The notations invented for the overflow

The clef is an integer counted the staff’s capacity exactly: eleven positions between the lowest line and the highest space, seven of them to an octave, so one clef covers an octave and a fourth and everything outside it goes onto ledger lines. It ended by naming the rung the sixth had already flagged beside it — the notations invented for music this system cannot hold.

There are dozens of them and they are not a miscellany. Every one is a response to a specific overflow, and the commonest response is the same: give every semitone its own position.

Forty per cent of the range, for all of the accidentals. That is the trade in one line, and it is arithmetic rather than opinion: eleven divided by twelve against eleven divided by seven.

Why the overflow is real #

The staff’s problem is not imaginary and the proposals are not cranks.

Seven of the twelve pitch classes have a position and five do not, so five need a symbol added — a sharp, a flat, or a key signature that applies one in advance. In diatonic music that is cheap: a key signature is five symbols once and the rest of the page is clean, which is what the time signature’s own claim is doing in the other axis — a statement made once that the rest of the page is read against. In chromatic music it is not, and the more chromatic the music the worse it gets.

So a chromatic staff is a reasonable proposal. Klavarskribo, Dodeka, the Express Stave and perhaps thirty others since the seventeenth century all do the same thing in different dress: twelve positions per octave, no accidentals, one symbol per sound.

What it costs in space #

The first cost is the one the hero figure computes and nobody proposing a chromatic staff seems to have.

Eleven positions is what a five-line staff gives, and it is not negotiable without changing how many lines a reader can count at a glance. Spread over twelve positions per octave instead of seven, those eleven cover 0.92 of an octave rather than 1.57.

The consequence for a real range is worse than the ratio, because ledger lines grow with the overflow rather than with the range. Writing a piano’s middle four octaves takes three staves and about fifteen ledger positions on a diatonic staff, and four staves and thirty-three on a chromatic one.

Most chromatic-staff proposals answer this by adding lines or by using two-colour line groups, and both answers cost the property the five-line staff was chosen for, which is that five lines can be counted without counting.

What it costs in what can be said #

The second cost is not about space at all, and it is the one this collection has already argued about at length.

Two names for one key established that the seven letters are not a lossy compression of the twelve. They are a contiguous window on the chain of fifths — F C G D A E B — and an accidental is a step along that chain. So a written pitch carries two independent things: which sound it is, and where on the chain it sits.

A chromatic staff cannot write the difference between C sharp and D flat, because the difference is not in the accidental but in the position, and it has only one position for both.

That is not a subtlety about spelling conventions. It is the loss of the information where the chain was never closed is about, of what a meantone keyboard’s split keys were for, and of the whole apparatus by which a leading tone is distinguished from a flattened supertonic — and a degree is where it goes next is the essay about a system in which that difference is the degree’s whole identity. A notation that cannot make the distinction cannot record a performance in any temperament but equal.

There is a third cost that follows from the first two and is easy to miss. A staff that covers less range needs its clef changed more often, and a clef change is a symbol and a discontinuity: everything the reader had learned about where the lines are is invalidated. The clef rung found that eight clefs a third apart do the work of fifteen on a diatonic axis; on a chromatic axis the same coverage needs more clefs closer together, and each of them is used for less music.

So the proposals compound. Fewer accidentals, more ledger lines, more staves, more clef changes — and the last three are all the same overflow arriving in three currencies.

Pricing the disease, which needs no corpus #

The case for a chromatic staff is that accidentals are expensive, and everything above prices the remedy without pricing the complaint. That looks like a job for a corpus and it is not: a passage’s accidentals are decided by which pitch classes it uses and which key signature is chosen for it, and both are enumerable.

For every subset of the twelve pitch classes, take the best of the twelve signatures and count how many members of the subset fall outside it. Averaged over all subsets of each size:

Five, at the very worst. Music that uses every pitch class in the octave needs an accidental for five of them and no more, because seven of the twelve always have a position; and the fraction of the pitch classes that need one therefore never exceeds five twelfths.

