Acoustics & Music Theory
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Scales and modes

The bridge is not a labelling problem

Seven of the key-finder's thirteen remaining misses are the thirty-two-bar song's bridge, and the essay before this one proposed that they might be an argument about labels rather than a failure: relabel the eight bars with the keys their dominants point at, and the reading's E and D majors might turn out to be right. They do not. The reading gets one bar of eight under either label, because four of the eight are read as the chord's own root taken for a tonic — a dominant seventh heard as a chord in its own right. And one of the bars the relabelling could have bought is unwinnable in principle: an applied dominant is the same four notes in the major and the minor of the key it points at.
17 min read 7 figures What the listener suppliesNotation hides thingsThe limits of the listener

Assumes: The ceiling is thirteen bars with names · The two parameters turn out to have a ceiling between them

The ceiling is thirteen bars with names listed every bar the key-finder still misses at its best setting and found seven of the thirteen in one place: the thirty-two-bar song’s bridge, eight bars of applied dominants that the scheme labels C major throughout and that the reading names in E and in D. It ended by pointing out that the disagreement could be priced from the other side.

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 relabelling is one line: the chords are untouched and only what counts as a right answer moves. Done, it buys one bar out of eight at best, and what it reveals is that the reading’s errors are the second kind.

Relabelling the bridge does not rescue itThe thirty-two-bar song's eight bridge bars at a key cost of 3 and a bass weight of 3: the chord in each, the key the scheme labels it in, the key its dominant points at, and what the reading names. III7: scheme says C major, the tonicization says A minor, the reading says E major degree 1 — the chord's own root as a tonic; III7: scheme says C major, the tonicization says A minor, the reading says E major degree 1 — the chord's own root as a tonic; VI7: scheme says C major, the tonicization says D minor, the reading says E major degree 4; VI7: scheme says C major, the tonicization says D minor, the reading says D major degree 5; II7: scheme says C major, the tonicization says G major, the reading says D major degree 1 — the chord's own root as a tonic; II7: scheme says C major, the tonicization says G major, the reading says D major degree 1 — the chord's own root as a tonic; V7: scheme says C major, the tonicization says C major, the reading says D major degree 4; V7: scheme says C major, the tonicization says C major, the reading says C major degree 5. Right under the scheme's label: 1 of 8. Under the tonicization: 1. Under the tonicization with the mode set aside: 2. Read as its own tonic: 4.III7bar 17CAmE Ias tonicIII7bar 18CAmE Ias tonicVI7bar 19CDmE IVin transitVI7bar 20CDmD Vin transitII7bar 21CGD Ias tonicII7bar 22CGD Ias tonicV7bar 23CCD IVin transitV7bar 24CCC Vrightthe scheme's keythe key it tonicizeswhat the reading namesbridge1 of 8 right under the scheme's label · 1 under the tonicization · 2 with the mode set asidethe chords are identical under every labelling; only what counts as a right answer moves
Fig. 1 The eight bridge bars: the chord’s Roman numeral, the key the scheme labels it in, the key its dominant points at, and what the reading names. One bar is right under the scheme’s label and one under the tonicization — the same bar, the last G7. Four are the chord’s own root taken for a tonic. The dial moves the one parameter that has any effect on the bridge at all, and its effect is to trade two of the four for one more right answer.

Four bars of a dominant heard as a chord

The reading’s answers, bar by bar, are not a chain of keys at all.

The first two bars are E7, and it names E major, first degree — the chord’s own root as a tonic, with the seventh treated as a passing dissonance in a key of E. The third bar is A7 and it names E major’s fourth degree, which is the reading carrying E forward across a chord that half belongs to it. The fourth bar is A7 again and it names D major’s fifth, which is exactly the tonicization label with the mode changed. The fifth and sixth bars are D7 and it names D major, first degree — its own root again. The seventh bar is G7 and it names D major’s fourth; the eighth is G7 and it names C major’s fifth, which is right under both labels.

So of the eight, four are the chord read as a tonic, two are the previous key carried one bar too far, one is right and one is a near miss of a kind the next section is about.

That settles the earlier question in the direction it thought less likely. The reading is not mistaking the key of a passage it has otherwise understood; it is mistaking the chord. A dominant seventh’s four notes fit a major key’s first degree tolerably — root, third, fifth and a flattened seventh that any key-finder’s collection overlap treats as a small penalty — and they fit the fifth degree of the key a fourth above exactly. Given no memory of what came before and a cost for changing key that has already been paid, the reading takes the near fit that needs no further change.

