How much music a page holds

The axis that is not a time axis measured the staff’s horizontal coordinate and found it to be a legibility layout with a weak duration term riding on it. The notations invented for the overflow measured the vertical one and found that eleven staff positions is a different number of octaves on every notation that has been proposed.

The ninth rung ended by naming what the two have in common and neither had asked. Every notation this ladder has measured is a page, and the count that would say something about a page is how much music it holds — because that decides how often it turns, and a page turn is a thing a player with two hands occupied cannot do.

The two axes, and why only one of them depends on the notation #

Vertically, everything depends on the notation. A staff has eleven positions — five lines, four spaces and the space above and below — which the clef fixes to a letter and and how much pitch those eleven cover is the whole of the eighth rung’s argument. Seven to the octave gives one and a half octaves; twelve to the octave gives eleven semitones, which is less than one; six to the octave gives nearly two. A compass wider than one staff needs several, and how many is the vertical cost of the notation.

Horizontally, nothing does. A system is justified to the margins, so the spacing rule decides where the notes fall inside a system rather than how many fit in one. What decides that is the other end: the shortest note has to be wide enough to read, and under a rule of exponent k a note of duration d is (d/d_min)^k times that width.

So the flatter the rule, the less a long note costs and the more music a system holds. A proportional layout gives a whole note sixteen times the width of a sixteenth; a columnar one gives it the same; Ross’s √2-per-doubling gives it four.

What the numbers say about the rules that exist #

Ross’s exponent is 0.5 and Gould’s is 0.42, and they are the two rules engravers actually use. They differ by eight hundredths and they produce page capacities within four per cent of each other, which is a small enough difference to be a matter of taste.

What they are both a long way from is proportional. A truly proportional layout — the page as a time axis, which is what a reader naively believes a score to be — costs about a third of the music on a page, and therefore a page turn every thirty-two seconds instead of every forty-six. Over a fifteen-minute movement that is twenty-eight turns against nineteen.

That is the practical case against proportional notation and it is not about legibility. The ninth rung’s finding is that engraved spacing departs from time by up to six per cent of a system on an ordinary phrase, and the obvious response — make it proportional and the departure goes to zero — has a cost nobody had put a number on. The number is nine extra page turns a movement.

The compass, which is the other half #

The vertical axis is where the notations differ, and the difference is bigger than the spacing rules’ difference.

Take a compass from E2 to C6, which is a cello’s or a modest keyboard’s. On the ordinary staff that spans about nineteen semitones per staff, so three staves cover it. On a chromatic staff each staff covers eleven semitones and four are needed. On a whole-tone staff each covers twenty-two and two suffice.

Four staves against two is a factor of two in the vertical, and the horizontal is unchanged, so it is a factor of two in the page. A whole-tone staff holds 2.33 times what a chromatic staff does, at the same legibility per note, for the same music — on this compass, and the qualification turns out to be load-bearing.

Both terms of that ratio are ceilings: a compass needs a whole number of staves, so the vertical cost jumps rather than sliding. Sweeping the compass from an octave and a half to six octaves, the ratio takes exactly four values:

The 2.33 is the maximum the ratio ever reaches, and the compass chosen for the figures sits at the top edge of the only band that produces it. E2 to C6 is forty-four semitones; forty-six gives 2.00 and a two-octave compass gives 1.75. So the honest statement is that a whole-tone staff holds between one and a half and two and a third times what a chromatic one does, that the figure depends on the compass in jumps rather than smoothly, and that the number this essay was about to quote is the best case.

That does not change the direction of the argument — the whole-tone staff wins on every compass — and it changes what can be said about the size. It is also a warning about the shape of the quantity: a ratio of two ceilings has no meaningful derivative, so a piece whose range grows by a semitone can lose a sixth of its page capacity and nothing about the notation has changed.

That is the argument the chromatic-staff proposals never had made against them in these terms. Their case is a good one — one position per semitone removes accidentals entirely, which is the eighth rung’s own measurement, and it removes the ambiguity two names for one key is about — and the cost is that the reader turns pages a third more often than they do now.

What the spacing rule is actually trading #

The two ends of the spacing exponent are two different things a page can be, and the ninth rung named them without pricing them.

At k = 1 the page is a time axis: a note’s horizontal position is proportional to its moment, and a reader can see rhythm as distance. That is the property a graphic score wants and it is what a naive reader assumes an ordinary score has.

At k = 0 the page is a list: every note gets one column, rhythm is carried entirely by the note heads and flags, and the horizontal axis carries nothing at all. That is what a chord chart is, what most tablature is, and — nearly — what a lead sheet is.

Ross and Gould sit near the list end. The ninth rung measured what that costs in fidelity: up to six per cent of a system’s width between where a note is and where its moment is. This measures what it buys, and the two numbers can now be set against each other: six per cent of a system’s fidelity buys thirty per cent more music on a page.

That is a trade an engraver has been making for a century and a half without either number, and both of them turn out to be small enough that the trade is genuinely close.

The page turn as a design constraint #

It is worth saying plainly why page capacity is the quantity rather than density, because the two are not the same argument.

Density is about the eye. A denser page is harder to read and there is a limit somewhere, and no measurement here says where.

A page turn is about the hands. A pianist has both occupied; a string player has one on the bow; a wind player has both on the instrument and their mouth on it as well. Turning a page is either free — there is a rest — or it is not available, and the composer either arranges for a rest or the player memorises or somebody else turns.

So a notation that turns pages a third more often than the ordinary one is a notation that imposes a third more of that problem on every piece written in it, and it does so invisibly: nothing about a chromatic staff looks as though it costs page turns, because the cost is in the vertical axis and the turns are a consequence of the whole page.

