Key-relations — the series · Acoustics & Music Theory

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.

part 1 · scales

The key plan is the shape. The key of a classical sonata-form movement against position in the movement, measured in steps along the chain of fifths from the home key. The positions are the proportions such a movement is described by rather than bar numbers from any one score. The furthest point is 4 steps out, sharing 3 of seven notes with home.

The key plan is the form

The large shape of a classical movement is not a shape at all, it is a journey — out to one key and back. Which key is not a matter of taste. Of the two keys that share six of their seven notes with home, only one introduces a note the home key does not use in any of its chords, and that note is the arriving key's own leading note. The departure is audible because of one accidental.

part 2 · form

How far every key is from C major. The 11 other major keys under three measurements. Notes in common and steps round the circle are the same measurement — the first is seven minus the second until it bottoms out at two — and voice-leading distance between the tonic triads is not. E major shares 3 notes and is 2 semitones away; A♭ major shares 3 notes and is 2 semitones away, against the dominant's 6 and 3.

Three ways to measure how far a key is

Notes in common and steps round the circle of fifths are the same measurement — among the twelve major keys the first is seven minus the second until it bottoms out. Voice-leading distance between the tonic triads is a different one, and it disagrees in exactly one place: E and A♭ are four steps from C, share three of its seven notes, and are two semitones away, which is nearer than the dominant.

part 3 · harmony

How many bars a key change takes to be heard. A twelve-bar progression that moves to G major at bar 6, read by the same correlation against all twenty-four profiles, with a window of 3, 4 and 8 bars. With 3 bars of history the new key is never the answer at all. With 4 bars of history the answer is G major from bar 7, one bar late, and it holds it from there. With 8 bars of history the answer is G major from bar 9, 3 bars late, and it holds it from there. The pivot bar is ambiguous by construction — it belongs to both keys, which is what makes it a pivot — so the lag is not a defect of the algorithm but a statement about how much evidence a key is.

How much evidence a modulation needs

Run a key-finder bar by bar over a progression that moves to the dominant at bar six. With four bars of history the answer becomes the new key at bar seven and holds. With three bars it never gets there at all, and reports E minor and B minor on the way. The window decides the lag as much as the music does.

part 4 · perception

The map is two-dimensional and the circle is one axis of it. Major keys along the top, each a fifth from the last; underneath each, its relative minor, which shares all seven of its notes. Moving sideways changes one note; moving down changes none at all and changes the tonic. Among the twelve major keys alone there is only the sideways move, which is why that map really is a circle — the second axis needs the minor keys to exist.

The circle is a circle, and the map is not

Among the twelve major keys, notes in common is a strict function of distance round the circle of fifths — one value for each step count, no exceptions — so there is nothing else to measure and the map really is one-dimensional. A second axis appears only when the minor keys are added, and it is a different kind of move: the relative shares all seven notes and the parallel is one semitone away.

part 5 · scales

A model that cannot say two keys does not say it is unsure. I – IV – V – I played in C major and in a second major key at the same time, with the second key moved round the circle of fifths. For each separation: the correlation the standard key-finder gives its single best answer, and the correlation reached by the best PAIR of key profiles — a hypothesis the finder does not have. The pair recovers both keys that are sounding at every separation, 7 of 7. The single answer names neither of them at 4 of the 7, and its confidence does not fall when it is wrong: at four steps apart it reports E minor at r = 0.886, against 0.959 for the same progression in one key.

Two keys at once

Every key figure so far assumes one key is sounding, and the standard key-finder has no value it can return that means two. Play one progression in C and the same progression a major third away at the same time, and the model does not report uncertainty: it reports E minor, at a correlation of 0.886, against 0.959 for the same progression in one key. It is as confident as it ever is, and neither key it names is being played. Give it the missing hypothesis — pairs of key profiles rather than single ones — and it recovers both keys at every separation, all seven of seven.

part 6 · harmony

The alternation a key-finder cannot follow

Two keys sounding together are not in the key-finder's vocabulary, and an earlier essay ended by pointing at the other case and saying what was missing: a passage whose alternation rate can be varied while everything else is held still. Built, it gives a rule with three numbers in it — the second key is never named below a block of three bars, the tracking clears ninety per cent above twice the window, and the reading is late by half the window throughout.

