Progression — the series
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A progression is a path, and the map can be drawn
Chords in a key are not a list. They sit at measurable distances from home, and a progression is a walk across that space — which explains why some sequences feel like journeys and others like wandering.
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The progression that never comes home
Just intonation is usually said to fail because a keyboard has only twelve keys. That is not the reason. Take one of the commonest chord sequences in tonal music, play every chord in exact whole-number ratios, and it arrives back on its starting chord a syntonic comma flat — on any instrument, including a voice.
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How often the chord changes
Two pieces can use the same chords in the same order and be nothing alike, because a progression says which chords and not how fast. Harmonic rhythm is the second variable, it runs over a factor of thirty between the styles that use it, and both of its limits are set by things that are not harmony — a listener's memory at the slow end and a building at the fast one.
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Which notes are the chord
A progression is a list of chords, and before there is a list something has to decide which of the notes sounding are chord tones and which are passing. Take the eight notes of a scale as eight quavers and score every triad and seventh at every root: barred as written the best reading is C major seventh, with the barline moved by one quaver it is D minor seventh, and with no metre at all three readings tie exactly and the passage has no best analysis. Same eight notes in all three. Harmonic analysis is a function of a variable that is not harmony.
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Three decisions that constrain each other
Every model here decides one thing at a time — the key from the pitch classes, the metre from the onsets, the chords from the metre — and an earlier essay ended by saying a listener does all three at once. Resolving them jointly costs a hundred and fifty-seven times the search and changes the reading of two passages in five. It never once changes the key, and the reason it cannot is the reason the whole account is built the way it is.
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The cadence as evidence
Every model so far infers a key from a bag of notes, and the thing that most obviously declares a key is a cadence — an ordered pair of chords in which the order is the whole content. The essays on closure built a five-component cadence vector long before this and nobody has used it for this. Two passages differing in one note, one cadencing in C and one in G, get opposite answers from the ordered pairs and the same answer from the profile.
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A count and a correlation
Cadence evidence is a count of ordered pairs and profile evidence is a correlation with a template, and the cadence essay refused to total them because they are not in the same units. Score each against its own chance and they are — standard deviations above chance are the same unit whatever produced them. Then the trouble moves: the obvious null does nothing at all to one of the two, because shuffling the bars leaves a pitch-class histogram exactly as it was.
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The passage built to make them disagree
A cadence count and a key-profile correlation have been put on one scale and run on a passage where the two agree, which tells nobody anything. Building one where they disagree — the notes of one key and the cadences of another — measures the exchange rate at about six bars per cadence, and finds something the agreement case concealed: the two statistics cannot be varied independently, because lengthening the profile evidence adds cadence evidence too, for a third key.
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The long note and the strong note
The segmentation that produces every object connected here has carried a free parameter since the day it was written: whether a note counts for its metrical weight or for how long it is held. Only the first has ever been drawn. The two name different chords on sixteen per cent of passages where the cues agree about the notes and forty-three per cent where they do not — and where they disagree most sharply a mixture of them picks a third chord neither one asks for.
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A fourth decision, and two that were never made
The joint search resolves key, metre and segmentation together and holds the segmentation's cue mixture at zero. Adding the mixture is one loop, and reading the search in order to add it turns up something worse than a missing axis: on the passages it is drawn on, the key it reads is the same key at all forty-eight of its hypotheses and the metre scores every barline identically. The fourth axis then cannot be ranked at all until each reading is measured against its own null, because a mixture changes the ruler and not only the answer.
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The chords never move the barline
Every hypothesis the joint search had drawn was one bar long, and on one bar with a note in every slot the metre cannot choose a barline at all. Four bars with rests in them make the barline a decision the metre and the chords both have an opinion about, and the prediction was that the chords would move the barline more often than the barline moves the chords. It is the other way round, completely: whenever the two prefer different barlines the search takes the metre's, on up to 72 per cent of passages, and in fifteen hundred passages the chords never once move it. What the chords decide is the one thing the metre cannot see — whether the bar starts on the downbeat or half a bar later — and they decide it right a little over two times in three at best.
