Acoustics & Music Theory
why it sounds like that
Concept

Voice-leading — where it appears

How the individual lines of a texture move from one chord to the next. Minimising the total motion is a computation over every assignment of voices, and its answers coincide with taught practice more often than by chance.

Named by 15 essays across 3 fields — each of them below, with the objects they name alongside it.

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.

harmony · Voice-leading 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.

harmony · The triad 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.

harmony · Voice-leading 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.

harmony · Voice-leading Five signals, computed separately, and no total. The five components of closure for 6 chord pairs. The first three are computed from the chords alone; the last two are properties of where the goal lands and how long it is held. There is no total column: the components are not commensurable and the ordering of these cadences depends on which is weighted.

What makes an ending an ending

Cadences are ranked. The authentic one is strong, the plagal weaker, the deceptive weaker still — and the solver here measured the quantity that ranking is usually explained by and found it says something else entirely. What survives is not a weaker version of the ranking but a different kind of object, with five components and no total.

form · Closure 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.

harmony · Tonal-expectation What each rule costs, in semitones of extra motion. Each prohibition priced on I – ii – iii – IV as the difference between the cheapest realisation that obeys it and the cheapest realisation of all, averaged over every one of the 144 melodies the progression admits inside an octave. The dearest rule costs 1.27 semitones a melody and the cheapest costs nothing, so the bars run from 1.27 to zero; the notes beside them give the share of melodies that pay nothing, or that cannot obey the rule at all.

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.

harmony · Voice-leading 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.

harmony · Voice-leading 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.

harmony · Voice-leading Four voices, placed by the arithmetic. The rules in force are: no parallel octaves; no parallel fifths; no voice crossing; no gap over an octave above the tenor; the leading note is not doubled; the leading note resolves, outer voices; no augmented melodic interval. The four parts are drawn lowest to highest — bass, tenor, alto, soprano. I – vi – ii – V – I in C major, realised in four voices by the cheapest set of voicings obeying 7 rules, at 16 semitones of motion in all. Every gap between adjacent voices narrower than the fission boundary of 5.2 semitones is marked, and below that boundary two parts cannot be heard as two however hard a listener tries.

A voice is a stream, and the ear decides which

Seven earlier essays have assigned voices to notes. Whether a listener follows the assignment is a separate question with laboratory numbers attached, and the numbers are unkind to it: two parts closer than about five semitones cannot be heard as two at any speed, a third of the gaps in the cheapest four-part writing are inside that limit, and in a third of chord changes the ear's own rule for continuing a line does not recover the parts as written.

perception · Voice-leading 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.

harmony · Modes 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.

harmony · Key-relations How much of the writing the prohibitions forbid. I – vi – ii – V – I in C major, written in 3, 4, 5, 6 parts, with every ensemble covering the same total compass. A triad has three pitch classes, so n parts double n − 3 of them and a duet cannot state one at all. Of every ordered pair of complete voicings of V and I, the share with no parallel fifth or octave between any pair of voices: 91.2% at 3, 75.0% at 4, 44.3% at 5, 17.9% at 6.

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.

harmony · Voice-leading What survives a change of encoding: verse and chorus. The same 32 bars of verse and chorus under five encodings, scored on the three things measured here measures. Mean off-diagonal similarity says how alike the piece looks to the arithmetic. Recall and precision are the novelty operator's boundaries against the 3 the section plan has, at a kernel of four bars. The period is the strongest peak of the lag profile, in bars. Under the bag of pitch classes every other figure uses, the piece is 90 per cent self-similar and the operator finds 0 per cent of the boundaries; under how far the root moved it finds 100 per cent. The period is the quantity that does not move.

The repeat that is not in the notes

Eight earlier essays compare bars by writing each one as a bag of pitch classes and taking a cosine. Nothing chose that encoding — the first used it and the other seven inherited it. Encode the same six schemes four other ways and one of the three findings survives untouched, one survives with different numbers, and one turns out to have been a statement about the encoding all along: the boundary operator finds none of the section edges in three schemes as a bag of pitch classes and every one of them as tonic, subdominant and dominant.

form · Repetition 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.

harmony · Notation

Named alongside it

The objects these essays reach for when they reach for this one.

Dominant seventhLeading noteCadenceCounterpointPart-writingTriadContrary motionNeo-Riemannian transformationParsimonyTonal functionClosureCommon-tone

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