One number for a page

Seventeen rungs have added terms to a reading load and the eighteenth spends the result. The pitch axis, the position axis, the duration term, the tie and what the first two share make one number a note, and a number a note makes a page comparable with another page.

Which was the point. The thirteenth rung said so in as many words — that a page’s difficulty could be quoted as one number instead of two, and that the trade a composer makes between a hard rhythm and a hard interval could then be priced.

The two axes are nearly the same size #

The six kinds of line span 5.26 bits between the easiest and the hardest, and the seven kinds of rhythm span 4.46. Their ratio is 1.18.

That is close enough to one to be worth stating as a result rather than as an approximation. A composer choosing to make a page harder has two knobs and they have almost the same range, so the choice between them is a choice about what kind of difficulty rather than about how much.

It was not obvious in advance and neither direction would have been surprising. Pitch has more distinct values than metrical position — twelve or more against eight — which argues for the pitch axis spanning more; metrical position has a far more unequal distribution, which argues the other way. The two considerations very nearly cancel.

The exchange rate that follows is the practical form of it. One bit of tune buys 1.18 bits of rhythm. A composer who wants to write a leaping line and keep the page at the difficulty it had must simplify the rhythm by about the same amount they complicated the tune, and the correction for the eighteen per cent is smaller than anything else in the measurement.

What the terms weigh against each other #

Before the trade, the four terms on one scale, because the ordering is not what the ladder expected when it started.

Where the note is costs more than which note it is — 2.23 against 1.89 — which is the fourteenth rung’s headline and was the surprise of that rung. The duration term is 0.67 and the tie term at a comparable density is about 1.05, so the two terms this ladder found last are together not far short of either axis.

That is worth reading as a criticism of the first thirteen rungs rather than of the last four. An anchor that spent nine rungs on pitch and one on duration was measuring what its notation makes conspicuous rather than what its reader pays for, and a staff makes pitch conspicuous by drawing it as a position and duration inconspicuous by drawing it as a shape.

The exchange rate below is between the two large terms because those are the ones a composer has continuous control over. A rest density and a tie density are properties of a texture rather than dials, which is why they enter the grid as an offset.

Pages that look nothing alike and cost the same #

The diagonals are the useful part of the drawing, because a diagonal is a set of pages a reader would call equally hard and a musician would not group together.

A scale in a syncopated rhythm and a leaping line on the beat sit near the same contour. So do a chromatic line in even quavers and a diatonic tune in a tresillo. Each pair is two pages that no editor, teacher or player would describe as similar, and the accounting says a reader spends the same on them.

That is the claim this ladder has been building toward since it started counting bits, and it is the one most exposed to being wrong. Nothing about a bit rate says a reader’s difficulty is a bit rate, and the two pages in each pair fail in different ways — a hard rhythm is missed as a rhythm and a hard interval is played as the wrong note.

What the measure predicts is the total error rate and not its kind, and even that is a prediction rather than a finding.

The pairing also has an uncomfortable consequence for teaching, which is where a difficulty measure would actually be used. A syllabus grades pieces by what they contain — the keys, the note values, the range — and the contour drawn here says two of those gradings are exchangeable at a stated rate. A student who can read a leaping line on the beat can, on this account, read a scale in a syncopated rhythm, and a syllabus that puts the two at different grades is asserting something the accounting denies.

That is a testable disagreement with an existing practice rather than a hypothetical, and the graded repertoire lists of any examination board supply the data for one side of it. It is also the one experiment on this whole anchor that a music school could run in an afternoon: two pages on one contour, twenty sight-readers, and count.

The extremes, and how far apart they are #

The hardest cell on the grid is a widely leaping line in a rhythm entirely off the beat, at 13.96 bits a note. The easiest is a scale on the beat, at 4.24.

A factor of three and a half between the hardest page a reader meets and the easiest is smaller than the difficulty feels. A scale on the beat is sight-read by a beginner in their first year; a chromatic leaping line in a syncopated rhythm defeats most professionals at tempo. Those do not feel like a factor of three and a half apart.

There are two readings of that and this rung cannot decide between them.

Either the measure is right and the experience of difficulty is highly nonlinear in the load — which is what a capacity limit does, since a reader with a fixed bit rate is comfortable below it and fails abruptly above it. Or the measure is missing terms that grow faster than these do, and the obvious candidate is that the chunking the twelfth rung found works far better for a scale than for a chromatic leap, so the easy end is easier than its entropy says rather than the hard end being harder.

The second is more likely and it is testable. A capacity limit predicts a threshold and a chunking failure predicts a curve, and a reading-error rate against this load would separate them on its shape.

The exchange rate a composer already uses #

There is a version of this trade in the practice and it is old enough that nobody argues about it.

A composer writing a fast passage simplifies its rhythm. Semiquaver runs are almost always even; the difficult rhythms in any repertoire sit at moderate tempos on longer notes. That is usually explained by what the fingers can do, and the fingers are certainly part of it — but a reader’s bit rate is a rate, so a passage at twice the tempo delivers twice the bits per second at the same bits per note, and the only way to keep the second quantity constant is to halve the first.

So the trade this rung prices between two axes has a third dimension the ladder priced six rungs ago, and the three together give a rule with content in it. A page is characterised by bits a note, and a performance by bits a second, and the second is the first times the tempo. A composer has three quantities and can hold any two.

Which is why fast music is diatonic and on the beat almost everywhere it exists, and why the exceptions — a Chopin étude, a bebop head — are described as difficult rather than as unusual. They are pages that spend all three at once.

What the number cannot be used for #

There is a use this measure invites and does not support, and it is worth refusing explicitly because the invitation is strong.

