The shortest move, which is what a chord change is

Two chords sound one after the other. Between them, every voice has to go somewhere, and there is more than one way to assign the destinations.

Take the shortest total distance over all assignments and a number falls out — a measure of how far a chord change actually is, in semitones, independent of what the chords are called.

C major to A minor: total 2 semitones. C major to F major: 3. C major to F-sharp major: 6. Those numbers are computed rather than asserted — every pairing of the three voices is tried and the cheapest wins — and they line up almost exactly with how closely related the chord pairs are held to be.

Why sharing notes is the whole of it #

The metric is dominated by one thing: how many notes the two chords have in common.

A shared note costs zero. Any note that has to move costs at least one semitone and usually two. So a pair of triads sharing two notes will always beat a pair sharing one, which will always beat a pair sharing none — before any of the details are considered.

The lattice makes the relationship geometric. Triads that share two notes are adjacent triangles; the three ways of flipping a triangle across one of its edges are the three chord changes that move a single voice by a single step.

Those three have names in neo-Riemannian theory. P (parallel) turns C major into C minor, moving the third down a semitone. R (relative) turns C major into A minor, moving the fifth up a tone. L (leading-tone exchange) turns C major into E minor, moving the root down a semitone. Each keeps two notes and moves one, and each is a reflection of a triangle.

What the metric explains #

Several standard features of harmonic practice fall out of minimising motion, and are usually taught as rules with no reason given.

Keep common tones. Every harmony textbook says to hold shared notes in the same voice. That is precisely the instruction to take the cheap assignment rather than an expensive one.

Move the other voices as little as possible. Also standard, also exactly the metric.

The dominant-to-tonic cadence. G major to C major shares one note. The other two voices move a semitone and a tone respectively — the leading note up into the tonic, the fourth degree down into the third. Total: 3 semitones, of which one is the semitone resolution that carries the whole gesture.

Why vi, iii and IV are the easy chords. They sit at radius one or two in the figure above, sharing two notes with the tonic. A progression that stays among them barely moves at all, and three quarters of I–vi–IV–V does — though not the whole of it, and a section at the foot of this essay counts the cost of the fourth quarter and finds the progression a good deal less economical than it looks.

The chromatic third relation #

The metric earns its keep on a case where conventional theory does badly.

C major and E major share one note. On the circle of fifths they are four steps apart, which is a long way, and traditional key relationships treat the move as remote. Yet the change sounds smooth and nineteenth-century music uses it constantly.

Compute the voice leading and the answer is 3 semitones — the same as C major to F major, which nobody calls remote. The circle of fifths is measuring shared scale content; the voice-leading metric is measuring shared chord content, and for chromatic harmony the second is the better predictor.

That is why Schubert, Wagner and film composers ever since move by major thirds so freely, and why the moves sound luminous rather than jarring. They are short moves that a diatonic map draws as long ones.

Contrary motion, and why it is preferred #

The metric counts total distance and says nothing about direction. Practice cares about direction a great deal, and the reason is not aesthetic.

Two voices moving in the same direction by the same interval — parallel fifths, parallel octaves — stop sounding like two voices. The ear groups components that move together, so two parts in exact parallel fuse into one part with a thicker timbre. For a texture whose whole point is independent lines, that is a failure.

Contrary motion, where voices move oppositely, keeps them separable. Oblique motion, where one holds and the other moves, does the same and is cheaper still — a held note costs nothing in the metric and provides maximum separation.

So the counterpoint rules are not arbitrary prohibitions. They are a set of instructions for keeping voices perceptually distinct, arrived at empirically several centuries before anybody could describe auditory grouping as a mechanism.

The metric on this page is blind to all of it. Two voices moving three semitones in parallel and two moving three semitones in contrary motion cost exactly the same, and one of them destroys the texture.

The bass is not a voice like the others #

A second thing the metric flattens: the lowest voice does a different job.

The bass determines which inversion a chord is in, and inversion changes a chord’s stability substantially — a second-inversion triad cannot end a phrase while a root-position one can. It also carries the harmonic progression: a bass moving by fifths states a functional sequence, and the same upper voices over a stepwise bass state something quite different.