Why a dominant seventh is a comfortable tonic to this reading

The four-bar result is worth one paragraph of mechanism, because it is not obvious that a chord with a flattened seventh should be read as a tonic at all.

The emission score is a collection overlap: how much of the bar’s pitch-class content belongs to each key’s seven notes. E–G♯–B–D against E major’s collection loses one note, the D, and keeps three. Against A minor’s it loses none. So on that cue alone A minor wins, and the bar should be read as the tonicization label says.

What overturns it is everything else in the sum. The degree term prefers a chord that is the key’s own first degree to one that is its fifth, because a first degree is what the reading’s transition model expects a key to settle on; the bass cue rewards a root-position chord that the key would put on that degree, and E major does put an E chord on its first; and the key cost has already been paid once, so having arrived in E the reading has no reason to leave for a key that fits one note better. Three small preferences against one small penalty, and the reading sits.

That is not a bug in any of the three cues. Each of them was added for a reason the earlier essays measured — a tonic bought with the function for the profile, a bass line is not a list of roots for the register rule — and each of them improves the reading on the other 180 bars. What the bridge shows is that a sum of cues that all describe a bar’s own content has no term for what a bar is for.

No setting rescues it

One setting is not an argument, and the listing’s whole method was to sweep the key cost and watch which misses move.

No labelling of the bridge gets more than half of it, and the scheme's gets one bar. How many of the eight bridge bars the reading gets right, against the cost of changing key, at a bass weight of 3, under three labellings — and how many it reads as the chord's own root taken for a tonic. key cost 0.5: scheme 1, tonicization 2, tonicization with the mode set aside 3, read as own tonic 2; key cost 1: scheme 1, tonicization 1, tonicization with the mode set aside 2, read as own tonic 4; key cost 1.5: scheme 1, tonicization 1, tonicization with the mode set aside 2, read as own tonic 4; key cost 2: scheme 1, tonicization 1, tonicization with the mode set aside 2, read as own tonic 4; key cost 2.5: scheme 1, tonicization 1, tonicization with the mode set aside 2, read as own tonic 4; key cost 3: scheme 1, tonicization 1, tonicization with the mode set aside 2, read as own tonic 4; key cost 4: scheme 1, tonicization 1, tonicization with the mode set aside 2, read as own tonic 4; key cost 5: scheme 1, tonicization 1, tonicization with the mode set aside 2, read as own tonic 4; key cost 6: scheme 2, tonicization 2, tonicization with the mode set aside 4, read as own tonic 2; key cost 8: scheme 2, tonicization 2, tonicization with the mode set aside 4, read as own tonic 2. The scheme's label is flat at one bar across almost the whole range. The tonicization label is flat at one or two. Only setting the mode aside as well reaches four, and it reaches four at the reluctant end, where the reading has stopped changing key for every chord.
Fig. 2 How many of the eight bars the reading gets right under each labelling, against the cost of changing key, with the count it reads as their own tonics drawn beside them. The scheme’s label sits at one bar across almost the whole range. The tonicization sits at one or two. Only setting the mode aside as well reaches four.

The scheme’s own label gets one bar at every key cost from 0.5 to 5 and two at 6 and above. The tonicization label gets one or two across the same range. The two labels are worth the same to within a bar everywhere, which is the strongest form of the finding: the bridge is not a place where the reading and the answer key disagree about how to describe something they both hear. It is a place where the reading hears something else.

The count of bars read as their own tonics is the mirror image. It is four through the middle of the range — the four bars where the reading has settled into E and into D — and falls to two at the reluctant end, where the reading changes key less and so arrives at fewer roots to sit on. That is the trade the earlier listing found for the relative minor, running in the bridge as well: a reluctant reading makes fewer tonics and stays behind for longer.

The ceiling is thirteen bars, and every one of them has a name. Every bar of the six schemes read by the key-finder at a key cost of 3 and a bass weight of 3, with the chord in every bar given exactly — no segmentation is involved. 175 of 188 bars have both the key and the degree right, 93.1 per cent. The misses: 7 in the thirty-two-bar song's bridge, where the chain III7 is read in E major, III7 is read in E major, VI7 is read in E major, VI7 is read in D major, II7 is read in D major, II7 is read in D major, V7 is read in D major; 4 bars of the rondo's A minor episode read as C major; 2 next to a change of key; and 0 of any other kind.
Fig. 3 The earlier listing, for reference: all 188 bars of the six schemes at the same setting, with the thirteen misses coloured by kind. Relabelling the bridge changes the total by nothing, because the bar it wins is the bar it loses.