The number is not a grading. A page’s bit rate is a property of its notation and a piece’s difficulty is a property of everything else too: what the hands have to do, how fast it goes, whether the ink is legible, whether the player has heard it. Every one of those is absent here, and the fifth rung’s tablature is the standing reminder that a notation which is harder to read can be easier to play.

Nor is it a ranking of notations. Two notations for the same music have the same information in them by definition, so their entropies per note are the same and the differences are entirely in what a reader has to do to extract it — which is a decoding cost this measure has no term for at all.

What it is for is exactly the trade in its title: two quantities in the same unit, so that a decision between them can be made rather than argued about. That is a small thing to get from eighteen rungs and it is the thing the ladder was built for.

It has one other use and it is the one an engraver would reach for. A page’s rate is a rate, so a passage’s total is its rate times its notes, and a system that is uncomfortably dense in bits can be identified before anybody plays it. Nothing in engraving practice measures that; what it measures is ink and millimetres, which the tenth rung priced and which is a different quantity entirely. A page can be spacious and expensive, and the two measures would disagree about exactly the pages a player complains about.

What eighteen rungs measured, in one sentence each #

An anchor that closes should be able to say what it found, and this one’s findings are unusually independent of each other.

The stave is not a pitch axis; it counts letters, and the clef is the integer that offsets the count. The horizontal axis is not a time axis either; it is proportional to nothing and is set by a spacing rule an engraver chooses. A reader does not read notes; they read figures, and a scale of eight is one object where eight leaps are eight. Four parts are easier than two, because parts are related and a reader who has read one has partly read the others. And where a note is costs more than which note it is, which no account of sight-reading contains and which is the thing this ladder is most confident of.

Four of those five are about the page being something other than what it looks like. That is the anchor’s through-line and it was not planned: a staff is a picture that a reader has learned to stop seeing as a picture, and every rung here is a place where the picture and the reading come apart.

The last four rungs are the exception and they are about arithmetic rather than about seeing. What they add is that the reading has a price, that the price is measurable, and that two of the four things it is made of cannot be measured on the corpus this anchor was built from.

What the pictures cannot show #

Every number on the grid rests on the corpus — three tunes, a hundred and one notes — and the axes are distributions measured over them. A different three tunes would move every cell; whether it would move them together is the question, and the interaction rung’s per-tune reading suggests it would not entirely.

The duration term and the tie term are parameters rather than measurements, so the grid’s offset is chosen and its shape is measured. That is why the exchange rate is the most secure number on this page and the totals are the least: the exchange rate is a ratio of two ranges and both parameters cancel out of it exactly.

The six lines and seven rhythms are constructed rather than sampled — a scale, a chromatic line, a leaping line, and seven rhythmic patterns chosen to straddle the interesting region — so the ranges they span are ranges over a chosen set. A real repertoire occupies a cloud rather than a grid, and where its cloud sits is a measurement this ladder has never made.

There is one more thing the grid hides by being a grid. Every cell pairs a line with a rhythm as though a composer chose the two independently, and the seventeenth rung’s whole result is that they do not — the interaction is the measurement of a habit, and a composer writing a leaping line writes a rhythm to go with it. So the cells in the corners of the drawing are the ones music least often occupies, and the hardest cell in particular is a page nobody has written. The grid is a space of possibilities and the cloud inside it is a great deal smaller.

And the whole accounting is per note, which assumes a reader’s cost is a sum over noteheads. The twelfth rung already showed that is false — a reader reads figures — and every rung since has used the per-note measure anyway, because a per-figure measure needs a model of what a figure is and this collection has one only for pitch.

The measure against the one thing that could falsify it #

Every number here is derived and none is validated, and it is worth saying exactly what a validation would look like since the anchor is closing without one.

The measure makes one prediction that no simpler account makes: the inversions. There are pairs of pages the pitch axis alone calls one way round and the joint number calls the other — a leaping line in an easy rhythm against a simple line in a hard one — and any account that ranks pages by their tunes gets those pairs wrong.

So the experiment is not “does difficulty correlate with the number”. Correlation would be got by almost any measure, because almost any measure puts a chromatic leaping line above a scale. The experiment is the inversions specifically, and a design that used ten pairs chosen to invert would settle in an afternoon what eighteen rungs have argued.

That is a cheaper test than anything else this anchor has proposed and it is the last thing it owes.

Closing this anchor #

notation closes at eighteen rungs. What bounds a model is having said something about every variable it has, and this one has four.

What does the page say about pitch? Rungs one, seven and eight: the stave counts letters rather than pitch, the clef is an integer offset, and the notations invented for the overflow are all the same device.

What does it say about time? Rungs three, nine and ten: the time signature is a claim about accent, the horizontal axis is proportional to nothing, and a page holds what its spacing rule lets it.

What does a reader do with it? Rungs eleven to fourteen and seventeen: an eye with a span, a reader who reads figures, several parts that are not independent, two axes with sizes, and an interaction between them.

And what does it cost? Rungs fifteen, sixteen and this one: a duration term the corpus could not hold, a tie that is charged twice, and one number.

Every variable the model has now has a rung, and this one is the last because it turns four measurements into one quantity and spends it. What is not on the list belongs elsewhere: what a notation is for beyond being read is what a tablature keeps; how a key is spelled is the comma; how many notes a page can hold physically is the tenth rung and is a fact about ink.

The debt the anchor closes owing is a corpus, and it has been owed since the ninth rung without being named until the fifteenth. Three tunes with no rests and no ties in them cannot supply two of the four terms, and the two they cannot supply are the two this anchor’s last four rungs are about. What is needed is not large: a few hundred pages of engraved music with rests, ties and barlines counted, from which the rest density, the tie density and a wider interval distribution all fall out at once. Until that exists, the shape of everything here is measured and two of its four constants are chosen.