Field

Harmony and voice leading

Chords as places, progressions as paths, and the smallest possible move between them.

Chords as stacked intervals. Each of 4 chords — major, minor, diminished, augmented — drawn as the semitone distances above its root, with the nearest simple frequency ratio beside each note where the 5-limit table has one. Where the chords are triads they differ only in the size of the two stacked thirds, and that difference is the whole of their character.

Three notes at once, and why these three

A triad is two thirds stacked, and the four ways of stacking them behave completely differently. The difference is entirely in whether the resulting ratios are simple.

How far the voices have to move. Three chord changes with their voices joined by the assignment that moves the least in total. The distance is computed over every way of pairing the notes, so a change that shares two of its three notes shows as two flat lines.

The shortest move, which is what a chord change is

Between any two chords there is an assignment of voices that moves the least. Compute it and the changes that composers actually use turn out to be the short ones.

The seven chords of a key, by distance from home. Each triad of the major scale placed at a radius equal to how far its voices must move from the tonic chord. The chords that feel closest to home are the ones that are closest, in the plain arithmetic sense.

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.

Every interval, and what it becomes when it is turned over. Each interval within the octave paired with its inversion — the interval left when the lower note is raised by an octave. The two always add to twelve semitones, so exactly one interval can be its own inversion, and it is the one at six.

The interval that inverts to itself

Six is the only number that divides twelve into two equal halves, so the tritone is the only interval unchanged by being turned over. That symmetry is not a curiosity — it is why one tritone belongs to two dominant chords, and why they resolve to keys a tritone apart.

The twenty-four triads, and the three moves between them. Every major and minor triad, placed on the single cycle that alternating two of the three transformations produces. Neighbours round the ring differ by one voice; the chords across the middle are the third transformation. Every edge's cost is the measured voice-leading distance, and none of them is more than two semitones.

Chords as a space

Three operations turn any triad into another by moving one voice. They generate all twenty-four major and minor triads in a single cycle, their costs are one, one and two semitones, and the map that results is a geometry rather than a list of rules.

Where a progression in pure ratios ends up. A short chord sequence taken in exact whole-number ratios, with the running difference from equal temperament measured in cents. The sequence returns to its starting chord and the pitch does not, which is the comma arriving in ordinary music.

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.

Which partials two notes have in common. The first 12 partials of a note and of the note an interval above it, on a logarithmic frequency axis, with every partial of the upper tone that lands on one of the lower marked. octave: 12 of 12, fifth: 6 of 12, fourth: 4 of 12, major third: 3 of 12, minor third: 2 of 12, minor second: 0 of 12. Sharing partials is what fusion is: two voices whose spectra mostly coincide stop being heard as two voices, which is a measurable property of the interval rather than a matter of taste.

Two voices that stop being two

The ban on parallel fifths is the most famous rule in Western music and it is usually taught as taste. It is not taste. At an octave the upper voice contributes no frequency the lower one did not already have, at a fifth it contributes half of them, and the number can be counted — which turns a prohibition into a measurement.

How surprising each chord is, in bits. Each step's information content, −log₂ of the probability the root-motion weights used here give it. a perfect cadence totals 6.4 bits over 3 steps; a deceptive cadence totals 7.3 bits over 3 steps; I – IV – V – vi totals 7.3 bits over 3 steps. The single most surprising move drawn is IV to V at 2.7 bits, which is 42 per cent of everything its passage spends. The eight weights are ordinal and stipulated rather than counted, so these are the numbers that ordering implies and not a measurement of any repertoire.

The chord that did not come

A deceptive cadence is described as a surprise, and the explanation offered is that the wrong chord arrived. Measured against the tonal hierarchy already in use, the wrong chord is the second best-fitting triad in the key — and two of its three voices do exactly what they would have done in the right one. The surprise is not statistical. It is one voice, and it is the bass.

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.

Every delay, scored — a round and a tune that is not one. Mean Plomp-Levelt roughness of the simultaneities a tune makes against a copy of itself entering a whole number of bars later, for Frere Jacques and Twinkle, twinkle. The two do not overlap: the worst delay of the first is smoother than the best delay of the second. The model sees roughness and nothing else — no voice leading, no parallels, no distinction between a passing dissonance and a structural one.