Put that beside the range figure and the two numbers turn out to be the same number. A chromatic staff loses 1 − (11/12)/(11/7) of its range, which is 5/12. The diatonic staff makes a writer add a symbol for 5/12 of the pitch classes, at most. Both are five twelfths, and they are the same five — the five that have no position of their own, counted once as extra symbols and once as extra space.

That makes the trade unusually easy to judge, and it judges badly. The remedy costs the worst case of the disease, unconditionally. A page of fully chromatic music pays five twelfths in accidentals and would pay five twelfths in range; a page of diatonic music pays nothing in accidentals, because the signature absorbs them all, and would still pay five twelfths in range. The saving is bounded above by the cost and equals it only for music that never repeats a key.

The distribution says the same thing less starkly. Even at nine pitch classes out of twelve, two thirds of the subsets need three accidentals and a third need two — so a chromatically active passage is typically buying back two or three symbols an octave, against a permanent forty-two per cent of the vertical.

The one system that has both #

There is a system with more range per staff than the diatonic one and it is worth computing because it shows the trade is a curve rather than a corner.

A whole-tone staff — six positions to the octave — covers 1.83 octaves in eleven positions, which is 17 per cent more than the staff and twice a chromatic one. It needs an accidental for six of the twelve rather than five. And it collapses twenty-nine distinctions rather than twenty-three, because six positions is a worse window on the chain of fifths than seven.

So the three systems are not ordered: going from twelve positions to seven buys range and buys spelling; going from seven to six buys more range and loses spelling. Seven is not the densest option and it is not the sparsest; it is the one that happens to be a fifths window, and that is a fact about the number seven and the chain of fifths rather than about vision or about paper.

Why seven is the essay about that coincidence from the other side, and it is worth reading as the answer to this one: the staff has seven positions because the scale has seven notes, and the scale has seven notes because seven fifths is where the chain nearly closes.

Which computation produced the numbers #

Two integers and a division, and then two counts.

The eleven positions are the clef rung’s own STAFF_POSITIONS: five lines and six spaces, counting the space below the bottom line and above the top one. The octave coverage is eleven divided by however many positions the system spends per octave.

The ledger cost is the range in semitones minus what one staff covers, converted back into positions at that system’s density. It counts positions rather than lines because a ledger line serves the position on it and the one above it, and comparing systems by lines rather than positions would flatter whichever has more positions per line.

The spelling capacity is writtenPitches, the site’s own enumeration: seven letters times five alterations from double flat to double sharp, which is thirty-five distinct written pitches with a computable pitch class and a computable position on the chain of fifths for each. A system with one position per sound is defined to have one spelling apiece, so what it loses is the difference.

Where the model stops #

Five lines is not sacred, and the hero figure’s slider is that admission. Every other count here assumes eleven positions, and a chromatic-staff proposal is free to use six lines or seven — at seventeen positions a chromatic staff covers an octave and a fifth and the range objection nearly goes away. What it cannot do is use seventeen and be read: the reason for five lines is that a reader identifies one by its place in a group without counting, and the number of things that can be done that way is small. So the real constraint is perceptual and this model has substituted an integer for it.

Ledger lines are not the only answer to an overflow. An octave-transposing clef, a change of clef mid-system, and an ottava mark all extend the range at a cost the arithmetic here does not price. The clef rung priced the first of those and found eight clefs a third apart do the work of fifteen.

And the count of what is lost is a count of distinctions, not of their value. Twenty-three enharmonic pairs collapse, and how much that matters depends entirely on the repertoire: in equal-tempered keyboard music written after about 1900 it may matter not at all, and in seventeenth-century vocal polyphony it is most of what the notation is doing.