part 7 · harmony

A key-finder that keeps the order

Every key-finding model until now begins by throwing order away — a histogram over a window, correlated against twenty-four profiles — and the thing that most obviously declares a key is an ordered pair of chords. A model that keeps the order costs 84 states and 7,056 transitions against 24 hypotheses and none, and what it buys is the one test a histogram was said not to pass: two keys in turn and two keys at once have histograms 98 per cent alike, and are two different sequences.

part 8 · harmony

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.

part 9 · scales

How sure the reading is, bar by bar. Every earlier essay reports one best reading. A dynamic program that finds a best path has, by construction, the best score into every state at every bar — so the gap between the best reading and the best reading in any other key is already computed and has never been printed. Here it is, in bits, for the thirty-two-bar AABA. The mean margin is 1.14 bits and 13 of 32 bars are inside one bit of a rival reading, which is where a listener would be genuinely undecided. The reading itself names C, E, B, D, G; the margin says what that naming is worth, and at the weakest bar — bar 31, C over F — it is worth 0.07.

The margin the dynamic program already had

Nine earlier essays produce a single best reading, and the passages worth arguing about are the ones where two readings are nearly equally good. What is needed for that has been inside the model from early on: a dynamic program that finds a best path has, by construction, the best score into every state at every bar — so the gap between the best reading and the best reading in any other key is already computed, and printing it turns every analysis here into a measurement of ambiguity.

part 10 · scales

How much of the reading comes from what has not happened yet. Every margin reported earlier is two-sided: the best path through a key at a bar is the best score into it plus the best score onward from it, and the second half uses bars a listener has not heard. Dropping that term is one line, because the dynamic program already had both halves separately. The mean margin falls from 3.90 bits with hindsight to 1.79 without it, so 54 per cent of this passage's certainty is retrospective. The two passes never disagree about which key is best here, so the hindsight buys confidence rather than a different answer. This is the quantity every earlier essay has assumed and none has measured.

How much of the reading arrives late

Every margin reported earlier is two-sided: the best path through a key at a bar is the score into it plus the score onward from it, and the second half uses bars a listener has not heard. Dropping that term is one line. On a thirty-two-bar song it removes more than half the certainty, and on a passage built to be ambiguous it changes the key named at nine bars out of eleven.

part 11 · scales

A listener has a quarter of an analyst's confidence and the same answer. The mean margin between a passage's best two key readings, against how many bars a listener's memory of the evidence takes to halve. The two-sided reading — the one that uses bars that have not happened yet — sits at 3.90; the forward pass with perfect recall at 1.79; a forward pass whose evidence halves every 6.6 bars at 1.00. The dots' size is how often that reading names the same key as the two-sided one: 100 per cent at perfect recall and 81 at a one-bar half-life. So forgetting costs a great deal of confidence and very little accuracy — the key is robust and the certainty is not.

The listener who forgets

Setting an analyst's reading of a key against a listener's measures what arrives late. Both passes assume perfect recall of their own half — which is as wrong going forward as knowing the future is going back. Put a decay on the forward pass and a listener with a memory of a few bars keeps a quarter of the confidence and nine tenths of the answers.

part 12 · harmony

Which bars the key is decided by, and which bars it is believed on. Every bar of a 32-bar scheme removed in turn, with what its absence costs. The column is how far the passage's mean margin falls without that bar — how much of the model's certainty it supplies. The dot is how many bars are then read as a different key — how much of the answer it supplies. The 18 bars an analysis would point at — a section opening or closing, a dominant, the chord a dominant resolves to — average 0.079 bits of certainty and 0.50 bars moved; the 14 ordinary bars average 0.047 and 0.07. So the structural bars carry 1.7 times as much certainty as the ordinary ones, and 7.0 times of the answer. Those are different quantities, and the second is the one an analysis is about: an ordinary bar can carry a great deal of a passage's certainty and none of its reading.