Rescored over all 188 bars, the relabelling moves the reading’s share from 93.1 per cent to 93.1 per cent. The one bar the tonicization label wins at the loose end it gives back elsewhere, and at the setting the plane calls best it wins and loses the same bar — the last G7, which is the dominant of C either way.

One bar nobody could get

The fourth bar is the interesting one and it is not the reading’s fault.

It is A7, the scheme calls it C major’s VI7, the tonicization calls it the dominant of D minor, and the reading names the fifth degree of D major. Key root right, degree right, mode wrong. Under a scoring that requires the mode it is a miss; under one that does not it is the second bar the tonicization label wins.

A secondary dominant does not say which mode it is pointing at. For each of the bridge's four applied dominants, the four pitch classes of the chord and the seven of the major and minor keys it could be pointing at. III7 is the dominant seventh of A: its notes E, A♭, B, D belong to both the major and the minor collection, which differ at C♯, F♯; VI7 is the dominant seventh of D: its notes A, C♯, E, G belong to both the major and the minor collection, which differ at F♯, B; II7 is the dominant seventh of G: its notes D, F♯, A, C belong to both the major and the minor collection, which differ at B, E; V7 is the dominant seventh of C: its notes G, B, D, F belong to both the major and the minor collection, which differ at E, A. In every case the two degrees that separate the two modes — the third and the sixth, the leading note being raised in the minor for this very chord — are absent from it. So a reading given the chord alone cannot name the mode of the key it tonicizes, and an answer key that names it is naming something the music has not yet said — which is why 0 of the eight bars are right only when the mode is set aside.
Fig. 4 Each of the bridge’s four applied dominants, against the major and minor collections of the key it points at. In every case the chord’s four notes belong to both, and the two degrees that tell the modes apart — the third and the sixth — are absent from it, because the minor’s leading note is raised for this very chord.

A dominant seventh on the fifth degree of a minor key requires the leading note to be raised, which is the whole reason the harmonic minor exists. Raised, the chord is E–G♯–B–D whether it is resolving into A major or A minor, and the two degrees that separate those collections — the third and the sixth — are not in it. The chord contains no evidence about the mode of the key it is pointing at. The evidence is in what comes next, and what comes next in the bridge is another applied dominant.

So one of the bridge’s eight bars is labelled with something the music has not said at the moment the reading has to say it. An answer key written by a theorist looking at the whole bridge knows the E7 leads to A minor because the A7 that follows is a dominant, not a tonic; a reading working bar by bar cannot know it without reading three bars ahead through two changes of key, which is more lookahead than the whole apparatus has.

That is worth stating as a limit on the measurement rather than on the reading. That 93.1 per cent contains at least one bar whose answer is not in the data, and any figure of merit computed against a theorist’s analysis will contain some of those. Finding them is the reason to relabel even when the relabelling buys nothing.

The scheme is the one that modulates, and it is the one that suffers

Reading this beside the earlier per-scheme breakdown puts the bridge in proportion.

Four of the six schemes are read perfectly at this setting, and both of the ones that are not are the two that change key. That is not a coincidence about these six passages: how much evidence a modulation needs measured, from the other side, how long a key-finder takes to commit to a new key, and found it a bar or two — which is the whole length of each tonicization in this bridge.

So the bridge is a modulation that never finishes, eight bars of a reading being asked to commit four times in the time it usually takes to commit once. Its misses and the two edge misses are the same failure at two speeds: a reading whose commitment lags the music by about a bar, meeting music that changes key every two.

That reframes the ceiling in a way worth carrying. What saturates is not the cue set’s accuracy; it is the cue set’s latency. A reading that named keys a bar late would score exactly this way — right in the middle of every stable passage, wrong at every boundary, and wrong throughout a passage made of nothing but boundaries. The key finder with no tonic tested what happens when a cue is removed entirely; nothing in these essays has tested what happens when the same cues are allowed to look forward.

What the two cues were doing instead

The reading’s cues were swept in the two essays before this one and neither of them touches this.