A melody that can accompany itself

Whether a tune works as a round is decidable before anyone sings it. Score every simultaneity it makes against a delayed copy of itself, at every delay, with the roughness model already in use, and two nursery tunes separate completely — every delay of one is smoother than every delay of the other. What the scan does not do is pick out the entry the tradition uses, and that refusal is the most informative thing in it.

The comma is conserved; only the place it is paid is chosen. Every policy available to a continuous-pitch ensemble on one line: drift per circuit along the bottom, worst mistuned interval up the side. The line is straight and its intercepts are fixed, because the 4 moves of the pump have to absorb 21.51 cents between them however they are shared. Spreading it evenly puts 5.377 cents on each interval — the same narrowing quarter-comma meantone applies to every fifth.

Somebody has to pay the comma

A choir has no frets and no keys, so it can tune every chord exactly — and a common progression sung that way arrives a comma flat, every circuit. The freedom a keyboard lacks turns out to be a freedom about where the error goes rather than whether there is one, and the two costs always add to the same number.

Where to put the third

Take three pitch classes, four octaves to put them in, and score all twenty-seven arrangements. The smoothest is root, fifth an octave up, third two octaves up — which is partials one, three and five of the harmonic series — and the roughest, at every register tried, is the chord in close root position at the bottom of the range. Every orchestration manual states that rule and none of them derives it.

What the rules cost

The prohibitions of counterpoint are constraints on a minimisation already computed here, so each one has a price in semitones of extra motion. Priced over every melody a progression admits, most of them turn out to be free, the dearest is not the famous one, and two of them cost no motion at all — they cost tunes.

Both criteria, measured from C major. The 23 chords other than C major placed by the smallest total voice motion and by how many notes are held in common. Agreement between the two criteria would be a staircase; the cells drawn in the discrepancy colour are the ones that break it, and over the whole set of 24 chords the two criteria order a pair of destinations oppositely in 72 of 6072 comparisons. A minimal assignment drops a common tone it could have held in 0 of the 552 pairs.

Two criteria that are taught as one

Move the least and keep the common tones arrive in one breath, as though they were one instruction. They are two different functions, and enumerating every pair of chords settles exactly where they agree: as a way of connecting two given chords they never once disagree, and as a way of saying which chord is nearer they disagree about one comparison in eighty.

Every triad, by how far it is from G7. All 24 chords ranked by the smallest total motion that takes G7 onto each, every note of the target sounded and one voice doubled where the sizes differ. The spread is narrow — 2 semitones at the nearest and 6 at the furthest — and the ranking is flat in the middle: 12 chords tie at four semitones.

The resolution the metric cannot find

Rank every triad by how far its voices are from the dominant seventh and the tonic is not at the minimum. It sits in a twelve-way tie in the middle, the chord a deceptive cadence goes to is the furthest away of all twenty-four, and the two nearest are chords no cadence ever uses. What selects the two resolutions is not a distance — it is a direction, and it picks out exactly two triads from the twenty-four.

The thirteen intervals at C4, ordered by roughness, against Fux's species, 1725. Every interval within an octave above 262 Hz, ordered by the Plomp–Levelt roughness of a string spectrum — from 0.0029 for the unison to 0.2635 for the minor second — beside whether Fux's species, 1725 counts it a consonance and how many of the thirteen may follow it under that rule set. The best single threshold on roughness misclassifies 2 of the thirteen, among them the major third and the minor third, and the gap it falls in is 4.2 per cent of the roughness either side of it.

A dissonance is what has to be resolved

The perfect fourth is a consonance between two upper voices and a dissonance against the bass, and the same three notes are involved either way. Score both arrangements with the roughness model and the one the rules call a dissonance comes out thirty per cent smoother. Whatever the rule is tracking, it is not the sound.

The triad on every degree of every rotation. The quality of the triad built on each degree of each rotation, computed from the set rather than tabulated. The triad on the fifth degree is major in 3 of the 7: Lydian, Ionian, Locrian. Of those, Locrian has no triad on its own first degree to resolve to, and Lydian has its fifth degree on the note the parent collection is built from, so the cadence would confirm the parent rather than the mode. What is left is Ionian — one rotation of the 7, and the others have to end some other way.

