The key plan is the form

Ask what the form of a classical first movement is and the answer comes back as a diagram of boxes: exposition, development, recapitulation, with subjects and transitions inside them. It is a description of a piece as a container.

There is a much older way to describe the same object, and it is the one the eighteenth century itself used. A movement of that kind is a departure and a return. It starts at home, it goes somewhere, the somewhere is unstable and gets less stable, and then it comes back and stays. The boxes are consequences.

The advantage of the second description is that it is about a quantity this site already computes. A key is a place, keys are neighbours in a measurable sense, and a movement’s key plan is a path across that neighbourhood which can be drawn.

The path #

The hero figure is that path. The horizontal axis is position in the movement; the vertical axis is distance from home, measured in steps along the chain of fifths — which is the site’s own key-distance measure and is the one that puts C and G one apart and C and F sharp six apart.

The shape is not subtle. Out by one step, stay there for a sixth of the movement, then a middle section that goes further and does not stay anywhere, then a return that arrives before the two-thirds mark and does not leave again.

Two features of that shape are worth naming because both are usually described in terms of the boxes rather than the journey.

The exposition’s second half is in a different key from its first. That is the single most important fact about the form and the boxes obscure it: a “second subject” is not primarily a second tune, it is a passage in another key. The tune is how a listener recognises it; the key is what it is.

The recapitulation’s second half is in the home key, where the exposition’s was not. So the recapitulation is not a repeat of the exposition — it is the exposition with its journey removed, and the whole point of the movement is that the material which was away is now home.

Why the dominant #

The destination in the hero figure is the key one step sharpwards. That is overwhelmingly the commonest choice in the repertoire, and the usual explanation — that it is the nearest key — is not quite true, because there are two keys equally near.

Seven keys either side of home, with their seven notes marked, say why the dominant is the first stop: one step round the chain changes exactly one note, so the nearest key is the one a listener can be moved to without being told. Two steps changes two, and so on down.

So the answer is not in how many notes are shared. It is in which one is not.

Sharpwards, the new note is the arriving key’s leading note. It belongs to no chord of the home key. It appears, it is heard as an accidental, and its only comfortable function is to resolve upward to the new tonic — which is precisely what a modulation needs. The note that marks the departure is also the note that establishes the destination.

Flatwards, the new note is a flattened seventh, and the home key has a use for it: it is the seventh of the tonic chord treated as a dominant, which is the ordinary way of approaching the subdominant without leaving home at all. So the same note that would signal a move flatwards is a note the home key uses regularly for something else, and a listener hearing it has no reason to conclude that the tonic has changed.

That asymmetry is arithmetic. Both keys are one step away, both share six notes, and only one of them announces itself.

It is worth noticing that this is the only place the two directions differ. The count of shared notes is the same, the count of shared triads is the same — four either way, by the census below — and the qualities of those triads are the same. Everything about the two moves is identical except which pitch class the destination adds, and that single difference is enough to make one of them the standard modulation of a whole repertoire and the other a device for going to the subdominant without leaving home.

The asymmetry is why the return is easy #

The same fact run backwards explains something about the second half of the movement.

Getting from home to the dominant requires introducing a note and treating it as a leading note. Getting from the dominant back to home requires dropping a note — the raised fourth simply stops appearing — and the home key’s own tonic becomes available again. That is not an event. It is a relaxation, and it is the reason the two halves of a movement are not the same length of work even when they are the same length of music.

Which is why a recapitulation can arrive without ceremony while an exposition’s modulation needs a transition passage to accomplish it. A departure has to be made; a return can be allowed. Composers of the period spend real effort on the transition and rather little on the retransition’s arrival, and the arithmetic says why.

And a modulation needs a hinge: a chord belonging to both keys, presented in the old 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; the further apart the keys, the fewer there are, and past five steps there are none at all.

A modulation needs a hinge #

The arithmetic above says which key to go to. It does not say how to get there, and the mechanism is worth a section because it is where the shared-tone count comes back into use.

Six of the seven notes are common to home and the dominant, so six of the triads built on them are available in both keys — a chord that is the supertonic at home is the submediant in the dominant, and nothing about the sound of it says which. Such a chord is a pivot, and a modulation of this kind is normally made by playing one, and then playing a chord that is only possible in the new key.

That is the leading note doing its work. The pivot establishes ambiguity; the new key’s dominant seventh, containing the raised fourth, resolves it. Two chords, and the second of them is the only one that has to be there.

The number of available pivots is worth counting rather than describing, because it does not fall off the way the shared-tone count does. Taking every diatonic triad of the home key and asking which are also diatonic triads of the destination:

The count collapses to zero at three steps, where four of the seven notes are still shared. Four common notes cannot contain a triad of either scale, because a triad needs all three of its notes present and the four that survive are never three-of-a-kind — so a three-step modulation has no diatonic pivot chord at all.

That is a stronger statement than needs a longer preparation, and it changes what the preparation is. A modulation of one or two steps is made by playing a chord that belongs to both keys; a modulation of three or more has to be made some other way entirely, and the ways are the ones the repertoire uses: a chromatic pivot borrowed from the parallel minor, a diminished seventh that belongs to no key, an augmented sixth, or simply a juxtaposition with nothing in between. The distinction is not between easy and hard but between two different devices, and the boundary between them is at exactly two steps.

It also says why the classical key plan stays where it does. The hero figure’s destinations are one and two steps out and its excursions go three; one and two are pivot territory and three is where the composer has to reach for something chromatic. That the development section is where the chromaticism lives is usually explained as a matter of expressive intensity, and the arithmetic offers a duller reason: it is the only part of the movement that goes far enough to need it.