The cues remove every miss but three kinds. The bars the key-finder misses over the six schemes, split by kind, against the cost of changing key, at bass weights of 1.5 and 3. Bass weight 1.5: cost 0.5, 102 missed (7 bridge, 0 relative, 4 edge, 91 other); cost 1, 73 missed (7 bridge, 0 relative, 4 edge, 62 other); cost 1.6, 55 missed (7 bridge, 4 relative, 2 edge, 42 other); cost 2.2, 15 missed (7 bridge, 4 relative, 2 edge, 2 other); cost 3, 17 missed (7 bridge, 8 relative, 2 edge, 0 other); cost 4, 17 missed (7 bridge, 8 relative, 2 edge, 0 other); cost 6, 19 missed (3 bridge, 8 relative, 2 edge, 6 other). Bass weight 3: cost 0.5, 102 missed (7 bridge, 0 relative, 4 edge, 91 other); cost 1, 73 missed (7 bridge, 0 relative, 4 edge, 62 other); cost 1.6, 55 missed (7 bridge, 4 relative, 2 edge, 42 other); cost 2.2, 15 missed (7 bridge, 4 relative, 2 edge, 2 other); cost 3, 13 missed (7 bridge, 4 relative, 2 edge, 0 other); cost 4, 13 missed (7 bridge, 4 relative, 2 edge, 0 other); cost 6, 16 missed (6 bridge, 8 relative, 2 edge, 0 other). At a low key cost most misses are of no particular kind and the cues buy them back; past a cost of about 2.2 what is left is the bridge, the relative minor and the edges of the rondo's episodes, and those do not respond.
Fig. 5 The misses by kind against the key cost, at two bass weights. The bridge’s seven are flat across the whole range; the unnamed misses, which the cues were built to remove, run from ninety-one down to none.

The bass cue rewards a key when the chord that key would put on the degree is the one sounding, and every bridge chord is a dominant seventh in root position — which is the correct chord for the fifth degree of two different keys and a tolerable chord for the first degree of one. The cue cannot separate them because it is a statement about the bass and the chord, and the bass and the chord are the same under all three readings.

The tonic cue fixes a minor passage and leaves the schemes' ceiling where it was. At a key cost of 3 and a bass weight of 3, the share of scheme bars with key and degree right, and the share of a natural-minor test passage read as A minor, against the probe-tone profile's weight in the reading. weight 0: schemes 93.1 per cent (4 relative-minor misses), minor passage 0 per cent; weight 0.2: schemes 93.1 per cent (3 relative-minor misses), minor passage 0 per cent; weight 0.4: schemes 93.1 per cent (3 relative-minor misses), minor passage 0 per cent; weight 0.5: schemes 93.1 per cent (3 relative-minor misses), minor passage 0 per cent; weight 0.6: schemes 93.1 per cent (3 relative-minor misses), minor passage 100 per cent; weight 0.8: schemes 93.1 per cent (3 relative-minor misses), minor passage 100 per cent; weight 1: schemes 93.1 per cent (3 relative-minor misses), minor passage 100 per cent. The profile is what tells A minor from C major when a passage is all minor, and it does not reach the rondo's episode, whose i and iv bars sit among C major bars on both sides.
Fig. 6 The profile cue, which decides that a passage is in A minor rather than C major, against its weight. It moves a wholly minor test passage from none to all at a weight of 0.6 and leaves the schemes at 93.1 per cent throughout.

The tonic profile is no better placed. It accumulates a preference over a passage, and the bridge gives it two bars of each key before moving on — which is shorter than the memory the profile needs and is, in the terms keys are neighbours sets out, a sequence of keys each of which is a step from the last on the same circle. Two bars of E7 look like two bars of E and like two bars of A’s dominant, and the profile has no third option to prefer.

The two parameters are not one, and the reason is a ceiling. The plane of the two parameters these readings have been swept one at a time: how much weight the bass cue carries, against what a change of key costs. Every cell is how often the reading names the right scale degree, and the lines are the contours of equal share. along the 50 per cent contour the product of the two coordinates runs from 0.10 to 0.50; along the 60 per cent contour the product of the two coordinates runs from 0.43 to 1.84; along the 70 per cent contour the product of the two coordinates runs from 0.66 to 2.56; along the 80 per cent contour the product of the two coordinates runs from 1.85 to 4.91; along the 90 per cent contour the product of the two coordinates runs from 2.71 to 15.20. If the two multiplied cleanly those products would be constant and the contours would be hyperbolae. They are not: every contour turns upward and then vertical, because past a bass weight of about 3 more of the cue buys nothing at all and only reluctance is left to buy anything with. The key cost has an interior best, at 3 on this grid, where the reading names 93 per cent of degrees — so a reading that will not change key at all is worse than one that will, which no sweep of a single parameter had found.
Fig. 7 The plane the ceiling was found on: the share of bars right over the bass weight and the key cost, with contours of equal share. The whole bridge sits inside every contour, unchanged by either axis.

