A tonic bought with the function

The key-finder with no tonic ends by naming a substitution and predicting what it would do. The ordered key-finder scores a bar by set overlap — the bar’s notes against the triad on a degree, shared tones over union — and a set has no shape, which is why the model finds a seven-note collection and cannot say which of its degrees is home. The histogram key-finder beside it has had a tonic since the first rung of this ladder, and the whole of the difference is that a probe-tone profile is not flat.

So the repair is one term. Score a bar’s notes against the profile rotated to each candidate key rather than against the triad on each degree, and the chain that keeps the order inherits the tonic the chain that throws it away has always had.

The prediction was stated so that it could fail, and half of it holds exactly. The passage above is the one the ladder built to be unreadable: a natural minor scale and its relative major are the same seven pitch classes, so a collection-finder has no observation that can separate them, and the twelve extra states the earlier essay added were shown to be worth precisely nothing there. With the profile in the emission the same thirty-two bars are named A minor at every one of them.

The other half of the prediction was that the substitution should lose margin on major material, because a graded emission separates fewer states than a binary one. That holds on four of the five major schemes and fails badly on the fifth — and the failure is the best thing on this page, because the fifth scheme turns out to have been read in the wrong key at every bar of it since the day it was written.

The substitution is one term, and it is the only term #

Everything else is held fixed, which is what makes the comparison a comparison. The states are still a key, a collection and a degree; the transitions are still the root-motion weights the ordered key-finder introduced; the cost of changing key is still 2.2, the value tuned in a modulation and a borrowing are one number apart. Only the emission moves.

The old emission is the log of a triad’s overlap with the bar. The new one is the log-likelihood of the bar’s notes under Krumhansl and Kessler’s probe-tone profile — twelve measured ratings, one per pitch class, rotated so that the key under test sits at the top of them — divided by the likelihood under a flat profile, so that the term is zero for a listener with no expectations at all and positive where a bar’s notes are the ones its candidate key rates highest.

Putting the two on one axis rather than running them separately is what the figure above is for. At a mixture of nought the emission is the published one and the chain reproduces the margins the ladder has printed bar for bar and bit for bit, which every figure here asserts before drawing anything. At a mixture of one it is the substitution the earlier essay asked for. In between each state’s emission is the weighted sum of the two.

The crossing on the minor passage is at a mixture of about 0.53 — the figure’s own grid puts it at 0.55 — and the interesting thing about it is the dip. The mean margin runs 3.38 bits under the overlap, falls to a third of a bit at the crossing and recovers to 4.56 under the profile. A model that changes its mind ought to be least certain at the moment it does, and this one is: the two readings are the same seven notes and the profile has to outvote a chain of root motions to overturn the label.

The scheme that has been in the wrong key since it was written #

The six schemes this ladder reads are not repertoire — they are the skeletons of six common forms, and the key plan is the form is the essay that built them. Five are written in C major and one, the four-bar ostinato, in C minor. Each is read here twice.

Four of the schemes lose margin exactly as predicted: the thirty-two-bar song by 18 per cent, the verse and chorus by 23, the rondo by 43 and the two eight-bar phrases by 52. Three of the four name C major at both ends of the axis, which is the key they are written in, so the loss is pure and buys nothing at all on that material. That is the second half of the prediction, and on those three it is correct and it is large.

The fourth is the rondo and it is the only scheme here that modulates, so it deserves a sentence of its own. Under the profile it reads G major over its first eighteen bars and C major over the remaining twenty-two — which is a modulation to the dominant that is genuinely there and starts sixteen bars too early. Bars nine to sixteen are the rondo’s G-major episode and the profile gets every one of them; bars one to eight are not, and it gets none. Counted against the key each bar is written in, the reading goes from 24 of 40 right to 22, so the profile trades eight correct bars for six. The set-overlap model, which reads C major throughout, never finds the modulation at all.

The twelve-bar blues does not lose margin. It gains 29 per cent — and it gains it while changing its answer at every one of its thirty-six bars.

The set-overlap emission reads a twelve-bar blues in C as F major. It does so at all thirty-six bars, at a mean margin of 3.78 bits — more confident than it is about the rondo or the sixteen-bar period, both of which it reads correctly. Nobody had looked, because the essay that added the minor collections reported how many bars each scheme renamed when the state space doubled and not which name any of them started from — and the blues renames none, because it is wrong in the same way with twelve collections and with twenty-four.

It is worth being exact about why, because the model is not malfunctioning. A blues is built on I7, IV7 and V7, and the union of those three chords is C, D, E♭, E, F, G, A, B♭ and B — nine pitch classes belonging to no major scale. The nearest seven-note collection is F major, which holds seven of the nine; C major holds six. So a model that finds a collection finds F, and it is right to. The passage is nevertheless in C, and every player, listener and lead sheet in the American vernacular repertoire from which the twelve-bar form comes says so.

That is the failure of the fourteenth rung’s own argument reappearing on material it had already read. That essay wrote that the missing states “have never produced a wrong answer that anybody was looking at”, and named the ostinato as the defect firing exactly once. The blues is a second firing and a much starker one: a major scheme, named a fourth away from its own key, confidently, on a figure that has been published.

