Keys are neighbours, and the map is computed

C major and G major share six notes. The only difference is that C major has an F and G major has an F-sharp.

That is not a curiosity. It is the generating fact of the entire system of keys, and nearly everything a musician learns about key relationships is a restatement of it.

Why one note, and not two #

The one-note relationship follows from the diatonic set being a contiguous run on the circle of fifths.

C major, written as a run of fifths, is F–C–G–D–A–E–B. G major is C–G–D–A–E–B–F♯. The two runs overlap in six positions; one note falls off the flat end and one is added at the sharp end.

Move two steps and two notes differ. Six steps — a tritone away — and six of the seven differ, which is as far apart as two major keys can be. The circle of fifths is therefore not a mnemonic device; it is a metric space, and distance along it is a genuine measure of how much two keys have in common.

What a key signature is #

Given the run picture, key signatures stop being a table to memorise.

A key signature records where the seven-note window sits on the chain of fifths. Zero sharps is the window F–C–G–D–A–E–B. One sharp moves it one step: the F drops off and F-sharp joins at the other end. Two sharps: C drops, C-sharp joins.

The order of sharps — F, C, G, D, A, E, B — is the chain of fifths itself, read from the flat end. The order of flats is the same chain read backwards. That is why the two orders are reverses of each other, a fact usually presented as a coincidence worth memorising.

It also explains the last-sharp rule. The last sharp in a signature is the seventh degree of the key, because the seventh degree is the sharp end of the window. And the last flat is the fourth degree, because a flat is added at the flat end of the window, which is the fourth.

Both rules are consequences of one picture, and neither needs remembering separately.

Relative and parallel, which are different distances #

Two keys can be close in two quite different senses, and the difference is instructive.

A relative minor shares every note with its major — A minor and C major have identical pitch content and differ only in which note is home. On the map of shared notes they are at distance zero. They are a rotation of the same set, which is to say the same mode question again.

A parallel minor shares only four notes with its major. C major and C minor differ in the third, sixth and seventh degrees — three notes — which puts them three steps apart on the circle of fifths, the same distance as C major to E-flat major.

Yet parallel major and minor feel closely related — closer, to most listeners, than C major and E-flat major do, though the pitch distance is identical. The reason is that they share a tonic, and shared tonic is a different kind of nearness from shared content. Modal mixture, the borrowing of chords between parallel keys, works precisely because that second kind of nearness exists and the first kind does not.

So there are two metrics, they disagree, and both are real. The circle of fifths draws only one of them.

What makes a modulation feel far #

A modulation to a neighbouring key needs one new note. That note can be introduced quietly, in a chord that both keys contain, and a listener may not register the moment of transition at all.

A modulation to a distant key needs several new notes at once, and there is no chord common to both to arrive through. It has to be asserted rather than eased into, and the assertion is the effect.

The technique for the first case is the pivot chord: a chord belonging to both keys, presented in the old key and reinterpreted in the new. C major and G major share four triads, so there are four available pivots and the change can be made almost imperceptibly. C major and F-sharp major share none, and no pivot exists — which is what six semitones of voice motion looks like in practice.

That is the whole of why some modulations sound smooth. It is a count of shared chords, and the count follows from the count of shared notes, which follows from the position of two windows on one chain.

The circle is a spiral wearing a disguise #

Everything above is true in equal temperament and misleading everywhere else.

The chain of fifths does not close. Twelve pure fifths overshoot seven octaves by 23.46 cents, so the twelfth step does not arrive back at the start. Drawing it as a circle asserts an identity — that G-sharp and A-flat are the same note — which is true only because equal temperament made it true.

In meantone tuning, G-sharp and A-flat are 41 cents apart, which is a fifth of a semitone and unmistakable. The far side of the circle is not a distant key but a broken one, containing the wolf, and composers avoided it because it was unusable rather than because it was remote.

The distinction matters historically. Before about 1750, key distance was partly a matter of shared notes and partly a matter of which keys the instrument could produce at all. Afterwards it was purely the first, and every key became equally available — which is what made nineteenth-century harmonic practice possible.

How a modulation is actually made #

The map says which keys are near. Getting from one to another is a procedure, and there are four of them.

Pivot chord. Find a chord belonging to both keys, present it as a member of the old one, then follow it with a chord that only makes sense in the new one. The listener’s interpretation of the pivot is revised retroactively, and the join is invisible. This is the standard method for near keys and requires shared chords, which requires shared notes.

Direct. State the new key without preparation. Effective, abrupt, and the only option for distant keys — which is why a lurch to a remote key reads as a gesture rather than as a transition.

Chromatic. Move a single voice by a semitone to convert a chord in the old key into one in the new. Cheap in voice-leading terms and disorienting harmonically, which is the nineteenth century’s favourite combination.

Enharmonic. Reinterpret a chord as its enharmonic equivalent — a German sixth as a dominant seventh, or a diminished seventh as any of four different chords. This exploits the fiction that the circle closes, and it only works in equal temperament: in meantone the two spellings are genuinely different chords and the trick is unavailable.

That last one is worth registering as a historical marker. Enharmonic modulation is impossible before equal temperament and ubiquitous after it, and the technique appears in the repertoire almost exactly when the tuning does.

The bass says which key #

There is a practical asymmetry the map leaves out: the note in the bass does most of the work of asserting a key.