Consequently, composers routinely accept an expensive bass leap in exchange for the harmonic clarity it buys, while keeping the upper voices as economical as possible. The metric, which weights all voices equally, scores that as a poor solution.

Voice leading came first #

There is a historical inversion worth being clear about, because the teaching order reverses it.

Harmony is normally taught as chords first and voice leading second: learn the chords, then learn the rules for connecting them. Historically it went the other way. Renaissance and medieval music was composed as simultaneous melodic lines, governed by rules about intervals between pairs of voices, and the vertical sonorities were what resulted.

The triad as an object of theory arrives in the sixteenth century with Zarlino, and functional harmony — chords with roots and progressions — is an eighteenth-century idea, Rameau’s. So for most of the history of European polyphony, what a composer was actually manipulating was voice leading, and chords were an emergent property.

That explains something otherwise odd: the rules of counterpoint are largely rules about motion, and only incidentally about which sonorities occur. Parallel fifths are forbidden not because two fifths are unpleasant but because the two voices stop sounding independent — a motion rule, in a system that was tracking lines.

Written out as three notes of one voice, harmony is several of these read together, which is what the metric assumes and what the notation says. And four triads as stacked intervals are four sets of distances: the metric treats them as unordered pitch classes, so a major triad and its inversion are one object to it and two objects to a listener.

What the metric misses #

The measure is a good predictor and it is not a theory of harmony, and its failures are informative.

It has no direction. The distance from G major to C major equals the distance from C major to G major. Musically these are completely different: one is a cadence and the other is a departure. Function is asymmetric and the metric is not.

It ignores which voice moves. Moving the bass by three semitones and moving an inner voice by three semitones cost the same in the metric and are entirely different events. The bass carries the harmonic identity; an inner voice does not.

It ignores the interval formed. Two voices moving in parallel fifths and the same two moving in contrary motion can cost the same and are treated completely differently by every practice that has an opinion.

It ignores register. All the figures work in pitch classes, so a voice that “moves one semitone” might be leaping an octave and a semitone in the actual music.

Whose music, and when #

Parsimonious voice leading is a strong feature of European practice from roughly 1500 to 1900, and it is a specific tradition rather than a general law.

It matters most where there are independent voices to lead — choral music, string quartets, keyboard counterpoint. It matters less where harmony is realised as block chords on a guitar, where the “voices” are whatever the shape puts under the fingers and the resulting leaps are nobody’s concern.

It matters differently again in traditions with a drone, where harmony is not a succession of chords at all, and the question of how voices move between them does not arise.

And the neo-Riemannian apparatus — P, L and R as operations on a lattice — was developed in the 1980s and 90s specifically to describe chromatic nineteenth-century harmony that functional theory handled badly. It is a good description of Wagner and a poor one of Bach, which is a point in its favour rather than against it: it was built for one job.

The geometry that generalises it #

The metric on this page compares chords pairwise. There is a way to see all of them at once, and it turns the discrete problem into a continuous one.

Represent a three-note chord as a point whose coordinates are its three pitches. Two chords that sound the same regardless of voice order should be the same point, so the space is quotiented by permutation; octave equivalence quotients it again. What comes out is a space with an unusual shape — for three-note chords, a triangular prism with a twist in it, closing on itself like a Möbius band.

In that space, voice-leading distance is ordinary distance. Chords that are close are chords whose voices need not move far, and a progression is a path. The triads turn out to lie along a central axis, and the smooth progressions of nineteenth-century harmony are short paths near it.

This is Dmitri Tymoczko’s construction, and its value is that it makes a question about chord relationships into a question about geometry, where the tools are better. It also generalises: four-note chords give a four-dimensional analogue, and the same central-axis observation holds for seventh chords.

Why parsimony and function disagree #

Two accounts of chord succession are on offer and they do not predict the same music, which is a useful disagreement rather than an embarrassment.

Functional harmony predicts progressions by role: subdominant goes to dominant, dominant goes to tonic, and the moves that matter are the ones between categories. It handles Bach and Mozart extremely well.

Parsimonious voice leading predicts progressions by distance: chords sharing two notes are natural neighbours, whatever their function. It handles Wagner and Schubert’s remote-key excursions extremely well, and it says almost nothing about why a cadence is final.