The bars a key is made of

A discount that treats every bar alike is the wrong shape for a memory, so take each bar away in turn and see what it was worth. The bars an analysis points at carry seven times as much of the answer as the ordinary ones and slightly less of the certainty — and a memory built only of them reads the passage worse than a memory with no structure in it at all.

part 13 · harmony

The twelve keys a key-finder has never had. Every scheme the key-finder reads, read twice: over the twelve major collections the model has always used, and over twenty-four with the harmonic minor added. The pale bar is the first and the dark one the second. On 5 of the 6 the extra twelve states change nothing a reader would see — the largest loss of certainty is 3.2 per cent, on the rondo — and no bar of any of them is renamed. The exception is the ostinato, every bar of which the twelve-collection model calls E♭ and the twenty-four-collection model calls C minor — the same seven notes, the right name. So the missing states were not costing the key-finder its answers. What they were costing is the ability to say which of a collection's seven degrees is home, and that is a different repair.

The key-finder with no tonic

Thirteen essays of key-finding have run over twelve major collections and not twenty-four keys, so a passage in A minor is read as C. Adding the missing twelve costs almost nothing and fixes almost nothing — because the model has no tonic in it at all, and below three raised sevenths in a passage the extra states are worth exactly zero.

part 14 · harmony

A tonic bought with the function

Putting the measured probe-tone profile inside the ordered key-finder is one term, and it does what was predicted: the natural-minor passage is named A minor at every bar instead of C major. It also does two things nobody predicted. It renames a scheme that has been read in the wrong key at every bar since the day it was written, and it destroys the chord's function while it is buying the key.

part 15 · harmony

Two cues meet in a corner

The profile finds a key's tonic and the bass finds a chord's degree, and until now each was swept with the other held at nothing. Swept together across 42 settings, the plane they make is not the ridge that was predicted. The tonic is a step in one direction, at a profile share of 0.55, and the bass cannot move it; the degree is a slope in the other, rising to 89 per cent as the bass is weighted, and the profile barely touches it. Every question is answered only in a corner of the plane — and the one place the two cues overlap is the one piece of music both can rescue.

part 16 · scales

A bass line is not a list of roots

Every bass note the key-finder has been given was its chord's root, and under that line a bass cue reads 89 per cent of scheme bars on the right degree. Give the same chords an economical bass that moves to the nearest chord tone, as a keyboard reduction would, and nearly half of them are inverted. The cue that rewards the triad rooted on the bass then reads 68 per cent at best and worse as it is trusted more; the cue that rewards any triad containing the bass cannot be fooled and stops at 61. The same inverted line does one thing the roots never did: it puts the leading note of each new key at the bottom, and finds the rondo's modulations.

part 17 · scales

Given the bar in octaves, the cue gets the degree back. How often the reading names the right scale degree, against how much the bass cue is worth, under three rules for what a bass note rewards. the bass is the root: 68 per cent at best; the bass is some chord tone: 67 per cent at best; the bar in octaves: 92 per cent at best; the bass is the root, on a root line: 89 per cent at best. The published cue rewards the triad rooted on the bass, which is right only when the chord is in root position; rewarding any triad containing the bass is right always and rewards three chords a bar instead of one. The third rule is not a bass cue at all — given the bar voiced in octaves a listener knows which three pitch classes are sounding, and that names the chord outright. It reaches 92 per cent on a real line against the published cue's 68 on the same line, and the dashed curve is what that cue manages on a line made entirely of roots — 89 per cent.

Given the bar in octaves, the degree comes back

A bass note is a bare pitch class in the usual figure, and a register in any realisation anybody plays. Voice each bar in octaves and a listener knows which three pitch classes are sounding and which is at the bottom, which names the chord outright — and the rule that uses it reads the right scale degree in 92 per cent of bars on a real bass line, against 68 for the published cue on the same line and 89 for that cue on a line made entirely of roots. The question was whether the register recovers the 89. It recovers it and passes it.

part 18 · scales

A sharper cue is worth nothing to a reading that will follow it anywhere. How often the reading names the right scale degree, against how much it costs to change key, at a bass worth 1.5 on a real bass line. the bass is the root: 20 per cent at a key cost of 0.5 and 68 at 4; the bass is some chord tone: 47 per cent at a key cost of 0.5 and 59 at 4; the bar in octaves: 46 per cent at a key cost of 0.5 and 91 at 4; roots, and a line of roots: 49 per cent at a key cost of 0.5 and 91 at 4. Where a key change is cheap the rule that names the chord outright reads no better than the rule that names three — 46 against 47 per cent — because a reading that will move key for one bar's evidence follows a sharp cue wherever it points. The sharper cue's whole advantage appears only once the reading is reluctant enough to stay put, and by a key cost of 2.2 it is 31 points ahead.

