent of eight-bar passages is not a failure; it is the best any of this achieves in the hole, and it needs a voicing convention followed on every occasion it can be.

Swept, the two curves are the same shape displaced. At one held change in eight the still bass is at 84 per cent and the walking one at 72; at one in five, 90 and 78; at the ceiling, 98 and 90. A walking bass needs about half as much convention again to reach the same reading, which is a usable statement: in a style whose bass moves constantly, a composer who wants the barline audible from the harmony has to voice more of the mid-bar changes over the bass than a composer whose bass holds.

Five points that should not have been there #

The hero’s curves wander by two or three points around the coin toss, and in the first version of this computation they wandered by five, always upward, always in the same direction, and reliably across seeds. Five points is small. It is also, in a figure whose entire claim is there is nothing here, exactly the size of thing that has to be accounted for.

It was not the harmony and it was not the bass. It was the counting.

A passage’s bass always sounds a note on its first slot, and a passage’s first slot is always the written barline, because that is how a passage is built. The move reading marked a slot as a move when the bass note there differed from the one before it — and for the first note there is no note before it, so the reading counted it as a move every time. That put a free move on the true barline of every passage and a free penalty on the candidate half a bar later, which over eight bars is a quarter of a move a bar. Against the margins the hold convention was measured with, which run from 0.04 to 0.28 moves a bar, a quarter of a move is not a rounding error; at the weaker strengths it is larger than the effect.

Treating that slot as carrying no information rather than as carrying a move removed most of it and introduced a smaller version of the same thing, because a reading that then averaged one candidate’s barlines over seven bars and its mid-bars over eight was comparing two means with different numbers of terms. At a move rate of 0.83 the values a mean of seven can take and the values a mean of eight can take do not interleave evenly, and the discreteness alone was worth three points. Requiring each bar to contribute both of its slots or neither leaves the residue at the scatter of the sample.

What the repair did to the published numbers is small and it is worth stating exactly, because the reading it corrects is the one the previous two essays rest on. At the convention’s ceiling that essay’s headline moves from 96 per cent to 96, and its margin from 0.28 moves a bar to 0.28. At the weaker strengths the bass alone loses between one and five points — 95 per cent becomes 91 at the ceiling, 73 becomes 69 at one change in six — and the proportionality constant between the margin and the held share falls from 0.83 to 0.81. That essay now carries the corrected numbers. Its argument is unchanged, which is the usual outcome of finding a bias of this kind in a result with a large effect; the reason to chase it was that this essay’s result has no effect at all, and a bias that does not matter at 0.28 matters entirely at 0.00.

One property was lost. A candidate half a bar later used to score exactly the negative of the candidate at the barline, which the figures checked on every passage. Under the repair the two candidates drop different bars from their averages, so the mirror holds to about a fiftieth of a move rather than exactly. That is a real cost of the fix and the figures now check the weaker statement.

Which computation produced the numbers #

The passages, the metre reading, the chord-change reading and the bass are the earlier essays’ own. Eight bars of eight quavers, a melody whose onsets favour the strong slots and whose notes are tones of the current chord, diatonic triads of C major never repeated consecutively, and a bass that states each new root and may move to another tone of the chord at any crotchet.

The convention is a probability applied at mid-bar changes only, and only when the sounding note belongs to the arriving chord; the figures are drawn against the realised share rather than the probability. A reading’s credit for a passage is one over the number of candidates it ties for best if the written barline is among them, and zero otherwise; every rate is over 200 to 300 passages from fixed seeds. The move and root readings both average a candidate’s barlines against its mid-bars over the bars where both slots carry information.

Where the model stops #

The chords are drawn at random and so cannot cadence. A real passage at two chords a bar reaches a cadence every few bars, and a cadence is a barline event of a kind nothing here has — the cadence as evidence priced one on the other side of the same question. The hole is therefore at its widest here.

The melody is symmetric about the half-bar too. Its onsets are drawn against metrical weight, which is the very symmetry that defeats the metre. A real melody has a contour, and a contour has its own period.

Only one voicing convention is modelled. A composer at two chords a bar has others — a suspension across the barline, a doubled root on the downbeat, a change of texture — and each would be an asymmetry of the same kind.

Nothing is heard imperfectly. Every note arrives, and the bass in particular is not masked by the texture above it.

What the numbers cannot say about a listener #

Whether the hole is audible. A model at chance is a claim that a particular set of cues carries no information, not a claim that listeners cannot find the barline. They plainly can, in fast harmonic rhythms, in every style that has one; the finding is about which cues that cannot be.

How often mid-bar changes are voiced over the bass in the repertoires that live at this rate. One in five is enough for a walking bass and one in three is the ceiling, and where a style sits between them is a count over scores that is not to hand.

Whether a listener weighs the two bass readings or runs one. The rule that asks the surer witness chooses between them from the passage; here there is nothing to choose, so the question does not arise in this texture and is not settled by anything above.

Still open: the leap, which the model says nothing about because it has no register #

The account of its own limits in a bass that holds through a change named a third thing a bass does and could not price it: 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. Nothing in any of these readings can see it, because a bass here is a list of pitch classes and the size of an interval needs octaves. Giving the line a register is a small addition — a compass, and a rule for which octave each pitch class goes in — and the reading is the same computation with semitones in place of counts. What makes it worth doing is that the rule for choosing the octave is itself a convention, of the same kind as the hold and equally invisible to a pitch-class model, so the answer may turn out to be a statement about register rather than about motion at all.