The bass errs fast where the content errs slow

Asked for the rate, it answers a multiple turned a reading that had been used to find barlines into what it really was, a detector of harmonic rhythm, and asked it how often the chords change. It errs, and it errs one way. A chord change every four slots is also a change at every eighth slot, so a slower grid inherits a faster rate’s evidence while a faster grid cannot inherit a slower one’s; asked for two chords a bar, it names one a bar a third of the time and a faster rate almost never.

That essay ended by naming what the passages lacked. Every reading so far is computed from pitch classes in slots, with no register, so a passage has no bass — and a chord change in the bass is a different event from a change in an inner voice. It predicted that a bass axis would be asymmetric in the other direction, because a bass moves at a chord change and also moves within a chord.

The prediction holds, and more sharply than it was put.

A bass for a passage that had none #

The passages are built as before, by the construction three decisions that constrain each other first used for a joint reading of key, metre and chords: a random succession of diatonic triads at a stated rate, notes in the slots of eight bars of eight slots, more of them on strong slots than weak, every note a tone of its chord. Under them is now a bass with two behaviours and no others. On the slot a chord begins, the bass states that chord’s root. Inside a chord, on each beat, it moves to another of the chord’s tones with a stated probability — the arpeggiation that walking basses, Alberti figures and every oom-pah accompaniment are made of. At a probability of nought the bass moves only where the chords change; at one it moves on every beat.

The bass notes also join the pitch-class content, since they are notes of the passage. So the content reading sees everything it saw before plus the bass, and the bass reading sees only the bass: which slots it moves on, meaning which slots hold a bass note different from the one before.

The bass reading scores each candidate period by how completely the bass’s moves land on its grid — the share of the grid’s slots on which the bass moves, taken at the best offset — and standardises that against the same moves put at random slots, exactly as the change reading follows the chords, not the bar standardised the content reading against its own shuffles. The period with the largest standardised score is its answer.

The two readings on their own #

The content reading’s performance is the earlier essay’s. At two chords a bar it names the rate 63 per cent of the time and a slower rate the other 37; at one chord a bar, 67 and 33; at a chord every two bars, 98. It never names a faster rate by more than a stray passage.

The bass reading is its mirror. At two chords a bar it names the rate every time. At one chord a bar it names it 18 per cent of the time and a faster rate 80. At a chord every two bars it never names it, and names a faster rate in all sixty passages. It names a slower rate in one passage of the 180.

The asymmetry has a plain cause, the complement of the content reading’s. A bass that moves on beats inside a chord puts moves on a grid faster than the chord rate, so a faster grid collects the chord changes and the arpeggiation, while a slower grid collects only some of the changes. The content reading is misled by the fact that slower grids contain faster changes; the bass reading is misled by the fact that faster grids contain moves which are not changes. Each reading is fooled by an event the other reading can see. An arpeggiation does not change the pitch-class content of a chord, and a change of content at a multiple of the rate does not move the bass.

Where each reading’s evidence sits #

Averaging the standardised evidence over the passages shows the two biases directly, as two different shapes.

The content reading’s evidence is negative below the true rate and positive at and above it. At one chord a bar it runs −2.36, −0.75, 3.39 and 2.14 across periods of two, four, eight and sixteen slots; at a chord every two bars, −2.39, −1.98, −0.21 and 3.65. Its peak is at the true rate, with its neighbour above not far behind, which is where its slow errors come from.

The bass reading’s evidence is positive everywhere — a bass that moves this often lands on every grid more than chance would put it — and it peaks at four slots in both rows: 8.61 at one chord a bar, 7.69 at a chord every two bars. The true rate is its second choice in the first case and its last in the second. Four slots is the half-bar, and it is where the bass’s evidence gathers because at that period the grid catches every chord change and half the beats, which is the best ratio of moves caught to slots checked that an arpeggiating bass offers. The bass reading is not reading the rate at all once the bass moves inside the chord; it is reading the grid its figuration fills best.

How much figuration it takes #

The arpeggiation probability is the one number the bass reading depends on, so it is swept.

With a bass that never moves inside a chord, the bass reading is perfect at every rate: its moves are the chord changes. At two chords a bar it stays perfect however much the bass arpeggiates, because a chord of four slots contains only one other beat: the four-slot grid then holds a bass move on every one of its slots, and no faster grid can do better than that, since the two-slot grid holds one on only three quarters of its slots when the bass moves on half its beats. At slower rates it falls apart quickly. At one chord a bar, a bass moving on a quarter of its beats is named correctly 60 per cent of the time and a bass moving on half of them 18 per cent. At a chord every two bars, a quarter already takes it to 23 per cent, and half takes it to nothing.

The content reading, drawn dashed, does not notice. At a chord every two bars it names the rate between 92 and 100 per cent of the time at every setting, because an arpeggiation adds chord tones to a window that already holds them. So the two readings are not merely biased in opposite directions; they are good in opposite places. The content reading is best at slow harmonic rhythms and the bass reading at fast ones, and both properties come from the same arithmetic of which grids contain which events.

Six ways to combine them, and none that works everywhere #

Two readings with opposite biases invite a combination that cancels them. The obvious candidates were all tried on the same passages.

Adding the two standardised scores does well at fast rates, 98 per cent at two chords a bar, and badly at slow ones, 67 per cent at a chord every two bars — worse than the content reading alone, because the bass’s evidence is larger in standardised units and its wrong peak outvotes the content’s right one. Adding rankings removes the scale problem and replaces it with another: it is the best rule at one chord a bar, 85 per cent, and falls to 37 at a chord every two bars, where the bass ranks the true rate third.