A sharper cue is worth nothing to a reading that moves

The rule that names the chord outright reads 92 per cent of scale degrees right where the published bass cue reads 68 — at the key cost these readings have always been run at. Sweep that cost and the advantage is not a property of the cue. Where a change of key is cheap the sharp rule reads 46 per cent and the vaguest rule 47, because a reading that will move key on one bar's evidence follows a sharp cue wherever it points. The cue's whole value is borrowed from the model's reluctance to be moved.

part 19 · scales

The two parameters turn out to have a ceiling between them

The essay before this one asked whether the bass cue's weight and the cost of changing key are one quantity with two names, and said the test was a contour: if they multiply, the curves of equal degree share are hyperbolae. They are not. Along the ninety per cent contour the product of the two runs from 2.7 to 15.2, because past a bass weight of about one and a half the reading saturates and more cue buys nothing. And the sweep finds something no single-parameter sweep here could: the key cost has a best value, and a reading that will never change key is worse than one that will.

part 20 · scales

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.

part 21 · scales

Relabelling the bridge does not rescue it. The 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.

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.

part 22 · scales

Told where each dominant goes, the reading splits every bridge chord between arrival and departure. The thirty-two-bar song's eight bridge bars — two each of E7, A7, D7 and G7 — at a key cost of 3 and a bass weight of 3, read without the resolution cue and with it at a weight of 3, where a bar read as the dominant of a key earns the weight when the next bar is rooted on that key's tonic. Without: E I, E I, E IV, D V, D I, D I, D IV, C V. With: A V, A V, A I, D V, D I, G V, G I, C V. Read as the key its dominant points at, root only: 2 of 8 without and 5 with; read as a dominant at all, 2 and 5.

Looking ahead rescues the relative minor, not the bridge

A key-finding reading that scores each bar by its own notes cannot hear an applied dominant, whose meaning is in the bar after it. Given one cue about a pair of bars — a dominant earns credit when the next bar is rooted on its key's tonic — the thirty-two-bar song's bridge is read as the chain of dominants it is, each chord both the arrival of the last and the departure to the next. But the same cue hears a blues's opening tonic seventh as the dominant of its own IV, and at the setting that read the listing best it costs five bars for three. The cue earns its place only in a reading reluctant to change key, and there what it rescues is the rondo's episode in the relative minor.

part 23 · scales

No setting of the two costs reads both the bridge's chain and the rondo's episodes. The six schemes read by a key-finder with a second slot for an applied dominant, at a bass weight of 3 and a resolution cue of 3, over every key cost and every cost of entering a tonicization: in each cell, the bridge's bars read as the scheme labels them (of 8), the rondo's (of 40) and the whole listing's (of 188). key cost 2: 0.5 4/24/163, 1 4/24/156, 2 1/24/153, 3 1/28/154, 4 1/34/160, 5 1/34/160, 6 1/34/160, 8 1/34/160, 12 1/34/160; key cost 3: 0.5 8/24/172, 1 8/24/172, 2 4/24/163, 3 1/24/160, 4 1/34/170, 5 1/34/170, 6 1/34/170, 8 1/34/170, 12 1/34/170; key cost 4: 0.5 8/24/172, 1 8/24/172, 2 4/24/168, 3 3/24/162, 4 1/24/160, 5 1/35/171, 6 1/35/171, 8 1/35/171, 12 1/35/171; key cost 6: 0.5 8/24/172, 1 8/24/172, 2 8/24/172, 3 8/24/172, 4 3/24/167, 5 2/24/166, 6 2/30/172, 8 2/35/177, 12 2/35/177; key cost 8: 0.5 8/24/172, 1 8/24/172, 2 8/24/172, 3 8/24/172, 4 3/24/167, 5 2/24/166, 6 2/24/166, 8 2/30/172, 12 2/34/176; key cost 12: 0.5 8/24/172, 1 8/24/172, 2 8/24/172, 3 8/24/172, 4 7/24/171, 5 5/24/169, 6 5/24/169, 8 2/24/166, 12 2/24/166. Every cell that reads the whole bridge reads the rondo at 24 of 40, its two eight-bar episodes heard as tonicizations of V and vi inside C; every cell that reads the episodes as modulations reads at most two bridge bars. The one-slot reading at a key cost of 6 reads 177.