What a theorist’s label is for

The relabelling was proposed as a possible correction to the answer key and it has turned out to be a demonstration that the answer key is doing something the reading is not trying to do, which is worth separating out.

A reading assigns a key to a bar. A theorist assigns a function to a chord, and a function is a relation between a chord and what follows it. The scheme’s label “III7 in C major” is not a claim that the bar sounds like C major — it plainly does not, since three of its four notes are outside C — but a claim that the bar’s job is to get to the chord after it, and that the passage as a whole is a departure from C and a return to it. The tonicization label “the dominant of A minor” is the same claim written one level down.

Neither of those is available to a machine that reads bar by bar, and neither is refuted by the machine failing to produce it. What the two labellings agreeing to within one bar shows is that the disagreement is not about which name to write; it is about what kind of thing a name is. A key-finder names a state and an analyst names a motion, and a passage made entirely of motion is where the two come apart.

The practical reading of that is narrow and it is the one to take. A figure of merit that scores a key-finder against a theorist’s analysis is scoring two different questions wherever the analysis is functional rather than sectional. On these six schemes that is seven bars of 188 — under four per cent, which is why the measurement is useful at all — and the four per cent is not noise to be reduced. It is the part of the analysis that is about what the music is doing rather than about what key it is in.

Which computation produced the numbers

The scheme is the key-relations essays’ thirty-two-bar AABA song, whose B section is eight bars of III7, VI7, II7 and V7 in pairs. Each bar’s chord is realised with the smooth bass line the earlier essays use, voiced in octaves, and read by the ordered key-finder: an emission score per key and degree from how far the bar’s notes fit each key’s own seven, the tonic profile at its weight, the bass cue at its weight, a cost for each change of key, and the best path found in both directions.

The relabelling changes each bridge bar’s answer key and nothing else. The pitch classes, the bass note, the voicing and the reading are identical under both labels; only the comparison at the end moves. A bar counts as right under the tonicization when the reading’s key, mode and degree all match; the mode-blind scoring drops the mode and keeps the other two. A bar is counted as read as its own tonic when the reading names the chord’s root as the key and the chord as its first degree.

Where the model stops

The reading has no memory of a chord’s resolution. It reads a bar’s key from the bar’s notes and pays to change; what an analyst uses to label an applied dominant is where it goes, which is the next bar. A reading that scored a pair of bars together would have the evidence and would be a different machine.

Eight bars of pure applied dominants is an extreme case. Real bridges mix them with the chords they resolve to, and every resolution supplies exactly the evidence about mode that is missing here.

The mode-blind scoring is a diagnostic and not a proposal. A key-finder that declines to name a mode is not answering the question all of these essays ask, and it is reported only to separate the bars the reading gets structurally wrong from the one it cannot get at all.

What a bar count cannot say

Whether a listener hears the bridge as a chain of keys. The reading does, and it is a reasonable model of somebody following bar by bar; a listener who knows the song is in the position of the theorist and not of the reading.

Whether four bars of a dominant read as a tonic is a mistake at all. A dominant seventh sounded alone for two bars is, in a good many styles, simply a chord on its own root, and a modulation and a borrowing are one number apart is the essay about how thin that distinction is. The scheme calls it a passing dominant because of the tradition it belongs to.

Whether a bridge is a key plan. The key plan is the form reads a movement’s keys as its structure, and on that reading a chain of applied dominants is one gesture and not four keys — which is a third description again, and the one a listener probably has.

Whether the ceiling would move under a third labelling. Two have been tried and they agree. A third — labelling each pair of bars with the chord’s own root, which is what the reading says — would score the bridge at four of eight and would be an answer key written from the reading, which is not a test of anything.

Still open: the lookahead the reading does not have

Every miss in the bridge and one of the two edge misses come from the same absence: the reading assigns a key to a bar from the bar’s own notes and the cost of having changed, and an applied dominant is a chord whose meaning is in the bar after it.

