Three ways to measure how far a key is

The first rung of this ladder is built on one fact: two keys a fifth apart share six of their seven notes, and the circle of fifths is the map that fact draws. It is a good map and it is a map of one quantity.

There are at least three quantities available and they do not agree. The disagreement is not noise: it is concentrated on one family of key relationships, and that family is the one whose use marks the change from the classical key plan to the nineteenth-century one.

The first two are one measurement #

Count the notes C major shares with each of the other eleven, and count the steps round the circle, and the two are related by a rule with no exceptions:

shared = 7 − |steps|, until it bottoms out at two.

One step: six shared. Two steps: five. Three: four. Four: three. Five and six: two, because a key six steps away shares only the two notes any two of these collections must share.

That is not an empirical finding; it is what a chain of fifths is. The diatonic collection is seven adjacent positions on the circle of fifths, so two collections k steps apart overlap in 7 − k positions, exactly, until the overlap runs out.

There is one wrinkle at the far side, and it is the reason the rule bottoms out. Five steps and six steps both give two shared notes, so the metric stops discriminating: by common tones, B major and F♯ major are equally far from C. By the circle they are five steps and six. The content metric saturates and the circle does not, which is a small disagreement between two things usually treated as one, and it is the only one available among the majors.

So the map of the twelve major keys really is one-dimensional. Any measurement of key distance that is a function of shared content is a function of position on the circle, and there is nothing else to find. That is worth saying because it is often presented as a discovery that common tones and the circle agree; they cannot do otherwise.

The third is a different quantity #

Now measure something else: how far the two tonic triads are apart, in total semitones, with the voices assigned to minimise the motion. That is the metric the voice-leading ladder is built on, computed by minimising over every assignment of three voices rather than by any rule of thumb.

The answers do not track the circle at all.

Read the extremes. D major is two steps from C, shares five of its notes, and its tonic triad is six semitones away — the largest distance in the table. E major is four steps away, shares three notes, and its tonic triad is two semitones away — the smallest, apart from C itself.

By content, D is close and E is far. By triad, E is close and D is far. Neither is wrong; they are answers to different questions.

The disagreement is the chromatic mediants #

The scatter is not random. The keys with small voice-leading distance and large fifth-distance are exactly E and A♭ — four steps sharp and four steps flat, sharing three notes each, tonic triads two semitones away.

Those are the chromatic mediants: the major keys whose tonics are a major third above and below C. Their triads are near because C–E–G and E–G♯–B share the note E and move the other two by one semitone each; C–E–G and A♭–C–E♭ share C and E and move G by one semitone. One or two common tones in the triad, and the rest moving by semitone, which is as economical as a change of triad can be.

The circle of fifths cannot see any of that, because a triad is three notes and a collection is seven, and moving between two collections says nothing about how their tonic triads are arranged.

Why the two cannot be reconciled #

It is tempting to look for a weighting that makes the two agree — some combination of common tones and triadic motion that ranks the keys sensibly. There is not one, and the reason is structural.

Shared content depends only on |steps|, so it is a function of one variable and it is monotone in it. Voice-leading distance between the tonic triads is not a function of |steps| at all: two keys at four steps out have distances 2 and 2, two at two steps out have 6 and 6, two at one step have 3 and 3, and two at three steps have 3 and 3. Written as a function of steps it goes 3, 6, 3, 2, 3, 6 — up, down, up, down.

One metric is monotone in the circle and the other oscillates with period three, because the triadic distance is really a fact about how many notes two triads a major third apart share, and major thirds divide the octave into three.

No monotone function of one can produce the other, so no weighting of the two is a refinement of either. They are two answers, and a musician using both is switching between them rather than combining them.

Which one a musician is using #

Both, in different centuries, which is the interesting part.

Common-tone distance governs the classical key plan. The pivot chord, which is the standard mechanism, exists only because the two collections overlap — though as the next section shows, “enough notes to build one” is a much sharper condition than the note count suggests, and how many bars of it a listener needs is the next rung’s subject. A sonata’s second subject goes to the dominant — one step, six notes shared, one accidental. The whole apparatus of pivot chords, of modulating to closely related keys, of the circle as a map of where a piece may go, is built on the collection metric.

Voice-leading distance governs the nineteenth-century one. Chromatic mediant relationships — C to E, C to A♭, C to E♭ — become a normal way to move a piece from one key to another in Schubert, Liszt, Wagner and after. They are far by the classical measure and cheap by the triadic one, and the change in practice is, in effect, a change of which metric is being minimised.

That is a strong claim and this site cannot corpus-check it. What it can say is that the two metrics genuinely disagree, that the disagreement is concentrated on a named family of relationships, and that the family is exactly the one whose use marks the change in style.

The pivot chord runs out three steps before the notes do #

The paragraph above assumed something worth checking: that a shared chord is available whenever the two collections share enough notes to build one. Four shared notes sounds like plenty — a four-note subset of a seven-note collection contains a triad more often than not.

It does not here. Counting the diatonic triads two major keys have in common gives a fourth column, and it collapses much faster than the note count.

A pivot chord exists out to two steps and no further. At three steps the keys still share four of their seven notes and share no triad at all: C and A hold A, B, D and E in common, and no three of those four stack in thirds. The overlap is real and it is the wrong shape.

That sharpens the classical claim rather than weakening it. A key plan built on pivot chords is not restricted to nearby keys because distant ones sound strange; it is restricted because past two steps the mechanism has nothing to work with. Tonic, dominant, relative minor and supertonic — the standard destinations — are exactly the keys inside the pivot’s reach.

And the direction is not symmetric either #

The other thing both metrics discard is which way the change goes, and the asymmetry turns out to be absolute rather than a matter of degree.

