The leap at the barline is a fact about register

A bass that holds through a change marks the barline closed with a list of the things its model could not hold, and the second item on that list has been waiting three essays:

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. Neither is visible to a reading of moves.

It is not visible to a reading of roots either. Every bass in these passages from the bass errs fast where the content errs slow onward has been a list of pitch classes in slots, and a pitch class has no size. C to G is a fifth up or a fourth down and the model cannot tell, because the model has no octaves — which is the same absence asked for the rate, it answers a multiple named when it said the passages here have no bass because they have no octaves, one level further down.

Giving the line a register is two decisions. The first is a compass, and E1 to E3 is the working range of the instruments that play these lines. The second is the one that turns out to matter: given a pitch class and a compass, which octave.

The realisation that has no leaps in it #

The default answer, and the one nothing in the model argues against, is least motion: place each pitch class at the instance nearest the note before it, ties taken downward. It is the assumption every figure about connecting one chord to the next is built on, it is what a singer does, and it is the only rule available to a model that has not been told anything about the instrument.

Under it the third device is worth nothing at all. The reading runs 100, 100, 96, 91, 83, 69, 52, 32 and 20 per cent across the busyness sweep, which is within a few points of the move reading’s 100, 100, 99, 99, 93, 89, 70, 54 and 20 at every setting. It is not adding a device; it is measuring the same one in different units.

And the margin it is measuring with goes the wrong way.

At a walking bass, a line realised at least motion moves further at the half-bar than at the barline, by nearly a semitone on average. The reason is the thing the rule is named for. A bass inside a chord moves between the chord’s own tones, which in a triad are a third and a fourth apart — three, four or five semitones. A bass at a chord change moves to the new root, and the roots of two successive diatonic triads are a second, a third or a fourth apart in pitch-class terms; placed at the nearest octave, that is one to five semitones and usually fewer than the arpeggiation’s. A rule that always takes the nearest octave cannot produce a leap, so a reading that looks for one finds the arpeggio instead.

This is the whole of why that limit was worth attacking rather than recording. Stated as the bass leaps at the barline, it sounds like a claim about harmony — root motion is wide, chord-tone motion is narrow — and as a claim about harmony it is simply false in this repertoire. Root motion is not wide. What is wide is the interval a player actually plays, and that is a different quantity.

The convention a pitch-class model cannot state #

The rule a bass player follows is not least motion. A bass part has a bottom, the roots live near it, and a line that has climbed through a chord’s upper tones drops back down to state the next root rather than taking whichever octave is nearest.

So: a note that states a new chord’s root goes to the lowest instance of that pitch class in the compass, and every other note goes to the nearest. That is a single rule and it is a convention of exactly the kind a texture in which nothing says where the bar begins found the voicing hold to be — a fact about performance that the chord sequence does not determine and a pitch-class model cannot express.

The pair is the argument in one picture. Sixteen notes — A, E, C, A, G, B, G, D, E, B, E, G, C, G, C, E — read left to right in both staves, in the same order, at the same instants. The upper realisation starts at A2 and climbs to E3 by the last bar, because every nearest-octave choice is a small one and small choices in one direction accumulate. The lower realisation returns to E1, G1, E1 and C2 at the four barlines and climbs away from each of them, which is what a bass line looks like.

And the same sixteen notes give the same margin with the opposite sign. Minus 1.67 semitones against plus 1.67: the least-motion realisation prefers the half-bar by exactly as much as the anchored one prefers the barline.

That coincidence of magnitude is a property of this passage rather than a general law, and it is worth saying which part is general. What is general is the sign: the anchored realisation is above zero at every setting of every sweep here and the least-motion realisation crosses it. What is particular is that in this passage the two happen to be the same size.

Why a bass does this, when a singer does not #

The anchored rule is stated above as a fact about performance, and it is worth a paragraph on why it is a fact about this performance and not about melodic lines in general, because the two rules are the two halves of a genuine division of labour.

A melodic line is heard as a line, and a melody is a walk, not a set measured what that costs: above about eight notes a second the ear stops being able to hold a large interval and a small one in the same stream, so a tune that leaps is two tunes. Least motion is not an arbitrary default for a melody; it is the condition on being a melody at all.

A bass line is not principally heard as a line. It is heard as the bottom of the harmony — as the thing that decides whether a chord is in root position, which is the quantity a bass line is not a list of roots found a key-finder’s bass cue depends on entirely. A bass note an octave lower is the same harmonic statement and a better one, because the chord above it is then spaced the way a chord is a register found the critical band requires. So a bass has a reason to take the low octave that a melody does not, and it pays a price in melodic coherence that a melody could not pay.

That is why the two realisations drawn above are not a modelling choice between equals. Least motion is right for the parts most of these essays are about, and wrong for the one part whose job is to be low — and the reason it went unnoticed for so long is that a model made of pitch classes cannot express the difference between them.

How much compass the convention needs #

The anchored rule is only a mark if there is somewhere distinctive to be pinned to, which is a question about how much room the line has.

At eleven semitones — a compass narrower than an octave, so that every diatonic root has exactly one place to be — the reading still works, at 83 per cent, because an anchored root and a nearest-octave arpeggiation are still different choices for the notes that are not roots. At fourteen semitones it is at 94 and its margin has risen from 1.50 semitones to 2.34. At seventeen it is at 96 with a margin of 2.95, and from there to a compass of three octaves nothing moves: 96, 96, 96, 96, with the margin fixed at 3.07.

The plateau is the interesting half. Beyond about an octave and a half the extra range is never used, because the arpeggiation places each note at the nearest octave and the nearest octave to a note inside the compass is inside the compass. The convention needs an octave and a half and is indifferent to anything wider, which happens to be about the range in which real bass parts spend most of their time, and which is a coincidence this account has no way to explain.

