Chords as a space

Take a C major triad and move exactly one of its three notes by exactly one semitone. There are six ways to do it, and two of them land on another major or minor triad: lower the E to E flat and the result is C minor; lower the C to B and the result is E minor.

Move one note by two semitones and one more becomes available: raise the G to A and the result is A minor.

That is the complete list of triads within two semitones of C major, and the three moves that reach them have names — P, L and R. The interesting question is what happens when they are applied over and over.

The three moves #

Each operation holds two notes still and moves the third. That is the definition, and everything else about them follows from it.

P, for parallel, keeps the outer fifth and slides the third. C major and C minor share C and G, and differ in whether the middle note is E or E flat. One semitone.

L, for Leittonwechsel — leading-tone exchange — keeps the upper third and slides the root. C major and E minor share E and G, and differ in whether the outer note is C or B. One semitone.

R, for relative, keeps the lower third and slides the fifth. C major and A minor share C and E, and differ in whether the outer note is G or A. Two semitones.

Each is its own inverse: apply P twice and the chord is back — the same self-cancelling property the tritone has as an interval. So three operations, each an involution, acting on twenty-four objects.

Which computation produced the number #

The costs above are not asserted. The generator measures them, using the same voice-leading solver the rest of this site uses: for two chords it tries every way of assigning voices, takes the shorter way round the circle for each voice, and returns the assignment with the smallest total.

Run on C major against C minor it returns 1. Against E minor, 1. Against A minor, 2. Those three numbers are printed in the middle of the figure and they come out of the solver rather than out of the caption.

The layout is measured too, and more strictly. The claim being drawn is that alternating R and L, starting from C major, visits all twenty-four triads and returns to the start. That is a strong claim and the generator refuses to draw the figure unless it holds: it walks the alternation twenty-four times, checks that the twenty-fifth chord is the first one, and checks that the twenty-four visited are all distinct. If either fails it throws.

It did fail, the first time. Defining L and R on minor triads with the same interval offsets as on major ones produced a walk that revisited eight triads and never reached the other sixteen — the picture would have been a plausible-looking ring with two-thirds of the chords missing and no visible sign of anything wrong. The check caught it because a wrong walk cannot close.

The correct offsets are not the same for the two qualities. A minor triad’s root sits at the bottom of its own stack, so the interval back to its major partner is the complement of the one going the other way: L takes a major triad up four semitones into minor and a minor triad up eight into major.

Why the alternation closes #

The walk that the figure is laid out on is worth following, because it turns out to be something already familiar.

Start at C major. R gives A minor. L gives F major. R gives D minor. L gives B flat major. Continuing: G minor, E flat major, C minor, A flat major, F minor, D flat major, B flat minor, and onward round the flat side until the twenty-fourth step returns to C major.

The major chords in that sequence are C, F, B♭, E♭, A♭, D♭ — the circle of fifths, descending. The minor chords are their relative minors, interleaved. So the LR cycle is the circle of fifths with each key’s relative minor inserted between the majors, which is the arrangement that appears on every circle-of-fifths diagram ever printed, and which neighbouring keys share six of their seven notes along.

That is a circle-of-fifths diagram, arrived at without being aimed for. Majors on the outside, their relative minors inside, and a reading that alternates between the two rings traces the neo-Riemannian cycle exactly — which is why the layout of the first figure is not a drawing decision but a consequence of what R and L do.

That is a satisfying result rather than a surprising one, and it means the neo-Riemannian machinery is not describing anything new at the level of which chords are near which. What it adds is that the nearness is measured — in semitones of voice motion — rather than asserted by a diagram that everyone has agreed to draw a particular way.

What the third move does #

P is the operation the circle of fifths has no room for, and what it contributes can be counted rather than described.

It adds no reachability. L and R alone already reach all twenty-four triads — that is the cycle the figure is laid out on — so P cannot connect anything that was disconnected. What it does is shorten:

Adding P more than halves the diameter and cuts the worst-case voice motion from eighteen semitones to seven. The furthest chord on the LR cycle is half a circle away and takes twelve moves; with P available nothing is more than five moves from anything, and no pair of triads is further apart than seven semitones of total voice motion.