A chord’s root in the bass states it plainly. The same chord in inversion is more ambiguous, and a piece establishing a new key almost always does so with the tonic in the bass at the moment of arrival. That is why the strongest cadence has the dominant root moving to the tonic root in the bass, and why the same chords over a different bass line do not settle anything.

Key character, and whether it survived #

If every key is a transposition of every other, then keys cannot differ in character, and the eighteenth-century literature describing them is describing nothing.

That literature is extensive. D major triumphant, E-flat major noble, F minor funereal, B minor patient — writers from Charpentier to Schubart produced detailed and reasonably consistent tables. In equal temperament these can only be conventions or synaesthesia.

In unequal temperament they were physically real. A well temperament gives near keys nearly pure thirds and far keys wide ones, so C major genuinely is a smoother sonority than F-sharp major on the same instrument. The descriptions were observations, and they became a tradition when the observations stopped being available.

Two residual effects survive equal temperament and are worth separating from the folklore: open strings, which make D and A major genuinely more resonant on string instruments; and absolute pitch height, since a key a semitone higher does sit higher. Neither accounts for the detailed character tables, and neither is nothing.

Whose music, and when #

The circle of fifths as a diagram is younger than the relationships it describes. Nikolai Diletsky published a version in Kyiv in 1679; Johann David Heinichen published the familiar circular form in 1728. The relationships were understood long before, in the form of the chain and the gamut.

Its dominance in modern teaching is worth being slightly suspicious of. It is an excellent map of diatonic key relations in equal temperament, and it silently asserts three things: that the octave has twelve notes, that enharmonic pairs are identical, and that the fifth is the right metric.

Chromatic harmony of the nineteenth century increasingly used relationships the circle does not represent well — chords a major third apart, sharing one or two notes, which are close in voice-leading terms and far on the circle. The lattice shows those relationships and the circle does not, and neither picture is complete.

Whether listeners hear any of this #

The map describes relationships between keys. Whether a listener perceives them is a separate question, and the experimental answer is more limited than the theory implies.

Listeners without absolute pitch cannot identify a key. They can, reliably, detect that a modulation has happened, and they respond differently to near and distant ones. What they appear to track is not the key itself but the stability of the current tonal centre, and a modulation registers as a disturbance whose size scales roughly with distance on the circle.

Over long spans the evidence gets thinner. Whether a listener notices that a movement ends in the key it started in — the structural claim on which a great deal of analysis of sonata form rests — has been tested, and untrained listeners largely do not, while trained ones do somewhat better than chance. That is an uncomfortable finding for a theory that treats long-range tonal closure as the organising principle of a form.

The safe position is that key relationships are real in the score, real in the sounding intervals, and perceived at short range with confidence and at long range weakly. Which is not nothing, and is less than the diagrams suggest.

Relative minor, and the ambiguity it creates #

A major key and its relative minor have identical pitch content, and that identity is exploited constantly and causes a specific problem.

Because the notes are the same, a passage can slide between C major and A minor without introducing a single accidental. Composers use this for shading — a phrase that begins in the major and cadences in the relative minor has changed mode without changing material.

The problem is that a key signature is therefore ambiguous by design. Two sharps means D major or B minor, and the page does not say which. A performer works it out from the harmony and from where phrases end, which is exactly the problem modes have and for the same reason: notation records pitch content and not which pitch is home.

The convention that resolves it is the leading note. B minor uses A-sharp, which is not in the signature, so the accidental’s presence announces the minor key. Notation solves the ambiguity by relying on the fact that the minor key does not actually use its own signature unaltered.

Where the model stops #

One metric only. The circle measures shared pitch content. It does not measure shared tonic, voice-leading distance, or common chords, and those disagree with it and with each other.

Equal temperament. The circle only closes because of it, and every claim about distant keys being merely distant depends on it.

Major keys. The inner ring of relative minors is drawn as an afterthought here, and minor keys have relationships of their own — through the harmonic minor’s raised seventh — that the diagram does not carry.

Modulation is a process. The figures show which keys are near. They say nothing about how a piece gets from one to another, which is where the actual craft is, and which happens in time.

Twelve keys, one scale #

The last consequence is the one that made the system worth building.

There is one major scale. There are twelve places to put it, and in equal temperament all twelve are exact transpositions — identical in every internal proportion, differing only in absolute pitch. A piece can be played in any of them and a listener without absolute pitch cannot tell which was chosen.

That is a remarkable amount of freedom and it was expensive. It required giving up pure intervals everywhere, which cost the beatless thirds that meantone had, and it required accepting that keys stopped having characters. What it bought was the ability to modulate anywhere and come back, and to write for fixed-pitch instruments that work in every key.

Every diagram on this page depends on that trade having been made. The circle closes because somebody decided it should, and the map of key relationships is a map of a decision as much as of a structure.

The ladder from here #

Later rungs: pivot chords and common-tone modulation. Chromatic third relations, and the harmonic space the circle cannot draw. Modal mixture and borrowed chords. The Tonnetz as an alternative map. Enharmonic modulation, which uses the fiction of the circle deliberately. Key character in unequal temperaments, tested. Tonal plans in large forms. And the question of whether key relationships are perceived at all by listeners without absolute pitch — which the experimental literature answers less encouragingly than the theory would like.

The order of sharps and the order of flats are the same sequence read in opposite directions, and generations of musicians have memorised them as two separate facts. They are one fact, and it is the chain of fifths, which is also the circle, which is also the key signature, which is also the reason a modulation to the dominant is the easiest move in tonal music.