Generator

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.

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. 6 of its 6 placements carry sound, built from the same numbers as the drawing, offering these buttons: C major → A minor, C major → C minor, C major → D major, C major → E major, C major → F major, C major → F♯ major, C major → G major, G major → C major.

Called by 5 essays

the blast radius of changing it

The triads each scale builds on its own degrees. Every chord that can be stacked in thirds on each degree of each scale, with its quality worked out from the intervals the scale actually supplies. The quality of the chord on each degree is a consequence of each scale's own step pattern rather than of a convention.

What a raised seventh is for

The harmonic minor is usually taught as a scale with an odd gap in it. It is better understood as a repair to a single chord — raising the seventh degree turns the dominant triad from minor to major, and everything else about the scale is the bill for that.

scales · Modes

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 seven chords of a key, by distance from home. Each triad of the major scale placed at a radius equal to how far its voices must move from the tonic chord. The chords that feel closest to home are the ones that are closest, in the plain arithmetic sense.

A progression is a path, and the map can be drawn

Chords in a key are not a list. They sit at measurable distances from home, and a progression is a walk across that space — which explains why some sequences feel like journeys and others like wandering.

harmony · Progression

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 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

All figures · What can be heard