Given the bar in octaves, the degree comes back

The essay that gave the reading a realistic bass line gave the key-finder a bass cue and then gave it a realistic bass line to use it on. The cue rewards the triad rooted on the bass note, which is the published form of it and which assumes every chord is in root position; a real bass line is not, and the degree share fell from 89 per cent to 68.

Its closing paragraph named what would be needed to recover it. The rule needs to know which chord tone the bass is, and that is a fact about register: a bar voiced in real octaves says whether the lowest note is the third of the chord above it or the root of a different one, by what lies between the bass and the upper voices.

That is a bigger change than it sounds, and the size of it is the essay.

What a voicing gives that a pitch class cannot #

Every bar in every figure on the account here has been a set of pitch classes and a bass pitch class. A voicing is the same bar with octaves in it: the bass in one register, the other two triad tones above it in close position, and each of them a note rather than a class.

Given that, a listener knows the chord. Not which degree of which key it is — that is still the question — but which three pitch classes are sounding, which is exactly what a bass note on its own does not say. A bass on C is consistent with C major, A minor, F major and several others; a bar sounding C, E and G over a C is one chord.

So the three rules are not three refinements of one idea:

• the root rule rewards the triad rooted on the bass: one chord a bar, right when the chord is in root position and wrong otherwise;
• the member rule rewards any triad containing the bass: three chords a bar, never wrong and never sharp;
• the register rule rewards the triad whose pitch classes are the ones actually sounding: one chord a bar, and the right one.

And what it buys #

On a real bass line the published cue reaches 68 per cent and the register rule reaches 92. The dashed line is the 89 per cent the published cue manages on a line made entirely of roots, which is the figure the seventeenth essay set out to recover.

It is recovered and passed. That is worth being careful about, because the obvious reading — that a register-aware bass cue is a better bass cue — is not what happened. The register rule beats the root rule on a root line, where the root rule has nothing to be wrong about. A cue that knows the chord is not a sharper version of a cue that knows the bass; it is a different and larger piece of evidence.

The member rule is the control that makes that plain. It is never wrong about the bass, it costs nothing in assumptions, and it reaches 61 per cent — worse than the root rule it was supposed to repair. Rewarding three chords a bar is rewarding no chord in particular.

Where the extra three points come from #

Beating the root-line figure by three points is small and it has a reason, which is worth having because it says what the ceiling is made of.

A line of roots and a root rule agree on every bar by construction, so the cue is correct everywhere and the remaining eleven per cent of errors are the key model’s — bars where the chord is genuinely ambiguous between two keys and the profile picks the wrong one. The register rule on a real line is also correct on every bar, and it is correct about more: it names all three pitch classes, so a bar whose chord is shared between two keys is at least reported as that chord rather than as a different one — the ambiguity a count and a correlation runs into from the profile’s side.

So the three points are bars where the root rule was right about the bass and wrong about the chord. Those are the bars where a real line puts a chord tone other than the root at the bottom and the triad rooted on that tone is also in the key — a first-inversion tonic under which the root rule reads a mediant, for instance.

Why the member rule loses #

The member rule is the one that ought to work and does not, and the reason is worth more than a line because it is a general point about cues.

It is the rule with no assumptions in it. The root rule assumes the chord is in root position and is wrong whenever it is not; the member rule assumes only that the bass is some tone of the chord, which is true by construction of every line here. It cannot be wrong, and it reads 61 per cent against the root rule’s 68.

A cue that is never wrong is a cue that never rules anything out. Rewarding three triads a bar and penalising the other nine sounds like evidence and is much less than it seems: at any moment three of the seven diatonic triads of a key contain a given pitch class, and so do three of the seven in several neighbouring keys. The rule moves the reading toward the keys that contain the bass note at all, which is nearly every key a fifth away in either direction.

So the member rule’s caution costs it. The root rule is wrong on a third of the bars of a real line and right about something specific on the other two thirds, and specificity turns out to be worth more than being safe — which is not obvious in advance and is the reason this essay needed the third rule to settle anything.

What the register rule has is both. It is never wrong, like the member rule, and it names one chord, like the root rule. That combination is available only because the voicing carries information neither of the other two rules is looking at, and it is why the gain is twenty-four points rather than a few.

The two cues stay separate #

One thing this does not touch, and the separation is worth recording because the account has been circling it since its fourteenth essay.

The bass cue and the profile weight are independent. Changing the bass rule moves the degree share by twenty-four points and moves the minor-tonic test by nothing at all: the minor passage is read right above a profile weight of 0.6 and wrong below it, under every one of the three rules and at every bass weight drawn.

So the register is evidence about the chord and the profile is evidence about the mode, and no amount of the first supplies the second. The key-finder with no tonic is the essay that established the profile’s own job, and this essay leaves it exactly where it was.

That is a tidier result than the account has usually managed, and it is the reason the plane the fourteenth essay introduced has two axes rather than one. They are not two settings of one dial; they are two kinds of evidence about two different questions, and the figures that sweep both at once are sweeping a genuine plane.

Which schemes the gain comes from #

Twenty-four points averaged over six schemes is not twenty-four points on each of them, and the distribution says something the average hides.

Three of the six schemes — the blues, the verse-and-chorus and the ostinato — are read correctly by every rule. They are short, they do not modulate, and their chords are unambiguous, so no bass cue of any kind has anything to contribute.

