Two cues meet in a corner

A tonic bought with the function gave the ordered key-finder two new cues and swept each one with the other at nothing. The probe-tone profile, mixed into the emission, gave the model a tonic: a natural-minor passage it had read as C major for as long as it had existed was named A minor at every bar. It also took the model’s grip on function away, so that at the far end of the mixture every bar of every scheme was read on the tonic degree. The bass note, rewarding the triad rooted on it, gave the function back — the share of bars read on the right degree rose from 47 per cent to 89 — and supplied no tonic at all.

So the model has three cues answering, as far as one-at-a-time sweeps can say, three questions: the triad overlap finds a collection, the profile finds which of its degrees is home, and the bass finds which chord a bar is. That essay ended by predicting the shape of the whole: sweep both added weights together and the best readings would lie along a ridge rather than at a peak, because the two cues answer different questions and neither should trade against the other.

Forty-two readings of the same music #

The plane is seven profile shares — nought, 0.2, 0.4, 0.5, 0.6, 0.8 and one — against six bass weights, from nothing to twice the log-weight of a triad match. At each of its forty-two points the same model reads the same material: the 188 bars of the six form schemes the key plan is the form built, and eight turns of i–iv–v–i in A minor with no raised seventh anywhere. Everything else is held where the earlier essays left it — the root-motion transitions, the cost of changing key at 2.2, the twenty-four collections of major and harmonic minor. The corner with neither cue reproduces the reading every earlier figure drew, bar for bar, and every drawing here checks that before it draws.

Four quantities are read at each point. The key share is the share of scheme bars named in the key they are written in; the degree share the share named in the right key and on the degree their roman numeral names; the minor share how much of the A-minor passage is named A minor; and the product of the last two, which is only high where the model is right about function and about the tonic at the same time.

The profile’s question is a step #

The tonic is a step and nothing else. Up to a profile share of 0.5 the A-minor passage is C major at every bar; from 0.6 it is A minor at every bar; and the step does not move by a single column at any of the six bass weights. The earlier sweep put the crossing at about 0.53, and the plane agrees that it lies between 0.5 and 0.6 and adds that the bass has nothing to say about where.

That is the invariance argument run over a whole axis rather than stated. A passage in the natural minor and the same passage read in its relative major are the same seven pitch classes, and rotating the degrees rotates the chord roots with them, so a bass line made of roots is the same bass line under both readings. Nothing a root-position bass note can say distinguishes a key from its relative, and on the plane that is a column of identical cells.

The bass’s question is a slope #

The degree is the other way round. Down the left-hand column it climbs from 47 per cent to 55, 77, 84 and then 89 at a bass weight of one and a half, and stays there. Across any row the profile moves it far less, and the pattern of how it moves is informative: at a bass weight of a half the degree share falls from 77 to 51 as the profile takes over the emission, because the profile scores every degree of a key alike and dilutes the triad overlap that was distinguishing them; at a bass weight of two it barely falls at all, from 89 to 87, because by then the bass is doing the distinguishing and has no need of the overlap.

The bass does not merely add function back; with enough weight it makes the profile’s cost to function disappear. The earlier essay’s finding that the profile spends the function is true at every point where the bass is weak and false where it is strong.

The key is where the two overlap #

The key share is the one quantity both cues move, and it moves the same way under either. In the corner with neither cue it is 72 per cent; with the profile at half the emission it is 91; with the bass worth a half and no profile at all it is also 91. Almost the whole plane sits at 89 to 91.

The exception is informative. Where the profile is most of the emission and the bass is weighted one and a half or more, the key share falls to 87. The loss is almost all in one scheme, the thirty-two-bar song, which a strongly weighted bass reads partly outside C: 88 per cent of its bars in C with the bass alone, 78 per cent once the profile joins it, with one bar of the verse and chorus going the same way. A strong root-position bass and a strong profile pull some of that song’s chords toward different keys, and it is the only place on the plane where the two cues disagree. It costs four points.