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The chords are a weak witness to the barline
Scaled by its own range, the metre overrules the chords every time the two disagree about where a bar begins. The obvious repair is to score each reading against its own chance — the metre against the same number of notes placed at random, the chords against the passage's notes shuffled across its bars — and add the standard scores. It changes very little: the search finds the barline within six points of where the product found it, and the chords gain the power to move the barline only on passages whose rhythm says nothing, where random notes move it nearly as often. The null's real result is the size of the two witnesses. At the written barline the metre stands up to 5.9 standard deviations above chance, and the chords, with every note a tone of its bar's chord, stand 1.55 above it at best.
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The chords mark the barline by changing there
Read bar by bar, the chords stood barely above chance at the barline and broke the metre's half-bar tie two times in three at best. Read instead by where they change — how different the chords are across a candidate's barlines against how different they are across the middle of its bars — the same notes break the tie right on 81 to 96 per cent of passages, and added to the metre they find the barline on up to 89 per cent against 61. The weakness was the question the old reading asked, not the harmony.
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The change reading follows the chords, not the bar
Every passage read until now changes chord exactly at the barline, which is the one harmonic rhythm at which 'the chords change here' and 'the bar starts here' are the same sentence. Pull them apart and the reading goes with the chords: at one chord a bar it stands 1.52 standard units above the other candidates and finds the barline half the time, at two chords a bar it stands 0.01 above them and is at chance, and at a chord every two bars its margin is exactly half — because half the barlines then carry no change at all.
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Asked for the rate, it answers a multiple
A reading that follows the chord rate rather than the bar can be asked what the rate is, and the shape of its errors is the whole of why it looked like a barline detector. Given two chords a bar it returns the right period a third of the time and something slower two thirds; given a chord every two bars it is right nine times in ten. It errs slow and essentially never fast, because a change every four slots also falls on every eighth slot and a slower grid inherits a faster rate's evidence — which is the same asymmetry that makes a pitch detector report an octave too low.
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The bass errs fast where the content errs slow
Asked how often the chords change, a reading built on pitch-class content names a slower multiple and never a faster rate. Give the passage a bass that states each new root and moves between chord tones inside a chord, and a reading built on the bass's moves errs the other way: it names a faster grid and never a slower one. At two chords a bar the bass is right every time; at a chord every two bars it is never right. Six ways of combining the two readings each trade one end of the range for the other.
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A bass that holds through a change marks the barline
At two chords a bar the chords change on the barline and on the half-bar alike, so a reading of where they change is at chance, and the metre ties the two. The bass has one more piece of evidence: a change inside the bar can be voiced over the note already sounding, and a change on the barline is voiced over its root. Hold the bass through one mid-bar change in six and the metre and bass together place the barline in 85 per cent of eight-bar passages; one in three, 97. The convention cannot be stronger than that, and the chord changes, asked first, only get in the way.
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A walking bass marks the bar by the note it is on
The reading that finds a barline from where the bass moves collapses to twenty per cent when the bass moves on every crotchet, and the bass the last essay called busiest moved twice a bar, because its beat was the half-bar. Given one that really walks, the same computation with sounds the chord's root in place of moves rises from nothing to a whole note a bar exactly as the other falls, and the two are equal at a bass moving on three crotchets in eight. Neither places a barline alone: the root is still sounding on the quaver after it. The metre, which cannot tell a barline from a half-bar, tells it from the quaver after it at once.
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A texture in which nothing says where the bar begins
Two chords a bar over a bass that moves inside the chord is the one arrangement in which every witness this account has stands together at the metre's coin toss. The metre ties the barline with the half-bar, the chords change at both, and the bass states a root at both and walks between both — at every busyness, and whichever question the bass is asked. The only thing that breaks the symmetry is the voicing convention, and what it is worth falls from 98 per cent to 90 as the bass fills in, not because fewer changes can be held but because the motion around them is evidence for nothing. Chasing five points of residue above the toss found a defect that had been flattering the reading since it was written.