The whole of the difference is in the other three, and it does not go one way. On the rondo the register rule reads 80 per cent of bars right against the root rule’s 68 and the member rule’s 70 — those are the inverted dominants the seventeenth essay named, where a bass on the leading note makes the root rule read a diminished triad. On the thirty-two-bar song the member rule reads 100 per cent and the register rule 78.

So the rules are not ordered. That is a surprise and it does not undo the finding — the register rule wins on the degree share, which is the measure the seventeenth essay set — but it means the right rule is a question about a repertoire rather than about cues.

Why the song should prefer the vaguest rule is not obvious from the mechanism and it is the kind of result that usually means a parameter is doing something unintended. Chasing it is the next essay’s.

Which computation produced the numbers #

The passages are the account’s own six schemes, each a list of bars with a Roman numeral and a key. A bass line is realised in one of three styles — the root of every chord, the chord tone nearest the previous bass note, or the fifth of every chord — and the smooth style is the realistic one and is what every figure here uses.

The voicing is built from the realised bass: the bass in the octave below the bass clef, and the triad’s other two tones stacked above it in close position, each raised by whole octaves until it clears the note below. That gives each bar a set of sounding pitch classes and a bass, and the register rule tests a candidate triad against the set.

The reading itself is unchanged from the essay that built the reading with no tonic in it: a dynamic program over key candidates, with an emission term from the profile weighted by one parameter and a term from the bass weighted by another, and a fixed cost for changing key. The key cost is 2.2 throughout, which is the plane’s own value.

The degree share is the share of bars on which the reading names both the right key and the right scale degree, over all six schemes, which is the essay that gave the reading a realistic bass line and is why its 89 and this essay’s 92 can be set side by side.

What the account has been measuring all along #

Stepping back over five essays, the bass axis has changed its meaning twice without changing its name, and it is worth saying what it now is.

the fourteenth essay introduced it as a second source of evidence beside the profile: a bass note is information about the key. The seventeenth found that the information depends on a realisation nobody had specified, and that the published form of the cue assumes a bass line made of roots. This one finds that what the cue is really reaching for is the chord — and that given the chord, the bass note itself contributes nothing beyond it.

So the axis is not a bass axis. It is a chord-identification axis with a bass-shaped proxy on it, and the proxy was costing twenty-four points of degree accuracy. That reframing is what makes the earlier essay’ numbers readable: the 89 per cent on a line of roots was never a statement about basses, it was a statement about a line in which the bass happens to name the chord.

It also says where the account’s remaining uncertainty is. If the axis is about chords, then everything on it is downstream of a segmentation the arithmetic here gives on a different account and does not feed in here — so the key-finder’s performance is conditional on a problem being solved that nothing here solves.

Where the model stops #

The register rule is given the chord, and that is a great deal. A listener does not receive a labelled triad; they receive notes, and identifying which three pitch classes constitute the chord is the segmentation problem which notes are the chord is about. What this essay shows is what a key-finder could do if that problem were solved, which is an upper bound rather than a model of listening.

The voicing is close position with no doublings. A real keyboard realisation doubles the root, spreads the upper voices, and moves them for reasons of voice leading. None of that changes which pitch classes sound, which is all the register rule uses — so the rule is insensitive to the voicing in exactly the way that makes “register” a slightly grand name for it.

And the bass line is one of three fixed styles. The smooth style is a stand-in for what a keyboard reduction tends to do, not a transcription of anything. It chooses the chord tone nearest the previous bass note, which produces a line that moves by step or third and puts a non-root in the bass on about a third of bars — plausible as a keyboard realisation and quite unlike a walking bass, a figured bass or an orchestral one, each of which would give a different census and therefore a different cost to the published cue.

What the picture cannot show #

It cannot show a seventh chord. The triads here are the first three tones of each bar’s chord, so a blues bar’s seventh is an upper voice and not part of the set the rule tests. A rule that tested four-note sets would be sharper still and would have more chances to fail on a passage that voices the seventh out.

The sounding set is the triad and not the bar. Where a scheme’s chord is a seventh, only its first three tones are voiced, so the register rule is tested against a triad in a bar that a player would sound as four notes. That makes the rule slightly vaguer than a real voicing would allow and is conservative in the direction of the finding.

Nor a passing note. Every note in every bar belongs to the chord, so the sounding set is never contaminated. A real bar has notes that are not chord tones, and telling a long note from a strong one is a piece of the same problem.

And it cannot show what a listener weights. The two axes are two numbers set by hand and swept. Nothing here says a listener uses either at any particular strength, and where the two cues meet is the essay that found the corner of the plane where both matter and neither dominates. What this essay adds to that corner is that its bass axis is now a chord axis, and the corner is in a different place because of it.

Still open: whether the sharper cue is the better cue everywhere #

The register rule wins by twenty-four points at the key cost used here, and the key cost is a parameter rather than a fact. It is how reluctant the reading is to change key, and it has been fixed at one value for four essays.

There is a reason to think the rule’s advantage depends on it, and the reason is legible before the sweep is run. A cue that identifies the chord outright tells the reading exactly which keys are consistent with this bar — and if changing key is cheap, the reading can follow that evidence anywhere, hopping to whichever key makes the named chord a tonic. A vaguer cue leaves more of the decision to the key model’s own preference for staying put.

So the prediction is that the register rule’s margin shrinks as the key cost falls, and may vanish: at a low enough cost a sharp cue about the wrong question could be worse than a vague one. Running the same four series across the key cost would say where the crossing is, and whether the account’s chosen 2.2 is on the side of it where the answer is interesting.