The circle is a circle, and the map is not

The circle of fifths is the first diagram anybody learns in this subject and one of the few that survives contact with everything after it. It is usually presented as a convenient arrangement — a mnemonic that happens to put related keys together, which is how the first rung of this ladder introduced it.

It is more than that and also less. Among the twelve major keys it is not a convenient arrangement; it is the only arrangement, because there is exactly one quantity to arrange them by.

One variable, therefore one dimension #

Here is the argument in full, because it is short and it is stronger than the usual presentation.

A major key’s notes are seven adjacent positions on the circle of fifths. Two keys k steps apart are two arcs of seven offset by k, so they overlap in 7 − k positions for k up to five, and in two positions for k of five or six.

Shared content is therefore a function of k and of nothing else. Not approximately: exactly, with no exceptions among the 66 pairs. Any measurement of key relatedness built out of shared notes — how many chords two keys have in common, how many accidentals differ, whether a pivot chord exists and how many there are — is a function of that same k, because all of them are functions of the overlap.

A set of twelve objects with one number between each pair, and the number depending only on how far round a cycle they are, is a circle. There is no second dimension to look for and no arrangement that would reveal more.

That is worth having as a positive result, because most of this ladder’s work has been showing that a familiar picture is incomplete. This one is not incomplete. Within its domain it is exhaustive, and the domain is the twelve major keys measured by content.

What one dimension costs #

A one-dimensional map is exhaustive and it is also very poor, and both halves of that are worth stating.

It is exhaustive because there is nothing else to measure. It is poor because one number is all it ever returns: asked how F♯ major relates to C major it says six, and asked how B relates to C it says five, and the difference between those two answers is one. There is no room in it for the fact that one of those keys is reached by sharps and the other by flats, that they are spelled differently, or that they are the same key on a piano and different keys in a string quartet’s intonation.

The saturation at the far side is the clearest symptom. Five steps and six steps both give two shared notes, so on the content metric the tritone-away key and the one next to it are equally distant, and the metric has run out of resolution exactly where the interesting question about remote keys begins.

Where the second dimension comes from #

Add the minor keys and something genuinely new appears.

The relative minor shares all seven notes. Distance zero on the content metric, and it is not the same key. That is a relationship the major-only map has no room for: every entry in that map at distance zero was C itself.

So content stops being a complete description. Two keys can now differ while sharing everything, and what distinguishes them is which note is home — which is precisely the quantity the scale ladder proved no property of a set can supply, arriving here as an axis on a map.

The two moves are independent in the strict sense: one changes the collection and not the tonic’s relationship to it, the other changes the tonic and not the collection. Neither can be expressed in terms of the other, so the map of twenty-four keys needs two coordinates.

And because both directions wrap — twelve fifths return to the start, and a relative minor’s relative major is where it began — the surface is a torus rather than a plane. That is the standard result, and it is worth deriving from the two moves rather than asserting.

The parallel is not on the grid #

There is a third relationship in constant use and it is not either of the two axes.

C major and C minor share four notes, which is a content distance of three steps — but the two are not three steps apart in any useful sense, because their tonics are the same note. And their tonic triads are one semitone apart, which is the smallest distance in the whole twenty-four-key table.

Put the parallel on the torus and it is a diagonal: three steps flatwards and down to the minor row. Nothing about that is wrong, and it is a poor description of a relationship that a musician experiences as staying in the same place and changing colour.

So the torus is the map of content and tonic, and it is not the map of everything. The mediant relationships from the previous rung are also diagonals on it, and they are also cheap by a metric the torus does not draw.

Why the relative pair is drawn one below the other #

The torus figure puts each major key’s relative minor directly beneath it, and that choice is not arbitrary — it is the only pairing under which the vertical move changes nothing about the collection.

Pair each major with its parallel minor instead and the vertical move changes three notes, so the two axes stop being independent: moving down and then sideways is not the same as sideways and then down. Pair it with the minor a fifth below and the same problem appears. The relative pairing is the unique one that makes the grid commute, and that is a fact about the arithmetic rather than a design decision.

It also explains why the inner ring of the standard circle-of-fifths diagram is drawn with the relative minors rather than the parallel ones, which is a choice most presentations make without comment. Any other choice would make the diagram’s two directions interact, and the diagram would stop being readable as a grid.

How many dimensions there actually are #

The honest answer is that it depends on what is being measured, and the count goes up as the metrics do.

• Content alone, majors only: one dimension, a circle, exhaustively.
• Content and tonic, all twenty-four: two dimensions, a torus.
• Add voice-leading distance between tonic triads: not embeddable in the torus, because the mediants and the parallel are near by it and far on the surface.

The third is the interesting one, and “not embeddable” is a claim strong enough to be worth checking rather than asserting. Give each of the twenty-four keys a torus coordinate — steps round the circle for its collection, plus one for a change of row — and measure both distances from C major to each of the other twenty-three.

Voice-leading distance is not a function of torus distance. Not merely a distorted one: at a torus distance of two it takes the values 1, 5 and 6, and at four it takes 1, 2 and 5. One position on the map covers nearly the whole range of the other metric.

That settles it in the strongest available form. A map is a placement of points, so any quantity a map can draw must be some function of the placement — and this one is not, so no rescaling, no stretching of the surface and no relabelling of the axes would let the torus carry it.

The size of the disagreement is worth a number too. Take all 253 pairs of the twenty-three keys and ask which of each pair is nearer to C by each metric: the two disagree on 91 of them, thirty-six per cent. That is not a metric with an occasional exception; it is a second, largely independent ordering.

