Acoustics & Music Theory
why it sounds like that
Concept

Circle of fifths — where it appears

The twelve pitch classes arranged so that each is a fifth from the last, which closes only because equal temperament makes it close. In any tuning with pure fifths the twelfth step overshoots the start by a comma and the circle is a spiral.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

The circle of fifths. The twelve pitch classes arranged so that each is a fifth above the last. Keys next to each other differ by one sharp or flat, which is why the notes of a key form a contiguous arc rather than a scattering.

Keys are neighbours, and the map is computed

Two keys a fifth apart share six of their seven notes. That single fact generates the circle of fifths, the key signatures, the shape of nearly every modulation, and the sense that some keys are far away.

scales · Key-relations 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 A model that cannot say two keys does not say it is unsure. I – IV – V – I played in C major and in a second major key at the same time, with the second key moved round the circle of fifths. For each separation: the correlation the standard key-finder gives its single best answer, and the correlation reached by the best PAIR of key profiles — a hypothesis the finder does not have. The pair recovers both keys that are sounding at every separation, 7 of 7. The single answer names neither of them at 4 of the 7, and its confidence does not fall when it is wrong: at four steps apart it reports E minor at r = 0.886, against 0.959 for the same progression in one key.

Two keys at once

Every key figure so far assumes one key is sounding, and the standard key-finder has no value it can return that means two. Play one progression in C and the same progression a major third away at the same time, and the model does not report uncertainty: it reports E minor, at a correlation of 0.886, against 0.959 for the same progression in one key. It is as confident as it ever is, and neither key it names is being played. Give it the missing hypothesis — pairs of key profiles rather than single ones — and it recovers both keys at every separation, all seven of seven.

harmony · Key-relations Four of the five are a whole number of semitones, and one is exactly half of one. Each mismatch in cents, against the ticks at whole semitones — which are the only places a transposing keyboard can put a player. 4 of the 5 land within six cents of a tick: the Chorton–Kammerton gap is 197 cents against a whole tone's 200, and Chorton against French pitch is 296 against a minor third's 300. The exception is an English organ against Handel's fork, at 50 cents — 50 cents from the nearest tick, which is as far as it is possible to be. So the small mismatches are the unsolvable ones, and the large ones were solved by shifting the keys.

The instrument that cannot be moved

A string is regauged and a woodwind is scaled. An organ's pitch is the length of its pipes, and metal can be cut off and cannot be put back — so an organ is a ratchet that only goes sharp. The mechanical answer was to shift the keyboard against the pipes, and its cost is not the transposition. It is that the temperament's key colours rotate out from under the notation, by an amount measured in fifths rather than in semitones.

tuning · Pitch standard

Named alongside it

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

ModulationKey-relationsTranspositionRelative minorChromatic mediantEnd correctionEnumerationKey colourKey-findingKey signatureOrganPitch standard

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