The blues is rescued twice #

Most of the key share’s rise from 72 to 91 is one scheme. The twelve-bar blues is built on I7, IV7 and V7, whose union is nine pitch classes that belong to no major scale, and the nearest seven-note collection is F major — so the overlap alone reads a blues in C as F major at every bar, and the earlier essay found that confident error and the profile’s correction of it.

The plane shows a second correction. With no profile at all, a bass worth a half names the blues in C at every bar as well — and, one step down the column, at a bass worth one, it slips to 92 per cent before recovering, which is the only cell on the plane where more of a cue buys less of this answer. The profile finds C because C is the pitch class a C-major profile rates highest and the blues keeps returning to; the bass finds C because the bass notes are C, F and G, and only a reading in C puts all three on degrees that root the chords sounding over them. The one piece of music both cues rescue is rescued for two unrelated reasons, and that is exactly the overlap a key share averages over — the one place where the two questions turn out to share an answer.

A corner, not a ridge #

Put the questions together and the shape is visible without any smoothing. The product of the degree share and the minor share is nought over the whole left half of the plane, because there the minor passage is never named A minor. On the right half it rises down each column as the bass is weighted, from 41 per cent with no bass to 87 at a weight of two, and the best cells — 87 per cent at profile shares of 0.6 to one with the bass worth two, and at 0.6 with the bass worth one and a half — fill the bottom-right corner.

A ridge is a line of best readings along which two parameters trade, so that more of one can be exchanged for less of the other at the same quality. There is no such trade here. The profile has to exceed about 0.55, and nothing the bass does changes that; the bass has to exceed about one and a half, and beyond a small dent the profile does not change that either. Two cues answering different questions produce a corner: each has a threshold of its own, and the good region is where both thresholds are passed.

So the prediction had the right reason and the wrong shape. The cues do answer different questions and neither trades against the other; that is precisely why the best region is bounded by two independent edges rather than stretched along a diagonal. What looked from the two edges of the plane like a ridge between them is the two edges of a corner.

The one irregularity inside the corner is worth naming. At a profile share of 0.8 with the bass worth one and a half the combined share dips to 80, against 87 either side of it. The key share in that cell is the same as in its neighbours, so the dip is in the degrees: the profile has made enough degrees alike that the bass has near-ties to break, and at that pair of weights it breaks some of them the wrong way. A finer grid would show a narrow valley rather than a single cell, and it would not change the corner.

The rondo, which no setting of either cue reads better #

One scheme does not move anywhere on the plane, and it is the one that most needs a model to move. The rondo is the only scheme that modulates — forty bars in five sections of eight, the first episode in the dominant and the second in the relative minor — and its key share is 60 per cent in the corner with neither cue, 60 per cent with the bass alone at one and a half, 55 per cent with the profile alone, and 60 per cent again in the corner where every other question is answered. Sixty per cent is twenty-four bars of forty, which is the three refrains at home: with the bass alone the model reads C major from the first bar to the last.

No combination of a tonic cue and a degree cue reads a key plan better than the overlap alone did. The profile, as a tonic bought with the function found, hears the dominant episode and then carries it back over the opening refrain. The bass, while every bass note is a root, has nothing to say about when a key changes: the roots of the dominant episode’s chords are G, D and C, all three of them notes of C major, and a bar rooted on D is the same degree-shaped object whether it is V of G or ii of C. What would time the modulation is the evidence neither cue carries: that a cadence is an ordered pair of chords rather than two sets, which is what the cadence as evidence set against the profile, and how many bars of the new key it takes before a listener should believe it, which is what how much evidence a modulation needs counted.

So the corner is a corner for the questions a single key and a single chord pose, and the one scheme with two keys in it sits in the same place on every cell of the plane. That is a third question — not which note is home and not which chord this is, but when home changed — and it is not on this plane at all, at least while the bass is a list of roots. A bass line that inverts its dominants puts the leading note of the new key at the bottom, which is the one bass note the episode has that C major does not, and a bass line is not a list of roots finds that it moves the rondo’s share from 60 per cent to 93.

Two kinds of knowledge, one threshold each #

The corner has a reading that is about listeners rather than about the model, and it is worth stating carefully because the model is not a listener.