One dominant among seven

The triad on the fifth degree is major in three of the seven rotations. One of those three has no triad on its own first degree, and one has its fifth degree on the note the parent collection is built from — so exactly one rotation of the seven has a dominant, and the other six end by moves that cost twice as much in voice leading.

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.

Every 3-note stack of one interval. Each chord built by repeating a single interval 2 times from the root, scored for summed roughness on a string spectrum and asked whether one low fundamental accounts for all its notes within 30 cents. 8 of the 11 are smoother than the major triad, which scores 0.288: stacked major thirds (an augmented triad) at 0.279, stacked fourths at 0.218, stacked tritones at 0.144, stacked fifths at 0.140. And 3 of them have no fundamental at all — stacked minor sixths, stacked major sixths, stacked major sevenths — meaning no series of harmonics up to the 16th accounts for their notes to within a 30-cent tolerance.

A stack that is not thirds

Build every chord that repeats a single interval and score them all. Eight of the eleven three-note stacks are smoother than the major triad; the four-note stack of fourths is smoother than a dominant seventh by a third and has no fundamental any matcher can find. Two absences, both computable, and both are exactly what the chord was adopted for.

The same eight notes, read three ways. The scale, eight quavers, scored against every triad and seventh at every root. Barred as written the best reading is C major7 at 0.850; with the barline one quaver later it is D minor7 at 0.850. With no metre — every note weighted the same — 4 readings tie at 0.625 and the passage has no best analysis at all. The notes are identical in all three. What changed is where the bar starts, which is not a fact about harmony.

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.

What one progression leaves open. Realisations of I–IV–V–I in four parts with no parallel fifths or parallel octaves, counted exactly by a dynamic programme over the voicings rather than sampled. The chord symbols admit 16,100,352,296; the Roman numerals 59,418,496; the figured bass 2,042,672. The three notations differ by four orders of magnitude, and every one of them was in daily professional use.

Three notations, one progression

A figured bass, a Roman numeral and a chord symbol are three professional notations for the same four chords, and the number of four-part realisations each of them admits can be counted exactly rather than argued about. With no parallel fifths or octaves the counts are sixteen billion, fifty-nine million and two million: a factor of eight thousand between the loosest and the tightest. What each one collapses is what its tradition thought a chord was, and the three do not agree.

Why the exercise is in four parts

Eight earlier essays move four voices, and nothing here ever chose four. Cut one choir's compass into three parts instead and the two most famous prohibitions in music cost exactly nothing — the cheapest realisation already obeys them. Cut it into six and they cost more than half a semitone per voice per chord change, because a parallel octave needs a doubled note and six voices sharing three notes can hardly avoid one. Four is the smallest number of parts at which the rules have a price at all, and it is the largest at which the price is small.

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.

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.

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.

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.

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.

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.

Surprise is a number

The chord that did not come was described rather than measured. Its measure is the information content of what did arrive, and a model of the probability has been to hand since the key-finding essays — eight root-motion weights, ordinal and stipulated. Reading them as a distribution prices a deceptive cadence at 2.71 bits against a perfect one's 1.85, and turns up the fact that the largest of the eight had never been read by anything.

Roughness and loudness do not rise together. One four-note chord, played at levels from 35 to 95 decibels, with both quantities drawn as multiples of what they are at the quietest. Roughness is quadratic in pressure, so 60 decibels multiply it by 1.0e+6. Loudness is compressive — about ten phons to a doubling of sones — so the same range multiplies it by 96. The gap between the two lines is the quantity: roughness per sone rises by a factor of 1.0e+4 between a pianissimo and a fortissimo of the same chord.

The ranking survives the dynamic and the chord does not

Roughness is quadratic in pressure and loudness is compressive, so sixty decibels multiply a chord's roughness by a million and its loudness by ninety-six. Roughness per sone therefore rises ten thousandfold between a pianissimo and a fortissimo of the same four notes — and yet the ranking of which doubling is smoothest, over four hundred and eighty voicings, does not move by a single place.