The same hole, from a third direction #

The obvious question is whether a device this different fills the gap the last essay found.

It does not. At two chords a bar the anchored reading sits at 54 per cent beside the other two at 50 and 53, which is the metre’s coin toss with the usual scatter on it. The reason is the reason for everything in that texture: both candidates carry a chord change, so both carry a stated root, so both carry an anchored root and a drop to the bottom of the compass. The bass does the same thing at each, and a reading of how far it did it cannot separate two identical events any better than a reading of whether.

So the three devices are three answers to one question and they fail together on the one rate where the question is ill-posed. Which is also, read the other way, a reason to be more confident in the last essay’s finding: three independent measurements, the third of them on a quantity the first two could not represent, all at chance in the same place.

What the figure does not settle about the rate #

One caution about the two-chords-a-bar column, because it is the third time this account has drawn a flat line there and a third flat line is a good moment to ask whether the flatness is being manufactured.

It is not, and the check is available in the same figure. If the passages at two chords a bar were simply harder than the others — noisier, shorter of evidence, more often tied — then every reading would be depressed there and depressed by different amounts, which is what a general loss of information looks like. What the figure shows instead is three readings at 54, 50 and 53 per cent, which is not depression but exactly the metre’s own two-way tie, split. A reading that had a little information left would sit at 60 or 70. A reading that had none sits at 50, and three of them sitting there together is the signature of a symmetry rather than of a difficulty.

The neighbouring column is the other half of the check. At a chord every two bars the same three readings give 85, 94 and 75 per cent — spread out, in an order, and each one degraded in a way its own mechanism explains. That is what these passages look like when they are merely hard.

Three devices, one fact #

Having three readings invites adding them, and it is worth doing once to see what it buys.

The sum runs 100, 100, 100, 100, 98, 99, 96, 94 and 91 per cent across the sweep, and at every point it is within six points of whichever device happens to be best there. It never beats the best; it only avoids being the worst.

That is the honest summary of the three together, and it is a modest one. They are not three independent witnesses whose errors cancel. They are three views of a single fact — where in the bar the bass restates its harmony — expressed as an event count, as a note identity and as a distance, and a listener who has all three has very little more than a listener who has the right one.

What the sum does buy is the thing the choosing rule bought, without the choosing: a reading that does not need to know which texture it is in. The best single device changes identity three times across the sweep — the moves at the still end, the root in the middle, the anchored distance at the walking end — and a listener has no way to know which. Adding all three costs at most six points and removes the question.

Which computation produced the numbers #

The passages, the metre, the two earlier bass readings and the bass itself are the earlier essays’ own and unchanged. The realisation takes the bass’s pitch classes in order and assigns each a MIDI note in the stated compass: the nearest instance to the note before it, ties downward, except that under the anchored policy a note stating a new chord’s root takes the lowest instance in the compass. The first note starts from the middle of the compass.

The leap reading scores a candidate barline by the mean size in semitones of the bass’s motion onto its barlines, less the mean onto its mid-bars, with a slot on which the bass does not move counting as a motion of nought — so that a reading of distance contains a reading of whether it moved at all. A bar contributes only when both of its slots carry information, which excludes the bar holding the passage’s first note. Rates are over 200 to 240 passages from fixed seeds, and each reading is used to break the metre’s own tie rather than to choose among all eight candidates alone.

Where the model stops #

The anchored rule is a caricature of a real bass part. A player drops toward the bottom of the compass at a structural arrival and not at every chord change, and drops further at a cadence than at a passing chord. What is modelled is the strongest possible version of the convention, so 90 per cent is a ceiling on it.

A passing note outside the chord is still not modelled. A walking bass reaches its next root by step through notes belonging to no chord, and stepwise approach is the commonest way a real bass marks an arrival. That would strengthen the anchored reading further and would also make the least-motion realisation less absurd.

Nothing in the compass is about the instrument. A double bass, a bass guitar and a left hand have different bottoms and different registral habits, and a compass here is two numbers with no physics behind them — unlike the same note is a different width, where the bottom of an instrument’s range is a consequence of its wire.

The chords are drawn at random, so there is no cadence, and a cadence is where the convention would be strongest.

What the numbers cannot say about a listener #

Whether a listener uses interval size at all. Everything above shows that the information is present in the signal under one convention and absent under another. Nothing shows that a listener reads it, and a reader should be aware that the same evidence supports a much duller account in which the anchored root is heard simply as low and the reading is really about register rather than about motion.

How often bass players anchor. That is a count over recordings or over figured parts with their realisations, and it is not to hand here.

Whether the octave choice is audible as such. A bass note an octave lower is not only further from the note before it; it is louder in the room, it masks more of the texture above it, and it sits where a chord is a register found the critical band doing something drastic. Any of those could be carrying the mark that this essay attributes to distance.

Still open: the cue that is not the harmony at all #

Three devices, three essays, and the same rate defeats all three, because every one of them is downstream of the chord sequence. At two chords a bar the chords are symmetric about the half-bar and so is everything computed from them, and the only asymmetries found anywhere in this account are conventions of voicing — which mid-bar change is played over a held bass, which octave a stated root goes in. Both are decisions a performer makes and neither is in a chord symbol.

What has never been asked of these passages is the melody. Every passage since the chords never move the barline has drawn its melody from metrical weight and the current chord, which makes it symmetric about the half-bar by construction and guarantees it says nothing. A real melody has a contour with a period, phrase ends that fall at barlines far more often than at half-bars, and long notes that arrive on downbeats — and the long note and the strong note already prices the last of those. Giving these passages a melody with a shape, rather than a melody with a distribution, is the one change that could put evidence into the texture where none of the harmony can, and it would say whether the hole is a fact about fast harmonic rhythm or only a fact about listening to the chords.