The other two pairings say why P has to be a supplement rather than a partner. P and L together reach only six of the twenty-four triads, and P and R only eight — they generate closed cycles rather than covering the space, which is what a pair of involutions can do when their product has small order. So the three moves are not interchangeable: R and L are the pair that gets everywhere and P is the pair-breaker that makes getting there quick.

That is the shape of the whole apparatus in three numbers. One cycle covers the space at a cost of up to eighteen semitones; one extra move, which costs a single semitone to take, brings the worst case down to seven. Nineteenth-century harmony’s fondness for the parallel major and minor is not an extra relation bolted onto the circle of fifths — it is the shortcut that makes the circle’s far side reachable in a few chords instead of a dozen.

On the ring alone, C major and C minor are seven steps apart — a long way. With P they are adjacent, and the ring becomes something that folds back on itself. The chords across the middle of the first figure are all P edges, and they are what turns a cycle into a genuinely two-dimensional space.

Musically, P is the operation that changes the mode without changing anything else, and its cheapness is why a shift from major to minor on the same root can be so abrupt and so small at once. One voice, one semitone, and the whole character of the chord — the same single note that distinguishes two adjacent modes.

The combination PL, applied repeatedly, produces the hexatonic cycle: C major, C minor, A flat major, A flat minor, E major, E minor, back to C major. Six chords, every step one semitone, and the roots are the three notes of an augmented triad. LP cycles of this kind are all over nineteenth-century harmony, and they are the standard modern account of chromatic passages that functional analysis handles badly.

How near everything is #

The map is small enough to measure completely, so it is worth measuring completely rather than describing the interesting corner of it.

Running the voice-leading solver from C major to all twenty-four triads gives the following distribution. Two triads at one semitone — C minor and E minor. Five at two — including A minor, and also C sharp major, E major, F minor and A flat major, which functional theory would not group together at all. Seven at three, three each at four, five and six.

Two facts fall out of that. The first is that the ring’s neighbours are not the only cheap moves: the five chords at distance two include several that the LR cycle places far apart, so the ring is a spanning structure rather than a complete account of nearness. The second is that the far side really is far — the maximum is six semitones, which for three voices means every one of them moving two, and there are only three chords out there.

Notice also that exactly three triads share two notes with C major — C minor, E minor and A minor, which are precisely P, L and R. That is not a definition being restated. The transformations were defined by what they hold still, and the count of chords sharing two notes was computed separately; they agree because the arithmetic makes them agree.

Why the triad and not something else #

A reasonable suspicion at this point is that any three-note chord type would produce a comparable map, and that the elegance is an artefact of drawing rather than a property of triads. It is not, and the check is quick.

Take a chord type, generate all its transpositions and inversions, and measure the voice-leading distance from the prototype to each. For the consonant triad the nearest other form is one semitone away, and there are two of them.

For the chromatic cluster $\{0,1,2\}$ the nearest other form is three semitones away. For the stacked-fifths chord $\{0,2,7\}$ it is three. For the diminished triad $\{0,3,6\}$, three. For a fragment of the whole-tone scale $\{0,2,4\}$, three. The augmented triad $\{0,4,8\}$ has only four distinct forms at all, and the nearest is three away.

Only one other trichord type in the system has a neighbour a single semitone away, and it is $\{0,1,6\}$ — a semitone and a tritone stacked, the sonority sometimes called the Viennese trichord. It is not a consonance, and its overall distribution is much worse: most of its forms sit five or six semitones off.

So the consonant triad is essentially alone in combining two properties: it is smooth, and it has close neighbours. Richard Cohn’s term for the second is parsimony, and the point of the transformational literature is that the combination is a coincidence of twelve-tone arithmetic rather than a necessity — the major third is four semitones and the minor third three, they differ by one, and that one is the entire reason chromatic harmony can move as smoothly as it does.

What “distance” means here #

There is a real conceptual shift buried in all of this, and it is easy to miss because the diagrams look like the old ones.