Take the notes that must be added to C major’s collection to reach each key, and ask what degree they are of the destination.

Sharpwards, the first note the music must borrow is the destination’s leading note, at every distance from one step to five; and the destination’s tonic is already in the old collection, at every distance from one step to five. The listener is handed a tendency tone that points at a note they already have.

Flatwards, the leading note of the destination is never among the borrowed notes at any distance, and from two steps out the destination’s tonic is one of them. The listener is handed the arrival itself, with nothing pointing at it.

So a sharpwards modulation announces where it is going and a flatwards one simply gets there, and that is arithmetic about arcs on the circle rather than a description of an effect. It is also why the classical key plan goes sharpwards first: the move that announces itself is the one a large form can afford to make early. It is also why the two feel different in a way no distance in this essay can express: they are the same distance and opposite mechanisms.

What each metric is a metric of #

The two are not rivals for one job. They answer questions about different objects, and naming the objects settles most of the confusion.

Common-tone distance is about what is available. It answers: how much of the material stays the same, how many chords belong to both keys, how many accidentals appear, whether a player has to think. It is a fact about the collection, and a collection is a resource.

Voice-leading distance is about what moves. It answers: how much has to happen at the moment of change, how many voices move and by how much, whether the two chords can be joined smoothly. It is a fact about a transition, and a transition is an event.

A modulation is both an event and a change of resources, so both quantities are real and neither is the modulation. The century that prized the first was writing music in which key was a large-scale structural resource; the century that prized the second was writing music in which the moment of harmonic change was the effect being sought. The same distinction shows up in what makes a cadence final, where total motion turned out to be the wrong statistic for exactly this reason: a small change is a poor ending, and a cheap transition is not a near key.

Adding the minor keys adds a direction #

Twelve major keys give a one-dimensional content map. Adding the twelve minors changes that, and the change is worth computing rather than describing.

The relative minor shares all seven notes — no major key does — and its tonic triad is two semitones off. The parallel minor shares four notes and its tonic triad is one semitone off, which is the smallest non-zero triadic distance in the table.

So the two most-used minor relationships sit at opposite ends of the two metrics — the relative maximises shared content, the parallel minimises triadic motion — and calling both of them “the minor” runs two different relationships together. The modes ladder found the same confusion from the other direction, where relative and parallel are two coordinate systems on one object.

Exactly two of the twenty-three are one semitone away: C minor and E minor. The first is the parallel and the second is the mediant minor, and they arrive there by different routes — the parallel keeps two notes of the triad and lowers the third, the mediant keeps two and lowers the root by a semitone. Below them, at two semitones, sit C♯ minor, E major, F minor, A♭ major and A minor, which is a list containing both chromatic mediants, both of their minors, and the relative. The triadic metric’s near neighbourhood is the mediant family and the parallel; the circle’s is the dominant and the subdominant, and the two lists share nothing at all.

A metric that agrees with neither #

There is a third family of measurements worth naming, because it is the one most often meant when somebody says two keys “sound” close: the number of accidentals that change.

That is not a new quantity. The number of accidentals is the fifth distance, exactly — one step round the circle is one sharp or flat — so it collapses into the first metric and adds nothing. Key signatures are a notation of the collection metric and nothing else, which is why they are useless as a guide to where nineteenth-century music actually goes.

The genuinely different third metric would be one that measured something a listener does in time: how long the change takes to be noticed, or how much of the old key survives it. That is the next rung, and it needs a running measurement rather than a distance.

What the picture cannot show #

A triad is not a key. Voice-leading distance here is measured between two tonic triads, which is a three-note stand-in for a seven-note collection and a whole tonal region. A modulation is not one chord change and the number does not claim to be a cost of modulating.

Neither of the three metrics knows about direction, which is why the asymmetry above had to be computed separately rather than read off any of them. The three columns of the hero figure are identical for a key k steps sharp and the key k steps flat, and the section above says the two are not the same operation at all. Nothing here proposes a fourth metric that would carry it; what it says is that a single number for key distance cannot, whichever number is chosen.

The census is over keys, not over passages. Nothing here has a piece in it. Two keys are compared as objects; what happens between them in a particular piece is a sequence of chords, and the sequence can make a distant key arrive gently or a near one arrive abruptly.

And voice-leading distance is one of several triadic metrics. Minimising total semitone motion is a choice; minimising the largest single move, or counting common tones in the triad, or counting moves on the Tonnetz all give slightly different orderings. They agree that the chromatic mediants are near, which is the part this essay relies on.

Whose music this is a claim about #

The classical key plan is a claim about European art music between roughly 1730 and 1830, and the chromatic-mediant claim is about the century after it. Both are the standard accounts and both are stated here without a corpus behind them.

The chromatic mediant is worth one more sentence, because its rise is usually explained by appeal to taste or to chromaticism in general, and the arithmetic offers something more specific. It is not that nineteenth-century composers became fond of distant keys; it is that they found the keys that are distant on one map and adjacent on another, and the effect they exploited is precisely that pair of facts holding at once — a chord change that costs two semitones and lands somewhere the key signature says is four steps away.

What is not a claim about any repertoire is the arithmetic. Shared content is a function of fifth distance in any music that uses diatonic collections; voice-leading distance between triads is what it is regardless of who uses it. The two metrics disagree whether or not anybody has ever exploited the disagreement, and the historical part of this essay is the suggestion that somebody did.

The ladder from here #

Both metrics above measure the distance between two keys as though a listener were handed both at once. A listener is handed one after the other, in time, with a passage in between — and the next rung asks how much of that passage has to go by before the change is detectable at all, with a pivot chord that is ambiguous by construction and a window that decides the lag as much as the music does.