The two agree on the common diatonic progressions, because those happen to be both functional and cheap. They diverge exactly where chromatic harmony lives — the chromatic third relation is a short move with no function, and a functional analysis of a Schubert modulation typically resorts to elaborate reinterpretation to explain a change that voice-leading distance calls trivial.

The right conclusion is that the two describe different aspects. Function is about expectation and closure; parsimony is about smoothness and continuity. Music that is doing both is well described by either, and music that abandons one is described only by the other.

Where the model stops #

Equal-sized chords. The metric as computed here pairs three notes with three notes. Chords of different sizes need a different formulation, and there are several with different answers.

Pitch classes, not pitches. Every distance is measured round a circle of twelve, so a voice can “move two semitones” by any octave-displaced route. Real voice leading happens in a register.

Optimal assignment is not what performers do. The metric finds the cheapest pairing. A written score assigns voices explicitly, and composers sometimes choose an expensive assignment deliberately — a leap for emphasis, a crossing for texture.

The figures show three cases. Whether short voice leadings really do dominate real repertoire is an empirical claim, and the pictures here illustrate it rather than test it.

What a listener actually tracks #

The metric assumes voices exist as separable strands. Whether a listener hears them that way is an empirical question with a qualified answer.

Auditory streaming research establishes that the ear groups a sequence into strands using proximity in pitch, continuity of timbre and rate. Two lines in similar registers and similar timbres fuse into one perceived stream regardless of how they were written; the same two lines separated in register stay distinct.

That has direct consequences for the metric. A voice leading that is short in pitch-class terms may not be heard as a voice moving at all if the parts have crossed or if the registers overlap. And a large leap that keeps a part in its own register may be heard as more continuous than a small one that crosses into another part’s territory.

So the honest position is that voice leading describes what a composer wrote and approximates what a listener hears, with the approximation getting worse as the parts get closer together in register and timbre. Which is, usefully, exactly the situation the counterpoint rules were written to avoid.

The ladder from here #

Later rungs: neo-Riemannian transformations in full. Parsimonious voice leading in chromatic repertoire. The rules of counterpoint as motion constraints. Parallel fifths, and what independence means. Contrary motion. Voice leading in chords of four or more notes. Tymoczko’s geometry of chords, where the space becomes continuous. Bass line as a separate structural voice. And the question of whether listeners track voices at all, which auditory streaming research answers with a heavily qualified yes.

The closing claim, checked #

The tempting last sentence for an essay like this one is that the most-used chord sequence in popular music is also the cheapest, and it was written here for four phases. It is not true, and the exhaustive check is one loop.

I–vi–IV–V costs 12 semitones over its four changes, not 8: two from the tonic to vi, one from vi to IV, six from IV to V, and three from V back home. The six is the whole of it — the subdominant and the dominant share no notes at all, so every voice has to move, and it is the most expensive single change in the progression by a factor of two.

Enumerating every four-chord cycle on the diatonic triads that starts on the tonic and never repeats a chord back to back gives 186 sequences, running from 4 semitones to 22.

It is at the median. Thirty-nine cycles are cheaper and fifty-four cost exactly the same. So parsimony does not pick it out, and the honest statement is the opposite of the one that was here: the commonest progression in popular music is unremarkable by the metric this essay is about.

What the enumeration does show is worth more than what it refuses. The four best-known four-chord loops all cost exactly twelve — I–vi–IV–V, I–V–vi–IV, I–vi–ii–V and I–IV–vi–V — which is a tighter cluster than chance would give over a range from four to twenty-two, and it says that the repertoire has settled on a particular amount of motion rather than on as little as possible.

And the cheapest cycles say why as little as possible is not what anybody wants. The 4-semitone winner is I–iii–I–iii, which is an alternation between two chords sharing two notes; the next several are the same shape. Minimising motion alone selects for going nowhere, which is the same result the closure ladder found when it discovered that the smallest moves available in a key are not cadences. Parsimony is a constraint on how a progression travels and not a reason for it to travel, and a metric that rewards standing still is measuring smoothness rather than harmony — which is what the section on function and parsimony says, arriving here as a rank rather than as an argument.