Acoustics & Music Theory
why it sounds like that
Concept

Neo-Riemannian transformation — where it appears

One of a family of operations that turn a triad into another triad by moving a single voice as little as possible. Composing them generates a space in which distance is voice-leading work rather than steps round the circle of fifths.

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

Also named here as parsimony — the same set of essays touches all of them, so they are one junction rather than several.

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 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 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

Named alongside it

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

ParsimonyVoice-leadingCommon-toneTriadContrary motionDominant seventhTonnetz

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