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The leap at the barline is a fact about register
A bass leaps to a root at the barline and steps inside the bar, and no reading of pitch classes can see it — so the essay that named it recorded it as a limit. Given octaves and the size of every motion, it is worth nothing: realised at least motion the reading tracks whether the bass moved at all and ends 0.87 semitones on the wrong side of zero, because a line that always takes the nearest octave has no leaps to take. Play a stated root at the bottom of the compass instead, and the same sixteen notes in the same order turn a margin of minus 1.67 semitones into plus 1.67 and hold at 90 per cent where the move reading has fallen to 20. The barline is not where the bass moves furthest. It is where the bass goes back down.
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A melody marks the barline by where it lingers
At two chords a bar over a walking bass, every harmonic witness stands at the metre's coin toss about where the bar begins. Give the passage a melody with a habit — long notes that prefer the downbeat — and the barline comes back: with a long note in half the bars and four in five of them on the downbeat, in 91 per cent of passages. The count of long notes barely matters; the preference is everything, and at even odds no number of long notes helps. A phrase end held through its bar is the strongest single cue. And a listener who adds the bass's readings to the melody's does worse than one who ignores the bass.
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A syncopated melody needs its phrases to land
Put a melody whose long notes all fall on the half-bar over a harmony that changes once a bar and a walking bass, and a listener adding up the evidence hears the bar begin in the wrong place three times in four. The harmony knows where the bar is, but only through one witness: the bass's root votes for the barline in every passage, and the bass's motion never does — a walking bass lands on each new root by step, so it moves less at the barline than inside the bar. One syncopated voice ties the root, and the motion breaks the tie its way. What rescues the barline is not the harmony but the melody's own phrase ends: hold half of them from the downbeat and the reading finds the bar in 86 per cent of passages.
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Two bars of drum hold the barline; a dozen bars of bass do not
A listener who settles where the bar begins and then holds it hears a syncopated tune against the bar only if the ground before it built enough confidence to spend. Read bar by bar, a walking bass under one chord a bar is worth less than half a vote a bar for the barline, and a syncopated bar takes an eighth of a vote back, so a tune needs about twelve bars of that ground to be heard against the bar nine times in ten. A drum on the downbeat makes each bar of ground worth a vote and a half, and two bars are then enough — but only for a listener who can bank more than four votes of confidence.
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A listener who weights its witnesses needs eight bars of bass, not twelve
A syncopated tune over a walking bass needed about twelve bars of ground before a listener with a memory heard it against the bar nine times in ten, with every witness given one vote a bar. Let the listener weight each witness by how often it has agreed with the reading it holds and the root earns a second vote, the bass's motion loses half of one, and the syncopation loses almost all of its. Eight bars are then enough, a tune that syncopates every bar becomes hearable against the bar at all, and reading the syncopation backwards buys nothing that ignoring it does not.
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A listener who knows the style needs no introduction
A listener who weights each witness to the barline by its record needed eight bars of walking bass before a syncopated tune was heard against the bar nine times in ten, and with no introduction at all it learned the syncopation's lesson instead. Finer records — one for each side a witness takes, one for each section — change nothing. What changes everything is arriving with the style's habits: a listener who already gives the bass's motion and the melody's long notes no vote hears the tune against the bar 93 per cent of the time from its first bar, and 94 per cent for a tune that syncopates every bar. The root need not be trusted more. The others need to be trusted less.
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The habit worth bringing is to distrust the tune
A listener who arrives with a style's habits — the bass's motion and the melody's long notes credited at a coin toss — hears a syncopated tune against the bar from its first bar. But a listener does not know the style in advance. Give it two sets of habits, one for tunes that mark the bar and one for tunes that syncopate, and let it weigh them by how well each predicts what it hears: it does better than a listener with no habits and far worse than one who simply knows, and it becomes sure of the style in fewer than one passage in ten. The reason is that the evidence runs one way. A tune that marks the bar never testifies against the habit of distrusting tunes, so the habit that serves every style is the syncopating one, kept always.