Under the profile the blues is C major at every bar, at 4.86 bits. The profile is not finding a better collection. It is finding a tonic — the note that is common, long and cadenced-to — and the tonic is C whatever the collection around it is doing.

What “no tonic” means, as a number #

The fourteenth rung’s central claim is a symmetry argument and it deserves to be run rather than believed. Inside one diatonic collection, the overlap between the triads on two degrees depends only on the difference between the degrees; the root-motion transitions were already indexed by how far the root moved rather than by where it moved from. So rotating a passage’s degrees within its collection ought to leave the model’s score for that collection untouched.

It does, and to the last bit the arithmetic carries: −14.628 bits at every one of the seven rotations, with nothing at all between the widest pair. The song read as written and the song with every chord moved a third up score identically. That is not a tendency or an approximation, it is a conservation law of the emission, and it is now something a figure asserts rather than something an essay claims.

Under the profile the same seven readings span 11.97 bits, and at one of the seven rotations the model names a different key altogether. A profile is a shape over absolute pitch classes; rotating the music slides the notes under the shape and the score moves. Breaking that symmetry is the whole of what the substitution does, and everything else on this page is a consequence of it.

This is the same statement nothing in the census knows which note is home makes about the diatonic set’s interval content, and the same one the same seven started later makes about the modes. What is new is that the model that was supposed to be immune to it is not immune to it, and the immunity was exactly the defect.

The tonic is bought and the function is spent #

A profile scores a key. It has nothing to say about which chord of that key is sounding, because the same twelve ratings apply to every bar. So at a mixture of one all seven degrees of a key share an emission, and the only thing left deciding which degree a bar is on is the transition weights — which prefer descending fifths and know nothing about the notes.

The key share rises from 72 per cent to 91 in one step, at a mixture of 0.45, and every bar of that step is the blues. The degree share falls from 47 per cent to 25, and at the far end it stops being a fall: the model reports one distinct degree over an entire scheme, naming the tonic chord at every bar of every form. Its apparent score there is not knowledge but arithmetic — 51 per cent of these bars happen to be tonic chords.

A substitution that answers one question by ceasing to ask the other is not obviously an improvement, and this is the finding the debt did not anticipate. The ordered key-finder was built because a histogram throws the order away and cannot tell an alternation from a simultaneity; its states carry a degree precisely so that a chord has a function inside its key. Replacing the emission with a profile hands the key back and hands the function away, and what is left is a histogram key-finder with a transition matrix bolted to nothing.

The bass answers the other question #

The fourteenth rung named a second missing cue in the same paragraph and expected the same thing of it: “the bass note at a cadence is the strongest tonic cue a listener has”. Every bar on this ladder is a set of pitch classes with no register in it, so the note at the bottom is not in the model at all — which is what a chord is a register is the collection’s account of.

Giving each bar the bass note its chord is voiced on, and rewarding a state whose triad is rooted on it, is one more term. The arithmetic refuses the expectation and offers something better in its place.

The bass repairs the function almost completely: the share of bars whose degree is read correctly rises from 47 per cent to 89 under the set overlap, and to 87 under the profile. That is the deficit the profile opens, closed by a cue that costs nothing but a register.

What the bass never supplies is a tonic. The natural-minor passage is named C major at every bass weight on the axis, exactly as it was without one, and the reason is the symmetry above. Rotating a passage’s degrees rotates its bass notes with it, so a root-position bass line is invariant under precisely the operation the tonic is missing from. A bass note says which chord this is. It does not say which chord is home.

That is a genuine correction to a recorded expectation, and it sharpens rather than weakens the case for putting the bass in the model. Two cues, two questions, and neither substitutes for the other: at a mixture of one with the bass worth twice a triad match, the model reads 87 per cent of these bars in the right key and 87 per cent on the right degree, against 72 and 47 for the emission this ladder has been using for six rungs. Worth once rather than twice it reads 91 and 77, which is the better trade if the key is what is wanted and the worse one if the function is.

Which minor, and a threshold that does not arise #

The minor collection in the state space is the harmonic minor, and the earlier essay flagged that as an untested choice with a prediction attached: the melodic minor differs from the major in its third as well as its seventh, a third is a stronger cue than a seventh, so the threshold of three raised sevenths would fall and the sweep did not say how far.

The threshold does not fall. It does not arise, and the reason given for it was the wrong reason.

“Differs from the major only in its third” is true of the parallel major: A melodic minor and A major share six of seven pitch classes and part company on C against C♯. The confusion this model actually suffers is with the relative major, and there the melodic minor is not close at all. A melodic minor is A, B, C, D, E, F♯, G♯; C major holds five of those seven. Two notes are outside it rather than one, and both of them are in the chords a melodic-minor passage uses at every turn — the raised sixth in its major subdominant, the raised seventh in its major dominant.