The apparatus already reads in both directions, so lookahead is already possible; what is missing is a cue that is about a pair of bars rather than about one. A resolution cue would score a candidate key for bar n partly by whether bar n+1 is that key’s tonic, which is one term in the emission score and needs no new data. What would come out is whether the bridge’s seven misses are a consequence of the cue set being per-bar — in which case the ceiling is an artefact of the machine’s shape and not of the music — or whether even a reading that knows where each dominant goes still prefers to sit on its root, in which case the applied dominants of a bridge are genuinely ambiguous and the theorist’s label is a convention rather than a reading.

Part 22 of 22

One essay in the series on Key-relations. The essays either side of this one:

The ceiling is thirteen bars, and every one of them has a name. Every bar of the six schemes read by the key-finder at a key cost of 3 and a bass weight of 3, with the chord in every bar given exactly — no segmentation is involved. 175 of 188 bars have both the key and the degree right, 93.1 per cent. The misses: 7 in the thirty-two-bar song's bridge, where the chain III7 is read in E major, III7 is read in E major, VI7 is read in E major, VI7 is read in D major, II7 is read in D major, II7 is read in D major, V7 is read in D major; 4 bars of the rondo's A minor episode read as C major; 2 next to a change of key; and 0 of any other kind. The ceiling is thirteen bars with names Part 21 — The key-finder's two cues stop buying anything at about 93 per cent of scheme bars read right, and the obvious suspect was the chord segmentation feeding it. The reading was never given a segmentation: every bar arrives with its true chord. What the ceiling is made of can be listed instead, and it is thirteen bars of 188 — seven in a bridge of secondary dominants, four in a minor episode whose chords C major also owns, and two at the edges of a modulation. No weight of any cue moves one of them.

Where this goes next

The ceiling is thirteen bars, and every one of them has a name. Every bar of the six schemes read by the key-finder at a key cost of 3 and a bass weight of 3, with the chord in every bar given exactly — no segmentation is involved. 175 of 188 bars have both the key and the degree right, 93.1 per cent. The misses: 7 in the thirty-two-bar song's bridge, where the chain III7 is read in E major, III7 is read in E major, VI7 is read in E major, VI7 is read in D major, II7 is read in D major, II7 is read in D major, V7 is read in D major; 4 bars of the rondo's A minor episode read as C major; 2 next to a change of key; and 0 of any other kind. The ceiling is thirteen bars with names The key-finder's two cues stop buying anything at about 93 per cent of scheme bars read right, and the obvious suspect was the chord segmentation feeding it. The reading was never given a segmentation: every bar arrives with its true chord. What the ceiling is made of can be listed instead, and it is thirteen bars of 188 — seven in a bridge of secondary dominants, four in a minor episode whose chords C major also owns, and two at the edges of a modulation. No weight of any cue moves one of them. The one number the ordered key-finder was tuned on. For each rate of alternation between two keys, the cost of changing key at which the model stops hearing two keys and starts hearing borrowed chords in one. The threshold rises with the period — 0.95 at 1 bar, 0.95 at 2 bars, 0.95 at 4 bars, 2.00 at 8 bars, 3.50 at 16 bars — so the parameter and the rate trade off against each other exactly. The value tuned earlier, 2.2, sits above every threshold on this axis, which means its verdict about fast alternation was a consequence of the tuning rather than a finding about the music. Filled means the model names two keys; hollow means it names one and calls the rest borrowings. A modulation and a borrowing are one number apart The key-finder that keeps the order has one tuned parameter, and said so. Sweep it and the parameter turns out to be the whole verdict: below a threshold the model hears two keys alternating, above it one key with borrowed chords. The threshold rises with how slowly the keys alternate — and the value it chose sits above every threshold in range, so its finding about fast alternation was a consequence of the tuning. The circle of fifths. The twelve pitch classes arranged so that each is a fifth above the last. Keys next to each other differ by one sharp or flat, which is why the notes of a key form a contiguous arc rather than a scattering. Keys are neighbours, and the map is computed Two keys a fifth apart share six of their seven notes. That single fact generates the circle of fifths, the key signatures, the shape of nearly every modulation, and the sense that some keys are far away.

The objects named here

The third way in, after the field and the series: the things themselves, and every essay that touches each one.

Dominant seventhEvidenceKey-findingModulationScale degreeTonal function