Traditional harmonic theory explains chord relationships in terms of function: a chord is a dominant or a subdominant or a mediant, it belongs to a key, and progressions are described by what those functions do. That account is powerful, has enormous explanatory reach in eighteenth- and nineteenth-century repertoire, and breaks down where key membership becomes unclear.

The transformational account explains chord relationships in terms of distance: how far the voices have to move. It says nothing about keys at all. A chord is not in a key on this map; it is at a place, and its neighbours are its neighbours whatever the surrounding music is doing.

The two accounts are not rivals so much as differently scoped. Where a passage stays in a key, function explains a great deal that distance does not — it explains why a dominant is unstable, which is a matter of the tritone inside it and not of how far anything moves. Where a passage abandons keys and moves by smooth voice leading through chromatic territory, distance explains what function cannot.

Where the model stops #

Twenty-four triads is a very small world. The whole apparatus is defined on major and minor triads only. Sevenths, augmented triads, diminished triads and everything with an added note are outside it. Extensions exist — there are transformational systems for seventh chords — and they are considerably less tidy, which is itself informative about how special the consonant triad is.

Voice-leading cost ignores register and doubling. The solver works in pitch classes and takes the shorter way round the circle for each voice. Real voice leading happens in a register, with a bass line that behaves differently from an inner part, and with notes doubled. A move that is cheap in pitch classes can be awkward to write for four actual voices.

Nothing here says which move will be made. The map gives distances. It does not predict a progression any more than a road map predicts a journey, and treating cheapness as a preference is a mistake this literature sometimes invites.

The parsimony is an artefact of twelve. Triads have this property — that two of them can share two notes and differ by a semitone in the third — because of the specific arithmetic of the major and minor thirds in twelve equal steps. In other divisions the analogous chords have different neighbours and the graph has a different shape.

Neo-Riemannian is a misnomer, and the literature says so. Riemann’s own theory was about function, and about the derivation of chords from a dualist account of major and minor. The transformations are named after him because his relations supplied the vocabulary; the geometry is a twentieth-century construction and does not commit anybody to his metaphysics.

Whose music, and when #

The transformations were developed to explain a specific problem in a specific repertoire.

Nineteenth-century chromatic harmony — Schubert’s late works, Liszt, Wagner, Franck — contains a great many passages in which every chord is a consonant triad, every change is smooth, and no key is in force for long enough to make functional labels useful. A functional analysis of the opening of Schubert’s B flat Piano Sonata, or of large stretches of Parsifal, produces a chain of increasingly implausible key assignments.

David Lewin’s Generalized Musical Intervals and Transformations in 1987 proposed replacing the question “what key is this in” with “what operation gets from here to there”, and the specific P, L, R system was developed through the 1990s by Richard Cohn, Brian Hyer and others. Cohn’s work on hexatonic cycles in particular showed that a large class of nineteenth-century chromatic passages traces PL cycles exactly.

The lattice itself is much older. Euler drew a version of it in 1739 for reasons that had nothing to do with voice leading — he was interested in the arithmetic of ratios — and Oettingen and Riemann developed it as a harmonic space in the late nineteenth century. Its reinterpretation as a voice-leading geometry, rather than as a map of tuning relationships, is the modern contribution and is a genuine change of subject: the same picture, answering a different question.

The reach of the account beyond that repertoire is real but limited. It applies well to any music that moves smoothly between consonant triads without settling in a key, which includes a good deal of film scoring and some popular music. It applies poorly to anything whose interest is in where a chord sits in a key, which is most of what was written between 1700 and 1850.

The ladder from here #

Later rungs on this anchor: hexatonic and octatonic cycles, and the repertoire that traces them. The extension to seventh chords, and why it is so much less elegant. Tymoczko’s continuous voice-leading spaces, in which chords are points in a genuine geometry rather than nodes in a graph, and the triad’s parsimony becomes a statement about where it sits in that space. Doubling and register, which the pitch-class account discards. And the relationship between transformational distance and functional expectation, which is the open question the whole programme raises.

Twenty-four chords, three moves, and the cheapest tour of all of them turns out to be a diagram every music student has already seen.