So a passage in the melodic minor is read as A melodic minor at every bar, at 4.29 bits, with no raised seventh needed because the raised seventh was never the scarce observation. Its nearest rival is not C major or A major but G major, which is the major scale that shares six of its seven notes — the same relative-major relationship one step further round the circle of fifths.

The natural minor is the other extreme and it is exactly zero. A model given the twelve major collections and the twelve natural minors separates nothing anywhere: every margin in that row of the figure is 0.00 bits, not approximately, because a natural minor scale is a rotation of a major scale and the two are the same states under two names. The earlier essay said the minor states were unreachable; the grid says they are not merely unreachable but formally identical, which is a stronger claim and a checkable one.

Which computation produced the numbers #

The dynamic program is the ladder’s own with a mixture parameter on its emission and an optional bass term, and it is checked against the published one rather than trusted to agree with it: at a mixture of nought and a bass weight of nought, every figure here asserts that it reproduces the twelve- and twenty-four-collection readings bar for bar and to within a billionth of a bit.

The profiles are Krumhansl and Kessler’s, measured in 1982 by playing listeners a key-defining context and asking them to rate how well each of the twelve chromatic pitches fitted it. They are two vectors, one major and one minor, rotated to twenty-four keys — the same two the Krumhansl–Schmuckler correlation has used on this ladder since its fourth rung. Using them as an emission rather than as a correlation target is the substitution; nothing about the numbers themselves is new here.

The test passage is eight turns of i–iv–v–i in A minor, one chord a bar, built to hold the collection fixed and vary nothing. The six schemes are the form skeletons, read with their sections and choruses expanded. The bass note of each bar is the root of its chord, which is the simplest possible bass line and the one that makes the invariance argument cleanest.

The threshold of three is what the profile makes unnecessary. Under the set overlap the raised seventh is the only observation in the apparatus that can distinguish a minor key from its relative major, so a fixed number of them has to accumulate; under the profile the distinguishing evidence is present in every bar, because the tonic triad of A minor puts weight on A and C and E and the C major profile does not rate those three highest.

Where the account stops #

A margin is not a probability. Every number here is a gap between the best reading and the best rival in bits of log-score, and the two emissions are on different scales — a percentage change in margin is comparable within one emission and is not a claim that one model is 29 per cent better than another.

The mixture is not a listener. Nothing says a listener weighs a triad match against a profile in any proportion, and the axis is a device for making one substitution visible rather than a parameter with a psychological reading. What the axis shows honestly is where each property appears and what it costs; where a real listener sits on it is not a question this collection can ask.

The profiles are a measurement of listeners who were told the key. Krumhansl and Kessler’s ratings come from a probe-tone experiment in which a context establishes a key and the listener rates a note against it. Using them to find the key inverts the direction of the original measurement, which is what the correlation has always done and is a known liberty rather than a new one.

And six schemes are not a corpus. They are constructions, one of them written specifically as a control, and the 91 per cent above is a share of 188 bars of skeleton and not of music. The blues correction survives that — it is a fact about the model on a passage anybody can write down — and the 91 per cent does not travel.

Where this ladder goes next #

Fifteen rungs: a map of twenty-four keys, a key plan read as a form, three metrics of distance that disagree, a count of the evidence a modulation needs, a circle of fifths that is not that map, two keys sounded together, two taken in turn, an ordered model beside a histogram, a tuned key cost, a margin at every bar, hindsight separated from what a listener could have, a memory that forgets, a measurement of which bars the answer sits in, the state space itself, and now the emission inside it.

What is owed after this is the combination, and it is arithmetic rather than a corpus. Three cues are now on the table — the triad overlap, which finds a collection; the profile, which finds a tonic; the bass, which finds a degree — and they have been swept two at a time. Nothing here searches the plane. The question is whether the best reading on that plane is the corner the last figures reach or somewhere inside it, and the way to settle it is to sweep both weights together and report the surface rather than two of its edges. The prediction is that the surface has a ridge rather than a peak, because the profile and the bass answer different questions and neither trades against the other; a peak would mean they overlap more than this page claims.

The second debt is a bass line worth the name. Every bass note here is the root of its own chord, which is the case in which the cue is weakest and the invariance argument is cleanest — and an inversion is exactly where a bass note carries information a root does not. A first-inversion tonic and a root-position mediant are the same three pitch classes with different notes underneath, and the collection has the machinery to voice them: where the chord actually lands prices a chord’s arrival and a chord is a register computes what a voicing does. Running the sweep over realised bass lines rather than roots needs no recording and no listener, and it would say whether the 89 per cent above is the cue’s ceiling or its floor.

The third is the one this page has now made unavoidable and cannot pay: the model still has no way to prefer a degree on grounds other than the notes. The profile gives it a tonic by making the pitch classes unequal; a listener also knows that a phrase ends on its tonic, that a tonic is held longer, and that a cadence is a specific pair of chords rather than a specific pair of pitch-class sets. Counting those needs analysed music — which is the corpus this anchor has recorded as owed since its seventh rung, and the one thing on this list that the arithmetic cannot supply on its own.