The surprise of each chord, against the uncertainty it arrived into. The information content of each step — minus the log of its probability under a distribution that multiplies the root-motion weight by how well the destination triad's notes fit the key — with the entropy of the moment before it drawn behind. I – IV – V – I: I→IV 1.71 bits, IV→V 2.80 bits, V→I 1.60 bits, against a mean uncertainty of 2.63; I – IV – V – vi: I→IV 1.71 bits, IV→V 2.80 bits, V→vi 2.61 bits, against a mean uncertainty of 2.63. A surprise larger than the entropy it arrived into is an outcome the model was not expecting even given how uncertain it was; one below it is an outcome the model had already mostly bet on. An earlier essay produced the first of those numbers and had no way to produce the second, because a set of preferences is not a distribution and only a distribution has an entropy.

A chord, given a key and a predecessor

A chord's improbability has been priced from its root motion alone, which left one multiplication unmade: a chord is also improbable because its notes do not fit the key, and that number has been available since the probe-tone profile. Multiplied and renormalised, the two give a conditional distribution — and a distribution has an entropy, which is the quantity a surprise has to be read against and which a list of preferences cannot supply.

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.

Expectation as a curve, and what a change costs where it lands. Every quantity so far is attached to a chord change: a list of surprises, one per event. A listener's expectation is continuous — it sharpens through a bar and collapses when the change arrives — and the two ingredients for it are already here, the harmonic rhythm and the metrical beat weights. The curve is the hazard: given that the chord has not changed yet, the chance that it changes on this beat. It runs from 0.043 on the weakest beat to 0.290 on the downbeat, a ratio of 6.7, against 0.125 if every beat were alike. The marked beats are where the changes actually arrive, and their timing bill is 3.6 bits against 6.0 for a listener with no metre — so these changes are 1.7 times cheaper to expect than a metreless listener would find them. That term is new: an earlier essay prices which chord arrived and this prices when, and a listener meets the sum.

Expectation is a curve, not a list

Every quantity so far is attached to a chord change: a list of surprises, one per event. A listener's expectation is continuous, sharpening through a bar and collapsing when the change arrives — and the two ingredients for it were already here, in two other accounts. What comes out is a second surprise, for when a chord arrives rather than for which one it is.

A loud chord is a smaller chord

Two earlier essays hold the level fixed, and the level decides how much of a chord a listener is given. At thirty decibels twenty of a triad's twenty-four partials stand above what the rest of it masks; at a hundred, nine do. Every roughness figure until now counts partials that are in the score, and a partial the chord masks is not a partial the listener has.

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.

Two surprises and one event

A chord change is surprising twice over — in which chord it is, and in when it comes — and a listener meets one event. Adding two surprises needs to know how much they share, which is a fact about a repertoire nobody has. Sweeping it instead: the total moves by half, the ordering does not move at all, and the most surprising moment in a passage is the same one whatever the answer turns out to be.

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.

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.

A third of the timing surprise is paid for by nothing happening. Every beat of a bar of 4/4 at 1 chord change a bar, with what it costs a listener in expectation. The lower block is the arrival — the chance the chord changes here times what that change costs to be surprised by. The upper block is the hold, which is paid when the chord does not change and is charged at every beat rather than at every event. Over the bar the arrivals come to 2.61 bits and the non-arrivals to 1.29, so the term never spent before is 33 per cent of the total rather than most of it. The two together are exactly the binary entropy of each beat's own hazard, which is why the bar's whole timing bill is 3.90 bits and cannot be raised by rearranging where the changes fall.

The surprise of nothing happening

A hazard charges a listener twice — once when the chord changes and once, quietly, at every beat it does not. The second term was expected to dominate, because there are more beats than changes. It is a third of the bill at one chord a bar and never reaches a half at any rate a metre survives, because the cost of a beat where nothing happens is second-order small.

What a chord change costs where it actually lands. Every beat of a bar of 4/4 at 1 change a bar, priced as a listener meets it: the pale block is what has already been paid waiting through the beats the chord did not come on, and the dark block is the arrival itself. Their sum is what it costs to be surprised by a change here. The last column is the remaining case, never priced before: no change in the bar at all, at 1.61 bits and a probability of 0.33. The 9 costs are a proper distribution — 1.000 — which is the check that this is one model rather than two. And the spread is the finding: the arrival term alone puts a factor of 6.7 between the best and worst beat, and counting the waiting makes it 19.