And the pivot count is symmetric. Four sharpwards and four flatwards, two and two, zero and zero — identical at every distance. So none of the asymmetry this essay found earlier is in the pivots; it is entirely in which note the move introduces, and the mechanics of getting there are the same in both directions. Key distance is a count of shared notes; the count of usable pivots is what that distance is for, and it has a cliff in it that the shared-tone count does not.

The development goes further and stays nowhere #

The middle of the hero figure has a different character from either end, and the difference is visible as a shape rather than as a label.

Its points are further from home — two, three, four steps — and no two consecutive points are the same. That is the whole definition of instability in this description: not that the music is dissonant or loud, but that it does not remain in one key long enough for a listener to settle.

There is a nice consequence for the boundary-finding machinery. A passage that changes key every few bars produces a great deal of local novelty, so a checkerboard operator run over a development section reports peaks everywhere — and reports the section boundaries at its edges as no more remarkable than its interior. That is not an error. A development really is a stretch of continuous change, and a change detector should say so.

It also explains why the retransition works. After several minutes of not staying anywhere, standing on one chord for eight bars is an event of the first magnitude, and the chord chosen is the dominant, whose resolution is the arrival. The stability is the signal.

The plan is slower than anything a listener holds #

One number puts this essay’s object in its place beside the rest of the field.

A classical first movement lasts something between six and fifteen minutes. Its exposition is a third of that, so the second key is in force for two to five minutes continuously. The window inside which a series of events is heard as one thing is at most eight seconds.

So the key plan is between fifteen and forty times too slow to be perceived as a shape. Nobody hears the hero figure. What a listener has instead is a sequence of local impressions — this is home, this is not home, this is a long way from anywhere, this is home again — held in memory and assembled afterwards, and the assembling is done over a span for which there is no perceptual mechanism at all, only recollection.

That is worth saying because it changes the status of the claim. The key plan is a real property of the music, it is public, and it is what the composer worked with. It is not a perceptual object, and any account of why the form works has to run through the local events — the transition that unsettles, the arrival that settles, the retransition that stands still — rather than through the shape of the path. The path is a description of the music; the events are what a listener has.

Whose music, and for how long #

Everything above is about European art music of roughly 1740 to 1830 and about the popular forms that inherited its harmonic language. It is a narrow claim and it needs to be, because both halves of it fail elsewhere in interesting ways.

In the seventeenth century the modulation plan is not a plan of the whole. Movements are shorter, key contrast is more local, and the large form is often a matter of texture or of a repeating bass. In the nineteenth century the plan widens: the second key is as likely to be a third away as a fifth, and by the end of the century a movement may visit keys that share three notes with home. What survives is the departure-and-return shape; what changes is how far out is thinkable.

And in the twentieth century the plan can be abandoned entirely, which removes the mechanism this essay describes and requires something else to do its work. That is not a decline; it is the reason so much twentieth-century form is built on repetition, register and timbre instead.

Outside Europe the whole apparatus is missing. A raga does not modulate — the pitch material is fixed for the duration, and what develops is register, speed and elaboration. A gamelan’s pathet is a mode rather than a key and there is no move between them within a piece. Neither tradition is short of large-scale form, which is a useful check on the temptation to treat modulation as necessary rather than as one available mechanism.

The same journey, one field along #

The departure-and-return shape is not unique to key, and noticing where else it appears makes it easier to see what it is for.

A comma pump is a progression that leaves home in pure ratios and does not come back — the same journey with the return broken, and the reason it is startling is that the return is the part everyone assumes. A chain of fifths walks away from a starting note and arrives a comma short of where it began, which is the same shape again at the level of tuning rather than of form.

In each case the object is a walk on a lattice, the steps are small, and the interest is entirely in whether the walk closes. A key plan is the one instance where closing is guaranteed by construction — the composer simply writes the home key again — and where the interest is therefore in the route rather than in the arrival.

That is worth stating because it locates what a key plan is doing. It is not solving a problem, as a temperament is. It is spending a resource: distance from home is a quantity a listener tracks, and a movement is a decision about how to spend it and when to stop.

What this cannot show #

The vertical axis is steps along the chain of fifths, and that measure does one thing badly on purpose. A major key and its relative minor share all seven notes, so on this axis a move to the relative minor is a move of zero steps — and the hero figure has to mark it by colour rather than by height.

That is a real limitation and it is the same one the novelty operator has: a description built on which notes are present cannot see a change of tonic within one collection. What separates a major key from its relative minor is emphasis, cadence and the raised seventh of the minor, and only the last of those is a pitch-class fact.

The positions along the horizontal axis are proportions rather than bar numbers, and they are the proportions such movements are described by rather than a measurement of any particular score. Nothing in the figure should be read as a claim about a specific piece.

And a key plan is silent about everything a listener notices moment to moment. Two movements with identical key plans can be entirely different pieces, and usually are.

There is also nothing here about the minor mode, which behaves differently enough to need its own treatment. A minor-key movement’s second key is normally the relative major rather than the dominant, which is three steps sharpwards on this axis rather than one — a much larger move by the measure used here, and one that a shared-tone count says is no move at all. Whichever measure is used, the minor mode’s plan is not the major mode’s plan transposed, and the two modes differ in more than a third.

Where the ladder goes #

The key plan is the slowest-moving variable in a movement. The fastest-moving one is how often the chord changes, and it turns out to be an independent variable with a floor and a ceiling of its own — set, at the top, by something outside music entirely.