The coincidence reading is the principled one. It counts a bass move at a grid slot only in proportion to how far the pitch-class content differs either side of that slot, on the reasoning that an arpeggiation moves the bass without changing the content and a chord change does both. It is the best compromise: 98, 82 and 77 per cent across the three rates. It is not the best rule at any rate, and it still misses a slow harmonic rhythm almost a quarter of the time — because at a short window the content either side of an arpeggiated beat differs too, by the accident of which chord tones the upper parts happened to sound in each half — the same small-window noise that made the chords a weak witness to the barline.

The sixth rule drops standardisation altogether, and it is worth having because it shows where the bass reading’s bias comes from. Unstandardised, the share of a grid’s slots on which the bass moves is highest at the true rate — every chord change lands on it — and equally high at every slower grid, since those contain chord changes too and nothing else. So the raw share, like the content reading, cannot tell the rate from its multiples; it is the standardisation against chance that breaks the tie, and it breaks it toward the half-bar, where a grid with many slots and a high share is many standard deviations above what random moves would give. Taking instead the fastest grid the bass fills completely names two chords a bar every time and a chord every two bars 88 per cent of the time — and one chord a bar only 52 per cent. It fails wherever a chord change does not move the bass, which happens whenever the arpeggiation happened to end on the next chord’s root: then the change is silent in the bass, the grid is not quite full, and the rule has nothing to say.

So the fast bias is not a property of basses. It is a property of asking a bass how improbable its moves are, and a bass that arpeggiates makes improbably regular moves at the beat.

So no rule here is the best at every rate, and the pattern of which is best — the bass at fast rates, the rankings in the middle, the content at slow ones — is the pattern of the biases themselves. Every combination trades one end of the range for the other.

Which computation produced the numbers #

Passages have eight bars of eight slots on a metrical grid of three binary levels. A slot holds a note with probability 0.875 if strong and 0.375 if weak; every note is a tone of the chord current at that slot, and the chords are drawn from the seven diatonic triads with no immediate repetition. Chord lengths are four, eight or sixteen slots. The bass states the root on each chord’s first slot and, on each other beat inside the chord, moves to a different tone of the chord with the stated probability.

The content reading is the earlier essay’s, run over the upper notes and the bass together, standardised against sixteen shuffles of the passage’s slots. The bass reading is standardised against thirty-two random placements of the bass’s own notes. The combination rules use those two readings unchanged; the coincidence reading multiplies each grid slot’s bass move by the content distance across it and is standardised against twenty-four joint permutations of the slots. Sixty passages at each setting, seeded, and the same passages for every rule at a rate.

Where the model stops #

A bass here moves only between chord tones, and only on beats. Real basses pass through non-chord tones, approach a new root by step, and syncopate. A passing tone moves the bass without being a chord tone at all, which would feed the bass reading’s fast bias further and would change the pitch-class content, which would feed the coincidence reading’s.

Every chord change is in root position. That is the strongest simplification here and it is doing a great deal of work. A change to a first inversion moves the bass to the third; a change over a held bass, such as a tonic six-four over the dominant, does not move the bass at all. The bass reading’s perfect score at two chords a bar depends on every change being stated by a root, and a convention in which mid-bar changes are more often inversions than barline changes would turn the bass from a rate detector into something else.

Duration and accent are not in it. A bass note is a slot, with no length and no weight, and the long note and the strong note found that which of those a segmentation counts changes the chords it names.

The bass is one part in a single register. An octave doubling, a pedal, or a second low part changes what “the bass moves” means.

What neither reading can tell a listener #

Whether a listener reads the rate from the bass at all. The readings are two ways of weighing evidence that is present in the notes. A listener has a trained ear, a style and a tempo, and how often the chord changes introduced harmonic rhythm as a property a listener hears rather than one that is inferred from counts.

What the figuration is for. An Alberti bass exists to keep a chord sounding on a keyboard that does not sustain, and which notes are the chord was the first essay to find that deciding which notes belong to a chord is a decision about metre before it is a decision about harmony. The bass reading being misled by figuration is the same finding reached through the bass.

Whose basses #

Harmonic rhythm that is fast, with the bass moving at every change, is the texture of chorales and hymn settings, where the bass is a line of roots and inversions and rarely arpeggiates. Harmonic rhythm that is slow, with the bass arpeggiating or repeating inside each chord, is the texture of Classical accompaniment and much popular song. The arithmetic says those are exactly the two textures in which each reading does well and the other does badly, which makes the textures, and not the readings, the thing a listener’s inference would need to know first. Nothing here has measured which reading listeners use in which style.

Still open: the barline, which the bass may mark by what it does not do #

This essay asked the bass for the rate. The earlier essays asked the chords for the barline and found they mark it only by changing there, which fails as soon as there are two chords a bar. The bass has a second kind of evidence the rate question could not use. In much tonal writing a change on the barline is voiced in root position and a change in the middle of the bar more often over an inversion or a held bass — the tonic six-four, the passing six-four, a first inversion on a weak beat — so the bass moves more at barlines than between them even when the chords change at both.

That is a convention and it has a strength, which is a single number: the probability that a mid-bar change is voiced over the bass already sounding. The computation that follows is the one the chords mark the barline by changing there ran, with the bass’s moves in place of the content’s changes and that probability swept. It would say how strong a voicing convention has to be before the bass finds the barline at two chords a bar, which the content reading finds at chance — and so whether the barline a listener hears in a fast harmonic rhythm is being carried by the bass.