It is worth putting the two costs beside each other, because they are not the same kind of thing. The range cost is a nuisance: more paper, more ledger lines, more clef changes, all of them tiring and none of them fatal. The spelling cost is a loss of expressive power — there are things the staff can write that the replacement cannot, and no amount of extra lines brings them back.

A notation that loses range is inconvenient and a notation that loses distinctions is smaller. The stave is not a ruler is the rung that established what the staff’s axis actually measures; this one says what happens to a system that measures something else.

Whose music, and what each proposal was for #

The staff is a European object of about a thousand years’ standing, designed for a diatonic repertoire, and the complaint against it is a complaint from a repertoire it was not designed for. That framing makes the history legible.

Klavarskribo (1931) was designed for keyboard players and is a picture of the keyboard: twelve positions, arranged as the black and white keys are, read vertically down the page. It buys exactly what a keyboard player wants — a note’s position is where the finger goes — and pays everything above, plus the ability to be read by anybody not at a keyboard.

Chromatic staves in general were proposed for chromatic music, and the proposals cluster in the periods when the repertoire got most chromatic: the 1890s to the 1920s, and again in the 1960s.

And the notations for music the twelve cannot hold at all are a different family with a different answer. A microtonal system needs more positions than twelve, not fewer accidentals, and every serious one has instead added accidentals — Sagittal’s array of arrows, Helmholtz–Ellis’s modified sharps — which is the staff’s own strategy applied harder. Those are the proposals that did not try to replace the axis, and they are the ones in use.

That is the pattern worth naming. Every proposal that changed the axis — the number of positions per octave — has failed to displace the staff, and every one that kept the axis and extended the accidentals has found a repertoire. The arithmetic above says why the first group has a hard problem: changing the axis costs range and costs spelling, and neither cost was what the proposal was about.

The graphic scores of the mid-twentieth century are the one family this does not describe, and they are outside the arithmetic entirely. A score that specifies a gesture rather than a pitch is not a competing axis; it is a refusal to have one, and there is nothing here to compute about it. What a tablature keeps is the rung about the other kind of refusal — a notation that fixes the action and lets the sound follow.

What the picture cannot show #

Whether any of this is why the staff survived. Notations persist for reasons of installed base, training and printing that have nothing to do with capacity. What the arithmetic gives is a cost that every replacement has had to pay, not a demonstration that paying it is what killed them.

The accidental count above is per pitch class and not per page. A section of this essay bounds the disease exactly — five twelfths of the pitch classes, at worst — and that bound is what a page pays only if every pitch class it uses appears equally often. How many accidentals actually reach a page depends on how often the chromatic notes recur, on whether they fall inside one bar, and on the conventions for cancelling them, and those are corpus measurements — the same debt the rhythm ladders recorded for a different reason. What is not a corpus measurement, and is settled above, is the ceiling.

And reading is not counted. The whole question is about what a reader can take in at a glance, and every quantity here is a count of symbols. A system with more symbols in a worse arrangement may still read faster, and nothing in this collection can say.

Where this ladder goes next #

Eight rungs, and all of them the same shape: a notation fixes some coordinates of a musical object and leaves the rest to be supplied, and which ones it fixes is a claim about what the object is. The staff fixes the letter and leaves the pitch; the signature fixes the bar and leaves the grouping; the dynamic mark fixes an order and leaves the spectrum; the tablature fixes the action and leaves the sound; the clef fixes where eleven positions sit; and now, the alternatives fix the sound and lose the letter, which is the coordinate everything else in this collection’s tuning arguments is written in.

What the ladder still owes is the coordinate none of the eight rungs has looked at, and it is the one every page carries and no essay here has counted: the horizontal axis. A staff’s vertical axis has been measured to a position; its horizontal one is proportional to nothing at all — a semibreve occupies the space the engraver gives it, and the space is set by how many notes are underneath rather than by how long the note lasts. That is a notation whose time axis is not a time axis, in a collection that has spent two ladders on the milliseconds it discards.