The ceiling is thirteen bars with names

The two parameters turn out to have a ceiling between them swept the key-finder’s bass weight against its cost of changing key over six harmonic schemes, and found that the share of bars read with the right key and the right scale degree levels off. Past a bass weight of about one and a half the reading stops improving, and the best cell of the plane reads 93.1 per cent of bars right, at a key cost of 3.

It ended by naming a suspect. A bass cue, it said, can only be as sharp as the chord it is reading, and each bar’s chord comes from a segmentation that names a different chord on sixteen per cent of passages depending on its cue. The proposed test was to run the plane again with the segmentation given the right answer.

That test cannot be run, because it is the computation already done. The plane reads the schemes bar by bar from their written harmony: each bar arrives with the pitch classes of the chord the scheme says is there, and no segmentation is anywhere in the pipeline. The ceiling is therefore not the segmentation’s. It is something the key reading does with chords it has been given exactly — and since the schemes are short and every bar can be inspected, the ceiling can simply be listed.

Six schemes, 188 bars, and a correction #

The schemes are the ones the key-relations essays have used throughout: a thirty-two-bar song, a rondo with an episode in the dominant and another in the relative minor, a twelve-bar blues, a verse-and-chorus song, a sixteen-bar period, and a minor ostinato. Each is a list of bars, each bar a chord named by its degree in a stated key. The reading realises each chord with a smooth bass line — the chord tone nearest the previous bass note — voices it in octaves, and reads it with two cues: how well the bar’s pitch classes fit each key’s collection, and whether the bass and its voicing name the chord the key would put there. A key change costs a stated amount, and the best path through the bars is found both ways, so every bar’s reading uses the bars after it as well as before.

The correction matters because it changes what kind of answer the ceiling can have. A segmentation error would have been noise — a bar whose chord is sometimes misread — and noise can be averaged down. Bars read wrongly although their chords were given exactly are a disagreement between the reading and the answer key, and those are either the reading’s fault or the answer key’s.

The thirteen #

At the plane’s best cell, the reading misses thirteen bars, and they fall into three groups with nothing left over.

Seven are the thirty-two-bar song’s bridge. The bridge is the familiar chain of secondary dominants that takes the song from its home key back to its dominant: III7, III7, VI7, VI7, II7, II7, V7, V7 — in C, E7, E7, A7, A7, D7, D7, G7, G7. The scheme labels all eight bars as C major, because that is how a harmony textbook describes the passage: dominants of the degrees of C, each pointing at the next. The reading names E major for the first three and D major for the next four, and only the last G7 as C. It hears a chain of dominants as a chain of keys, and on the evidence of the chords alone it is not wrong: E7 contains G♯ and D, A7 contains C♯, D7 contains F♯, and none of those is a note of C major.

Four are the rondo’s A minor episode. The episode runs i, iv, i, V7, i, iv, V7, i in A minor. The V7 bars contain G♯ and the reading gets them right. The i and iv bars are A–C–E and D–F–A, which are also exactly C major’s vi and ii, and the reading names three of the four bars before the first V7 and the last bar after the second V7 as C major. A reading that pays to change key keeps the passage in C wherever the notes allow it, and in a relative minor the notes allow it everywhere except on a leading note.

Two are edges. The rondo moves to G major for an episode and back. The bar before the episode begins is read as G, anticipating it, and the bar after it ends is read as G, holding it over. Both are a single bar and both come from reading the passage in both directions: the evidence for G in the bars after the boundary reaches back one bar, and the evidence after the return reaches forward.

Every other bar in the six schemes — all of the blues, all of the verse and chorus, all of the period and the ostinato, the whole of the thirty-two-bar song outside its bridge, and the rondo outside its two bars at episode boundaries and four minor bars — is read with the right key and the right degree.

What the cues buy, and what they cannot #

The listing is at one setting, and the earlier essay’s plane is a family of settings. Counting the misses by kind across the key cost says which of them the cues were buying back.

At a low key cost the reading changes key freely and misses 102 bars, and 91 of them are of no named kind: they are bars in a single key read as a neighbouring one because one chord happened to fit it better. That is the error the cues were built to remove, and they remove it. By a key cost of 2.2 the unnamed misses are down to two, and from 3 upward there are none.

What does not move is the named thirteen. The bridge contributes seven misses at every key cost from half to four; the edges two from 1.6 up. The relative-minor misses are the only group that responds, and it responds in the wrong direction for a single answer: four at the best setting, eight when the key cost rises to 6 or the bass weight falls to 1.5. A more reluctant reading holds C major through the episode’s V7 bars as well, and a weaker bass cue lets the ii–V pattern of C outvote the leading note.

That is the anatomy of the saturation the earlier essay drew. The cues work until only disagreements with the answer key are left, and then they have nothing to buy. The contour turning vertical is the unnamed misses running out.

It also explains the earlier essay’s interior optimum on the key cost. Below 3 the reading still makes a few unnamed errors; by a cost of 6 it has started reading the episode’s V7 bars in C as well, and misses sixteen. The best key cost is the one that has just removed the unnamed errors and not yet started adding relative-minor ones.

Where each scheme stands #

The thirteen are not spread over the schemes; they are concentrated in the two that change key. At the best setting the blues reads all 36 of its bars right, the verse and chorus all 32, the sixteen-bar period all 16 and the minor ostinato all 32. The thirty-two-bar song reads 25 of 32, and the rondo 34 of 40.