The extreme case is the one to keep. E♭ minor sits at the far corner of the torus, seven steps away — the greatest distance any of the twenty-four reaches — and its tonic triad is four semitones from C major’s. D major sits two steps away and its triad is six. The most remote key on the map is nearer, by the metric a voice actually travels, than one of the map’s own neighbours.

Two smaller assertions in this essay were checked at the same time and both hold exactly. The overlap formula max(0, 7 − k) + max(0, 7 + k − 12) reproduces the observed shared-note counts 7, 6, 5, 4, 3, 2, 2 at every step count, and the pentatonic’s counts come out 5, 4, 3, 2, 1, 0, 0 — the flat tail arriving by exhaustion rather than by wrapping, as claimed.

Two maps, two metrics, and each of them exhaustive within its own domain. That is the shape of the answer, and it is better than a single map with everything on it would be, because a single map would have to distort one metric to accommodate the other.

The same shape, twice, in two fields #

There is a structural echo here worth naming, because it is the second time this phase has run into it.

The modes ladder found two coordinates on one object: relative, which changes which degree is home while the collection stays; and parallel, which changes the collection while home stays. That was a statement about modes of one collection, and it was made without any reference to keys.

This essay finds the same two coordinates on the map of keys. Moving sideways on the torus changes the collection; moving down changes which degree of it is home. They are the same two coordinates and the same object seen at two scales — a mode is a choice of tonic within one collection, and a key is a choice of collection with the tonic fixed by convention at the first degree.

That explains an asymmetry in the vocabulary. There are seven modes of one collection and twelve transpositions of one mode, and both facts are usually taught as though they were about different things. They are the two axes of one grid, and the grid is this one with all seven degrees drawn rather than only the first and the sixth.

A test the argument could fail #

The claim that shared content is a strict function of fifth distance is a claim about all 66 pairs of major keys, and it is the kind of statement that would be easy to assert and wrong in one corner.

The check is cheap and the result has one interesting feature. Every step count from zero to six maps to exactly one shared-note count — 7, 6, 5, 4, 3, 2, 2 — with no pair anywhere disagreeing. The mapping is not injective, because five and six both give two, so the function is monotone and not strictly monotone.

That single collision is the whole of the metric’s imperfection, and it has an exact cause. Two arcs of seven on a cycle of twelve, offset by k, overlap in max(0, 7 − k) + max(0, 7 + k − 12) positions: a direct overlap at one end and a wrap-around overlap at the other. At five steps that is 2 + 0. At six steps it is 1 + 1. The direct overlap falls by one and the wrap-around overlap rises by one, and they cancel exactly.

The formula is worth having because it says when the effect happens. The wrap-around term is nonzero only once k exceeds 12 − m, so a collection larger than half the universe wraps and a smaller one does not — a fact this phase has now met three times, most recently in the census that forces a deep scale to be about half its universe.

Run the same count on the pentatonic and the picture is different: 5, 4, 3, 2, 1, 0, 0. Five of twelve is less than half, so the wrap-around term never fires; the collision at five and six steps is there anyway, because the overlap has simply hit zero and cannot go lower. Two different mechanisms produce the same flat tail, which is the sort of coincidence worth catching before it becomes an explanation.

What the picture cannot show #

Everything here is equal temperament. The circle closes because twelve fifths are seven octaves, and they are not — in any unequal temperament the chain does not close, the twelfth key is not the first, and the circle is a spiral. Every claim in this essay about the shape of the map is a claim about a tuning decision made in the eighteenth century.

“Key” here means a diatonic collection with a tonic. A passage in C major with borrowed chords is not a point on this map; it is somewhere between several. The map has no way to represent a mixture and most real music is one.

The map has no direction. Sharpwards and flatwards are symmetric here and they are not symmetric in practice: a sharpwards move introduces the new key’s leading note, which belongs to no chord of the old key, and a flatwards move introduces a note the old key already contains inside its own subdominant. The circle draws those two as the same move in opposite directions.

The torus coordinate used in the embeddability test is one of several. It counts steps round the circle for the collection and adds one for a change of row, which is the natural graph distance on the grid the figures draw. A different weighting of the two axes — two for a row change, or half — moves the numbers in the table and does not move the finding, because the finding is that one torus position holds several voice-leading values, and no weighting of the axes can separate positions that are already the same.

And distance is not difficulty. Nothing here says a distant modulation is harder to write, harder to hear or rarer. The previous rung found the opposite of the naive expectation on the timing question, and there is no reason to think a spatial metric predicts anything about a listener without a separate argument.

Whose music this is a claim about #

The circle of fifths as a pedagogical object dates from the early eighteenth century and it arrives at the same time as the practice it describes: a repertoire in which pieces modulate to adjacent keys, in which key signatures are stable, and in which the enharmonic equivalence that closes the circle is available.

Before that the picture is a spiral rather than a circle, and the far end of it is a wolf rather than a return. After the nineteenth century the map’s dimensions stop constraining anything much, because the relationships that get used are the ones the map draws as diagonals.

So the circle describes about a hundred and fifty years exactly, is a distortion of the two centuries before it, and is an under-description of the century after. That is a good run for a diagram, and its persistence is earned by the argument at the top of this essay: within its domain, there is genuinely nothing else to draw.

Where the ladder goes next #

Five rungs in, this ladder has a map of keys, three metrics on it, and a measurement of how long a move between two of them takes to notice. What it does not have is any account of what a key change does to a piece — whether a modulation to the dominant at the same point in two different movements is the same event, and what a listener has to remember for it to be one.

That is a question about form as much as about harmony, and it needs the repetition machinery rather than the pitch-class machinery. It is the obvious next rung and it is not written.