Where the chord actually lands

Every timing surprise so far is evaluated on a downbeat: the curve spans a factor of 6.7 and every number is read at its peak. Charge the waiting as well as the arrival and the costs over a bar become a proper distribution, the spread between the best and worst beat rises to a factor of 19.5, and a third of the probability sits on a bar in which nothing changes at all.

A long note and a strong note disagree, and the winner is neither. The same 8 notes scored against every triad and seventh at every root, with the weighting run from the metrical one always used to a durational one never drawn. On the left each note counts for its metrical weight; on the right, for how long it is held. The long notes here are on beats 2, 4, 6, 8, which are the weak ones. The two cues point at different chords — C major7 on the left and D minor7 on the right — turning over at a mixture of 40 per cent. And at the crossing the winner is A minor7, which is neither cue's answer — a chord that shares three notes with each and is not the reading either rule asks for. Nothing about the notes changed. What changed is which of two cues a theorist would call obvious is being believed.

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.

A notehead in four parts costs 1.30 bits and one in two parts costs 1.89. What one notehead asks of a reader, against how many parts are on the page, for a progression realised by the voice-leading solver used here at 2 semitones of motion a voice a chord. The horizontal rule is a note of a single melody under the measure established earlier, 1.89 bits, which is what a texture costs when its parts have to be read one at a time. The bars are the harmonic reading: the chord, charged at the worst case of 2.81 bits for one of seven diatonic degrees, plus the logarithm of how many voicings of it the previous chord could legally have moved to. At two parts there is no bar, because a duet has no complete voicing of any triad — it cannot state the harmony and has to be read as 3.79 bits of two independent lines. Every thicker texture is cheaper a notehead than the thin one, and the four-part figure is an upper bound.

Four parts are easier to read than two

Twelve earlier essays read one line, and a score is several at once. Measured through the voice-leading model, a notehead of a four-part chorale asks a reader for 1.30 bits and a note of an independent line asks 1.89 — so twice the ink is less than three quarters of the load. The reason is a boundary those essays already established: a duet has no complete voicing of any triad at all, so two parts cannot be read from their harmony and have to be read as two melodies.

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.

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.

Eighty-one chords the expectation model cannot tell apart. Every voicing of a dominant seventh on G inside the three octaves above its own root, placed by how rough it is and how far its outer voices are apart, and coloured by which member of the chord is at the bottom. The roughness runs from 0.269 to 1.449, a factor of 5.4, computed from each voicing's own spectrum under Plomp and Levelt's roughness model. The identity surprise the expectation model assigns is 3.51 bits for every one of the 81, because it is a function of a scale degree and its predecessor and there is no register anywhere in it. What separates them is spacing rather than inversion: roughness falls as the outer voices spread apart, correlating -0.42 with the span, and is indifferent to which member of the chord is at the bottom at 0.02. The seventh in the bass is not what makes a chord rough; a fourth and a third packed together at the bottom of the range is.

Eighty-one chords, one number

A dominant seventh has eighty-one arrangements inside three octaves and their roughness spans a factor of five and a half. The tonal-expectation model gives every one of them the same 3.51 bits, because its states are scale degrees and there is no register anywhere in them. Conditioning the surprise on the voicing costs no corpus — and the arithmetic says the conditioning belongs beside the probability rather than inside it, for three reasons that can each be computed.

How often the metre, the chords and their product find the barline, chords at 1. Constructed passages of four bars of eight quavers, 100 at each setting, with the barline at the first slot. Rhythm regularity is how much likelier a note is on a strong slot than a weak one; chord regularity is how much likelier a note is to be a tone of its bar's chord than a random scale tone. rhythm 0: metre finds it 10%, chords find it 41%, product finds it 16%; rhythm 0.25: metre finds it 34%, chords find it 35%, product finds it 56%; rhythm 0.5: metre finds it 49%, chords find it 21%, product finds it 66%; rhythm 0.75: metre finds it 50%, chords find it 17%, product finds it 56%; rhythm 1: metre finds it 50%, chords find it 11%, product finds it 45%.

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.

