Acoustics & Music Theory
why it sounds like that
Concept

Tonnetz — where it appears

A lattice whose two axes are the fifth and the third, so that every major and minor triad is a triangle on it. Neighbouring triangles share two notes, which makes voice-leading distance a distance on the page.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

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.

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 The map is two-dimensional and the circle is one axis of it. Major keys along the top, each a fifth from the last; underneath each, its relative minor, which shares all seven of its notes. Moving sideways changes one note; moving down changes none at all and changes the tonic. Among the twelve major keys alone there is only the sideways move, which is why that map really is a circle — the second axis needs the minor keys to exist.

The circle is a circle, and the map is not

Among the twelve major keys, notes in common is a strict function of distance round the circle of fifths — one value for each step count, no exceptions — so there is nothing else to measure and the map really is one-dimensional. A second axis appears only when the minor keys are added, and it is a different kind of move: the relative shares all seven notes and the parallel is one semitone away.

scales · Key-relations

Named alongside it

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

TriadChord qualityCircle of fifthsInversionKey-relationsModulationNeo-Riemannian transformationParsimonyRelative minorStacked thirdsTranspositionVoice-leading

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