There is a cost to the horizontal axis that the capacity figures hide, because they count notes and a reader reads moments.

So the page holds what it holds by being not a time axis, and the density this essay has been counting is bought with that distortion. A reader following the page as a clock is most wrong exactly where the music is slowest — which is the opposite of where a time axis would be least useful, and is the price of every note the page fits.

Which computation produced the numbers #

The vertical is ledgerCost, which the eighth rung uses: for a stated compass it reports how many staves each notation needs and how many ledger positions are left over. A system is that many staves, each five spaces tall, with a gap between systems, and a page holds as many systems as fit in its height.

The horizontal is a normalisation the ninth rung’s engravedPositions does not do, and the essay’s whole argument turns on noticing that it does not. That function justifies a run of durations to a stated width, because a real system is justified to the margins — so it cannot answer “how many notes fit”. What answers it is the minimum legible width of the shortest note, times the spacing rule’s own exponent applied to each duration.

The units are staff spaces, which is the only length notation has: a page is quoted as 120 spaces wide and 160 tall, which for a five-space staff is about A4 with ordinary margins. Every ratio in the essay is independent of that choice and only the absolute seconds depend on it.

The seconds come from a tempo and a note density — a hundred beats a minute at two notes to the beat — and both are stated arguments. The ratios are the result; the absolute figures are an illustration, in the way this collection keeps having to record about its own numbers.

The one notation that escapes both axes #

There is a notation in this collection that the arithmetic treats very differently, and it is worth putting beside the three.

A tablature has no pitch axis at all. Its vertical positions are strings, so their number is a property of the instrument rather than of the compass, and a six-string tablature covers a guitar’s whole range in six lines however wide that range is. There is no ledger line, no second staff and no overflow — what a tablature keeps is the essay about what it gives up to get that, which is any representation of pitch a reader can transpose.

So a tablature’s vertical cost is constant and its horizontal cost is the same as everybody’s, which means its page capacity is the best available by a wide margin. The margin is computable under the same model: six lines is two and a half staff spaces against the ordinary staff’s five, one staff always rather than three, so a page holds eighteen systems against four and lasts three and a half minutes against forty-seven seconds. Four and a half times, and — the part that matters more than the size — the same on every compass, because nothing about a guitar’s range enters the arithmetic.

That is the trade the fifth rung named, seen from the page. A tablature buys a factor of four and a half in page capacity, and constancy in the compass, with the ability to say anything about pitch that is not tied to one instrument. The two halves of that sentence are usually argued separately and they are one decision: what makes the vertical axis cheap is exactly that it is not a pitch axis, so there is nothing for a wide range to overflow.

That is a third position in the same trade the chromatic staff and the whole-tone staff occupy, and it is the only one of the four that has ever been widely used alongside the ordinary staff rather than instead of it.

Where the model stops #

A page of music is not a run of notes. Real engraving carries clefs, key signatures, rests, dynamics, slurs, text and barlines, all of which take horizontal room, and a real system is not full — three notations, one progression is about how much of a page is not notes. The count here is an upper bound on a page’s capacity and the ratios between notations survive that because the overhead is nearly the same for all of them.

The minimum legible width is asserted. Two and a bit staff spaces for the shortest note is a stated number, and every absolute figure scales with it. It does not affect the comparison between rules, because all four rules are given the same minimum.

And a system’s height is not only its staves. A real score leaves room above and below for dynamics, and a piano score’s two staves are braced with a fixed gap. The model has one gap parameter and it is the same for every notation, which flatters the notations that need more staves.

What the picture cannot show #

It cannot show reading speed. A page that holds more music is a page that is denser, and a denser page is read more slowly by an amount nobody here has measured. The whole trade might be neutral, and the figure is silent about it.

Nor can it show where the turns fall. A page turn is not a constant cost: it is free at a rest and impossible in the middle of a run, so what matters to a player is not turns per minute but whether a turn lands somewhere it can be taken. That is an engraving decision and it is the reason a real page is not full.

And it cannot show what a page is for. An orchestral part, a chamber part and a solo piano score have completely different capacities and completely different constraints, and the model has one page and one density.

Whose notation, and when #

The compass, the page proportions and the two spacing rules are those of Western engraving since about the middle of the nineteenth century — the point at which plate engraving and then photolithography made a page a standardised object. Ross’s manual is 1970 and Gould’s is 2011, and they are codifications of a practice that had been oral.

The chromatic and whole-tone staves are proposals rather than practices, and they belong to a long line of them: the ladder’s eighth rung counts several, mostly from the century between 1850 and 1950, and none of them is used. The reasons usually given are conservatism and the weight of existing repertoire, and both are surely right. What this figure adds is that one of them — the chromatic staff, which is the proposal with the best argument — carries a measurable cost in page turns that the others do not, and that the proposal with the best page arithmetic is the whole-tone one, which is the one with the worst argument about accidentals.

The two axes push in opposite directions, and the notation in use sits between them. That is not evidence that the staff is optimal; it is evidence that it is a compromise, which is what the whole of this ladder keeps finding.

Where this ladder goes next #

Ten rungs. The stave is not a ruler; two names for one key; the time signature is a claim; a mark that is not a level; what a tablature keeps; three notations for one progression; the clef is an integer; the notations invented for the overflow; the axis that is not a time axis; and now the two axes multiplied.

What the ladder owes now is the reader. Every quantity here is a property of the page, and a page exists to be read by somebody whose eyes move across it in saccades of a measurable size at a measurable rate — the published figures for sight-reading put the eye a fixed distance ahead of the sounding note, measured in notes rather than in centimetres. That converts a page’s density into a reading distance, which is the quantity that decides whether a layout can be sight-read at all, and it is the only thing in this ladder that would put a listener’s limits and a page’s arithmetic in one number.