The product and the sum of standard scores, finding the barline, chords at 1. Constructed passages of four bars of eight quavers, 100 at each setting, with the barline at the first slot. Rhythm regularity is how much likelier a note is on a strong slot than a weak one; chord regularity is how much likelier a note is to be a tone of its bar's chord than a random scale tone. rhythm 0: product finds it 16%, sum of z finds it 22%, metre finds it 10%, chords, z 35%; rhythm 0.25: product finds it 56%, sum of z finds it 58%, metre finds it 34%, chords, z 38%; rhythm 0.5: product finds it 66%, sum of z finds it 61%, metre finds it 49%, chords, z 19%; rhythm 0.75: product finds it 56%, sum of z finds it 58%, metre finds it 50%, chords, z 14%; rhythm 1: product finds it 45%, sum of z finds it 48%, metre finds it 50%, chords, z 12%.

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.

Four bars read by where the chords change, barline by barline. A constructed passage of four bars of eight quavers, its barline at the first slot and its chords C, F, Em, Dm. Notes: slot 1 C, slot 3 E, slot 4 G, slot 5 E, slot 6 C, slot 9 C, slot 10 C, slot 11 F, slot 13 A, slot 16 C, slot 17 B, slot 19 B, slot 20 E, slot 21 E, slot 24 G, slot 25 F, slot 29 F. For each of the eight places the barline could fall: as written metre, z 3.97, chords, z 2.00, change, z 4.30, metre + change, z 8.27; 1 quaver late metre, z -2.45, chords, z 2.14, change, z -0.87, metre + change, z -3.32; 2 quavers late metre, z -1.38, chords, z 2.15, change, z -0.54, metre + change, z -1.93; 3 quavers late metre, z -0.31, chords, z -0.27, change, z -2.64, metre + change, z -2.95; 4 quavers late metre, z 3.97, chords, z -1.63, change, z -4.30, metre + change, z -0.33; 5 quavers late metre, z -2.45, chords, z 1.27, change, z 0.87, metre + change, z -1.58; 6 quavers late metre, z -1.38, chords, z 1.18, change, z 0.54, metre + change, z -0.84; 7 quavers late metre, z -0.31, chords, z 1.05, change, z 2.64, metre + change, z 2.32. Best metre, z: as written and 4 late. Best chords, z: 2 late. Best change, z: as written. Best metre + change, z: as written.

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.

Level does not dilute the register's roughness, it multiplies it. The mean roughness of the I – vi – IV – V – I arrivals at 4 registers, each relative to the register as written, read three ways. Level-free, the bass is 8.6 times rougher than the treble. With every note at 70 dB it is 8.6 times, the same factor, because one level rescales every pair alike. With each chord played at the level that makes it as loud as the written register's chords — 82.8 dB −2 octaves, 75.5 dB −1 octave, 70.0 dB as written, 66.8 dB +1 octave — the bass is 343 times rougher than the treble, because roughness grows with the square of the pressure and the bass needs more of it to be heard at the same loudness.

A rough arrival is rough because of its spacing

The pair the expectation essays report for every chord — how surprising it was, how rough its voicing is — has no level in it. Putting level back in answers the question it left open, and not the way it was framed. At one written dynamic the arrivals keep their order from 40 to 90 dB at three registers of four, and the bass stays 8.6 times rougher than the treble. Made equally loud, the bass has to be played 12.8 dB harder, and it is 343 times rougher: level does not explain the register's roughness away, it multiplies it.

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.

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.

The content errs slow and the bass errs fast. Passages of eight bars built at three harmonic rhythms, each with a bass that states every new chord's root and moves to another chord tone on a beat 50 per cent of the time. Two readings are asked the chord rate: one from how much the pitch-class content changes across a grid, one from how completely the bass's moves land on it. At two chords a bar the content reading names the rate 63 per cent of the time, too fast 0 and too slow 37; the bass reading 100, 0 and 0. At one chord a bar the content reading names the rate 67 per cent of the time, too fast 0 and too slow 33; the bass reading 18, 80 and 2. At a chord every two bars the content reading names the rate 98 per cent of the time, too fast 2 and too slow 0; the bass reading 0, 100 and 0. The two readings miss in opposite directions and at opposite ends of the tempo range: the content reading at fast harmonic rhythms, by naming a multiple, and the bass reading at slow ones, by naming its own arpeggiation.

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.

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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