A second slot hears every modulation as a tonicization

A key-finder that names one key per bar reads a chain of applied dominants by moving the key with every chord. Give its state a second slot — the key a bar is in, and the key its dominant points at — and the thirty-two-bar song's bridge is read exactly as the scheme labels it, every bar, for the first time in these essays, and the blues's opening stays in its tonic. At every setting that does that, the rondo's two eight-bar episodes are heard as tonicizations inside C, and no setting of the three costs reads both. A modulation to a closely related key and a long tonicization are the same chords; what separates them is how long they last, and a cost for entering a state cannot count.

part 24 · scales

A modulation is a dominant that keeps coming back

A key-finder with a slot for an applied dominant read the thirty-two-bar song's bridge as its scheme labels it and heard the rondo's two episodes as tonicizations inside C, and no setting read both. The obvious repair — charge a tonicization by the bar, so a long one costs more than a short one — makes it worse: the reading gives up the bridge before it gives back a single episode, because the bridge spends six bars in applied chords and the episodes three and two. By the bar, the chain is the longest tonicization in the listing. What the episodes have and the bridge does not is a dominant that resolves to the same tonic again and again.

part 25 · scales

Three ways of remembering where the dominants went, and none reads both the bridge and the episodes. At a key cost of 6, a resolution cue of 3 and a tonicization cost of 0.5: the song's bridge (of 8 bars), the rondo (of 40) and the whole listing (of 188) read as the schemes label them, for a reading that charges an applied dominant resolving again to the target the last one reached — with that memory lasting the whole home key, or four bars — and for one that instead pays a bonus whenever a key's own dominant resolves to its own tonic. a return charged, remembered through the key: 0: bridge 8, rondo 24, all 172; 2: bridge 8, rondo 24, all 136; 4: bridge 8, rondo 24, all 155; 8: bridge 8, rondo 16, all 147; 12: bridge 8, rondo 16, all 147. a return charged, remembered four bars: 0: bridge 8, rondo 24, all 172; 2: bridge 8, rondo 24, all 172; 4: bridge 8, rondo 24, all 172; 8: bridge 8, rondo 24, all 172; 12: bridge 8, rondo 24, all 172. a confirmation paid: 0: bridge 8, rondo 24, all 172; 2: bridge 8, rondo 24, all 149; 4: bridge 8, rondo 24, all 116; 6: bridge 2, rondo 29, all 115; 8: bridge 2, rondo 28, all 104. The rondo's episodes are recovered only once, at a confirmation of 6, and by then the bridge has fallen to 2.

A return is cheaper to avoid than to pay for

The essay before this one found what separates a modulation from a tonicization: an episode's dominant keeps resolving to the same target, and the bridge's dominants never do. So the key-finder was given a memory of where its last dominant went, and charged for a return — the prediction being that one setting would at last hear the episodes as keys and the bridge as a chain. None does. Every return the charge is aimed at is avoided for less than a nat, by hearing a D7 as a minor chord or by moving the passage into another key, while the reading the charge was meant to force costs twelve.

part 26 · scales

The chromatic notes the reading already hears, and the ones it does not. Every note in the listing's six schemes that lies outside the key the reading hears it in, counted once a bar, by scheme, and how many of them are notes of the chord the reading hears in that bar: 32-bar song 6 of 6, rondo 5 of 5, blues 0 of 30 — 11 of 41 in all. Every chromatic note of the song's bridge and the rondo's episodes is the third of an applied dominant the reading already hears, and so already accounted for. All 30 it leaves unaccounted for are the blues's sevenths.

The leading note is already inside the chord that hides the key

A key-finder kept hearing a rondo's episodes as passages inside C, and the escape it used was to hear a D7 as a D minor chord, throwing away its F♯. Charge a chromatic note by how strongly it points at another key, and that escape closes. Nothing else moves, because every F♯ and G♯ in the episodes was already inside the chord the reading heard — as the third of V of V. The charge takes a chromatic note seriously everywhere except where it was needed, and in the blues it mistakes a flattened seventh for a leading note.

part 27 · scales