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A late root is believed because the record is kept by the reading
The habit worth bringing to a tune turned out to be trusting the bass's root and doubting everything else. A bass that anticipates the next chord does not break that habit — the root is still tied across the barline, and it is the bass's motion that lies. A bass that puts the root at the half-bar does break it, and nothing a listener learns can repair it, because each witness is judged by whether it agrees with the reading, and the reading was made by the liar.
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Two honest bars convict a lying root
A bass that states its root late defeats every listener who learns from agreement, because the reading forms before any honest witness speaks. Give the passage a drum on its first downbeats and the same record that ratified the liar convicts it — but only a listener holding a habit that doubts the root can act on the verdict. Two strokes are enough for that listener, for as long as the tune lasts. One is not, and no amount of drum is enough for a listener that credits the root and the drum alike.
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An honest drum has to come first
Two strokes of an honest drum at the very start of a passage protect a listener from a bass whose root arrives late in every bar. The same two strokes a bar later do nothing at all, and neither does a drum that then plays every downbeat to the end. A listener that keeps a record of its witnesses scores the late drum against the reading the root has already made, and convicts it within two bars. Every bar the lie is believed first multiplies the trust the drum would need, and by the seventh bar no trust short of certainty is enough.
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A break rescues a late drum only by ending the reading
A drum that enters after a late root has made the reading needs odds in the hundreds to undo it. Bring it in after a break in the texture, and the obvious guess is that the break helps by making the listener forget whom it trusted. It barely does. What rescues the drum is the listener forgetting what it heard: with the old reading gone, a drum on bar 9 needs odds of nine to one, and with the record gone too it needs less than the three to one it has always had. A break that leaves even a hundredth of the old reading behind rescues nothing at all.
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A silent break has to outlast the reading's lead
A listener waiting through a break goes on expecting downbeats at the reading it holds, and each one that fails to come can be charged against that reading. Charged at a witness's vote apiece, the silence ends the reading only after as many bars as the reading has votes in hand: three bars after two bars of tune, eight after four, twenty-one after eight. A one-bar break would need a missing downbeat to cost twenty votes. And a silence that does end the reading leaves the record standing, which convicts the drum anyway.
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A fill cannot end a reading, but it can win the next one
A drum fill in the break before a late drum is a witness that speaks while the old reading still stands, so every stroke it plays is scored as disagreeing, and its credit falls to a coin toss within two or three strokes: at three to one it can cast 1.37 votes in all, against a lead of twenty by the middle of a tune. A fill cannot end a reading. But once a silence has ended it, the same fill sets the next one, and the record that convicted the drum through every length of silence starts convicting the bass instead: after twelve bars of silence before bar 5 the drum goes from 7 per cent of passages to all of them.
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A cadence has to empty the reading, not weaken it
A drum fill after a long enough silence wins the next section for an honest drum, because it speaks into a reading that is empty. A cadence might do the silence's work in a bar. It does, but only on one condition: before bar 9, a cadence that removes ninety-eight per cent of the listener's conviction leaves a two-stroke fill winning one passage in a hundred, and one that removes all of it lets the fill win every one. The fill turns the reading either way. What the last sliver decides is whether the fill's first stroke is recorded as a disagreement — and that record is what the late root then uses to win the reading back.
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A cadence that votes must not vote as the drum
A cadence lands on a downbeat, so it could cast a vote for the barline at the moment it loosens the listener's grip. Whether that vote helps depends on whose it is taken to be. Heard as the tune's long note, it speaks with credit the tune spent long ago and is worth almost nothing. Heard as the drum, it is exactly one more stroke of fill, and is convicted as the fill's first stroke was. Heard as a witness of its own, it carries a third of a vote and no record to lose — and that moves the line from an emptied reading to a fiftieth of one.