Acoustics & Music Theory
why it sounds like that
Generator

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.
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.

Drawn above with its standard settings, which is almost never how an essay draws it: an essay states the numbers it is arguing about, so the figure a reader meets there is about that argument rather than about the drawing in general. Every option a placement passes is checked against the ones this function actually reads, because an option it does not read is silently ignored and the figure quietly draws what is above instead.

It makes a noise. 7 of its 7 placements carry sound, built from the same numbers as the drawing, offering these buttons: L — C major to E minor, P — C major to C minor, R — C major to A minor, Six steps round the ring.

Called by 3 essays

the blast radius of changing it

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 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 The surprise of each chord, against the uncertainty it arrived into. The information content of each step — minus the log of its probability under a distribution that multiplies the root-motion weight by how well the destination triad's notes fit the key — with the entropy of the moment before it drawn behind. I – IV – V – I: I→IV 1.71 bits, IV→V 2.80 bits, V→I 1.60 bits, against a mean uncertainty of 2.63; I – IV – V – vi: I→IV 1.71 bits, IV→V 2.80 bits, V→vi 2.61 bits, against a mean uncertainty of 2.63. A surprise larger than the entropy it arrived into is an outcome the model was not expecting even given how uncertain it was; one below it is an outcome the model had already mostly bet on. An earlier essay produced the first of those numbers and had no way to produce the second, because a set of preferences is not a distribution and only a distribution has an entropy.

A chord, given a key and a predecessor

A chord's improbability has been priced from its root motion alone, which left one multiplication unmade: a chord is also improbable because its notes do not fit the key, and that number has been available since the probe-tone profile. Multiplied and renormalised, the two give a conditional distribution — and a distribution has an entropy, which is the quantity a surprise has to be read against and which a list of preferences cannot supply.

harmony · Tonal-expectation

All figures · What can be heard