At a key cost of half, the same reading gets the ostinato wholly wrong — none of its 32 bars — and the verse and chorus 5 of 32, the period 10 of 16 and the blues 18 of 36, while the two schemes that actually modulate score 24 and 29. That inversion is worth dwelling on. A reading that changes key cheaply fails worst on the music that never changes key, because every bar gives it a small reason to move and nothing gives it a reason to stay; a reading that changes key reluctantly fails only on the music that does change, and only at the places where it does. The saturation is the point at which the single-key schemes are perfect and the key-changing ones have been reduced to their contested bars.

So the 93.1 per cent is two numbers averaged. For music that stays in one key this reading is at 100 per cent from a key cost of 3, and no further cue could raise it. For music that modulates it is at 82 per cent on these two schemes, and what separates that from 100 is a set of bars on which the question “what key is this in” has more than one defensible answer. That is the situation how much evidence a modulation needs described from the other side, when it found a key-finder with a short memory naming the new key a bar or two late, and it is the reason keys are neighbours — the relative minor and the dominant share six of their seven notes with the tonic, so a boundary between them is always a matter of degree.

The tonic cue does not reach the episode #

The reading at the plane’s corner uses no tonic cue at all: the probe-tone profile, which is how a tonic bought with the function taught the key-finder to tell A minor from C major, has a weight of nought. The obvious repair for four relative-minor bars is to give it weight.

It is the right repair for a passage that is all minor and the wrong one for this episode. A natural-minor test passage, read at the same setting, is named C major in every bar with no profile and A minor in every bar from a profile weight of 0.6. The schemes’ share does not move: 93.1 per cent at every weight from nought to one. Three of the four relative-minor misses stay relative-minor misses, and the fourth becomes a different kind of miss.

The difference is context. The test passage is minor from its first bar to its last, so the profile’s preference for A as a tonic has every bar to accumulate over and no competing key on either side. The rondo’s episode is eight bars between C major sections, and its i and iv bars are chords C major explains exactly as well; at this key cost, the profile’s preference is not enough to move them, and whether a longer episode or a cheaper key change would is a separate computation. A tonic cue decides which relative a passage is in; it does not decide whether a short passage is in the relative at all.

At a reluctant setting #

The same listing at a key cost of 6 shows the one group the key cost trades.

Doubling the key cost buys one bridge bar and costs four minor bars. The reading now follows the bridge one bar less, and it refuses to follow the episode into A minor at all, reading all eight bars — including both V7s — as C. The edges are unchanged. So the three groups are not equally fixed: the edges are a property of reading in both directions and do not move at all, the bridge moves by a bar at most, and the relative minor is a trade the key cost makes in either direction.

Which computation produced the numbers #

The six schemes are the key-relations essays’ own, 188 bars in all, with each bar’s key and degree taken from the scheme’s labels. Each chord is realised with the smooth bass line and voiced in octaves; the bass cue rewards a key only when the chord that key would put on that degree is the one actually sounding, which is the register rule a bass line is not a list of roots introduced. The reading is the ordered key-finder: an emission score for each key and degree in each bar from the collection overlap, the profile at its stated weight and the bass cue at its stated weight, a cost for each change of key, and the best path found in both directions. A bar is right when the key and the degree both match the scheme. A miss is classed as the bridge if it lies in the thirty-two-bar song’s B section, as relative if the bar is labelled minor and read in the major key three semitones up, and as an edge if the key changes on one side of it; anything else is unnamed.

Where the answer key stops being an answer #

The bridge is labelled by convention. A theorist calls the chain of dominants “in C” because the passage is heard as a return to C’s dominant, and a listener hearing the bridge on its own might well hear E, then A, then D. The seven bridge misses are the reading disagreeing with that convention, and whether they are errors depends on whether key means where the passage is going or which notes it is using. A modulation and a borrowing are one number apart found the same boundary in a single parameter: a tonicization is a modulation the key cost refuses.

The edges are the price of reading both ways. A listener who reads forward only cannot anticipate a key change, and how much of the reading arrives late measured what a forward-only reading loses. The two edge bars here are the analyst’s reading, not the listener’s, and a listener would miss different bars.

The schemes are few and simple. Six schemes, two of which are single-key, is enough to name what a ceiling is made of and not enough to say how large each group would be over a repertoire. A repertoire with more secondary dominants has a lower ceiling by exactly those bars.

What a list of misses cannot tell a listener #

Which key a bridge is in. The arithmetic can say that the chords of a chain of dominants each belong to a different major key and that a reading which charges for key changes follows them in steps of several bars. Whether a listener hears the bridge as travelling or as staying home while pointing elsewhere is a question about hearing, and there is evidence in the key-finding literature for both.

Whether a short minor episode is heard as minor. Eight bars of A minor inside a rondo in C is read as C by a cost-paying reading for the same reason a listener might hear it as a darkening of C rather than a new key. The profile does not decide it, which means the question is about the cost of changing key — and the key-finder with no tonic is where that cost first became a free parameter.

Still open: a bridge read as the keys it tonicizes #

The largest group of misses is a disagreement about labels, and it can be priced by changing the labels instead of the reading. If the bridge’s bars are labelled with the keys their dominants point toward — E7 as the dominant of A minor, A7 of D minor, D7 of G major, G7 of C — the answer key describes the passage as a chain of brief tonicizations, and the reading’s E and D majors are wrong in a different way: a dominant read as a tonic.

The computation is the listing above with one scheme relabelled, and it separates two things the ceiling currently mixes. If the reading names E major for a bar labelled “the dominant of A minor”, it is hearing a dominant seventh as a tonic, which is the blues grammar leaking into a song through the collection cue. If it names A minor, the bridge misses were purely a matter of labelling, and the ceiling of this key-finder on these schemes is not 93 per cent but whatever is left after the relative minor and the edges — about 97.