The profile is a measurement of what listeners raised inside a tradition expect: counting produced the hierarchy is the account of how a rating of which pitch classes fit a key can be learned from how often they occur. The bass note is a fact about register — which note of a chord is at the bottom — and a chord is a register is the account of what register does to a chord’s identity. One is schematic knowledge a listener brings; the other is information in the sound. The plane says that on this material the two enter the reading as two thresholds rather than as substitutes, so that neither can be traded for the other and each has to be present at a sufficient strength.

That is also why the natural-minor passage is the right test and a severe one. It is the case in which the sound offers no evidence for the tonic at all, since the collection is shared with the relative major and a root-position bass is invariant under the exchange. The key-finder with no tonic found that three raised sevenths are enough to overturn that under the overlap alone, and real minor-key music carries raised sevenths constantly. On real music the profile’s step would therefore matter less often than it does here, and the corner’s left edge would be softer; on this passage it is the only thing that can name the key.

The one-at-a-time edges #

The two sweeps that preceded this plane are its top row and its left column.

Read as lines, both edges suggest a trade that is not there. Along the left column the degree climbs and the minor tonic stays wrong; along the right column the degree climbs and the minor tonic stays right. Nothing on either line says that crossing from one to the other is a step with no cost, and the plane is what shows it: every cell on the right half has the tonic, and the degree share it carries is set almost entirely by the bass. A trade inferred from edges is a trade with the interior missing, which is the whole reason the plane had to be computed rather than inferred.

The reading the plane rests on #

The model is the key-finder that keeps the order: a dynamic program over states that are a key, a collection and a degree, with transitions weighted by root motion and a cost of 2.2 for changing key, the value a modulation and a borrowing are one number apart tuned. Its emission at each bar mixes the log of a triad’s overlap with the bar and the log-likelihood of the bar’s notes under Krumhansl and Kessler’s profile rotated to the candidate key, and adds a bass term that rewards the triad rooted on the bar’s bass note against a floor of a fifth. Every bass note here is the root of its chord.

The shares are shares of bars, not margins. The margin the dynamic program already had is a separate quantity, and a cell can be right on every bar with a thin margin or wrong on a few with a thick one; the plane reports which reading wins, not by how much.

What forty-two cells do not show #

The schemes are skeletons. Six constructed forms and one constructed minor passage are not a repertoire, and the percentages are shares of 188 bars of skeleton. The corner is a fact about the model on this material; where the corner’s edges fall on real music is a question for analysed music.

The bass line is a list of roots. Every bar’s bass is its chord’s root, which is the one realisation under which the bass cue is strongest and the invariance argument cleanest. A real bass line inverts chords, and an inverted chord’s bass note is not its root — so the bass cue’s 89 per cent is measured on the line most favourable to it.

The grid is coarse where it matters. The tonic’s step lies somewhere between 0.5 and 0.6, and the bass’s threshold somewhere between one and one and a half. The corner’s shape does not depend on where exactly, and its area does.

And the cues are not a listener. The two weights are not parameters of hearing; they are dials on a model, and the plane says where on those dials the model reads skeletons of tonal forms correctly. Whether a listener’s inference has anything like two independent thresholds is not a question a key-finder can answer about itself.

Whose analysis #

The analysis is the common-practice reading of tonal forms — keys, degrees and functions assigned to the harmonic skeletons of songs, rondos, periods and blues choruses — and the claim is about what an automatic reading of those skeletons needs. It needs a tonic from somewhere other than the collection and a degree from somewhere other than the pitch classes, and on this material those two needs are met by two cues that do not interfere.

Still open: a bass line that is not a list of roots #

The bass cue was the easy half of the corner because every bass note in the plane is its chord’s root. An economical bass line puts roughly one chord in two in inversion, and a cue that rewards the triad rooted on the bass will reward the wrong triad on every one of them: a C major chord with E underneath rewards E minor. Whether the degree share of 89 per cent survives a realised bass line, whether a cue that rewards any triad containing the bass note does better, and whether the corner survives at all once the bass is a line rather than a list of roots, is the same plane computed over realised basses.