Where the note is costs more than which note it is

Four parts are easier to read than two put a whole texture on the reading measure and found that a notehead of a chorale costs 1.30 bits against an independent line’s 1.89. It ended by naming the thing every rung of this ladder has left out:

Every line measured here and on the rung before it is a run of equal notes, and every teacher’s account of sight-reading difficulty puts duration at least level with pitch.

That is not a small omission. The three rungs that price a reader at all price only what is on the vertical axis, and a student at a stand stumbles over a note that arrives in the wrong place at least as often as over one at the wrong height.

The ingredient has been on this site since the third rung of this same ladder, unused for this purpose. The time signature is a claim works with a metrical hierarchy — a number for every position in a bar, saying how strong that position is. A set of non-negative numbers over a set of positions, divided by its own sum, is a probability distribution. And a probability, on this ladder, is a number of bits waiting to be taken.

A weight is a probability, and one had already been read as one #

The substitution is the twelfth rung’s, applied to a different coordinate. A reader does not read notes prices a note at minus the log of the probability of the interval that reached it, where the probabilities come from the distribution of melodic steps the melody ladder measured over these tunes. The rhythmic version prices a note at minus the log of the probability of the position it landed on.

Both are first-order and both are categorical. The melodic one asks which of a set of interval sizes; this one asks which of the eight quaver positions in a bar of four. Nothing else changes, and that is the whole of the mechanism.

What is surprising is that this collection had already done it, on a ladder about harmony, and called it something else. Expectation is a curve, not a list joins the same metrical weights to a rate of harmonic change and produces a hazard — the chance that the chord changes on this beat given that it has not yet. It then takes minus the log of that hazard and calls it a timing surprise, which is what a listener pays for a change arriving when it does.

At one chord change a bar the scaling in that construction is exactly the normalisation used here, and the two quantities are the same numbers to fourteen decimal places. The figure above checks it rather than asserting it. So the substitution proposed here was already made once in this collection, for a chord and never for a notehead, and the reason it looked new is that the two live on different ladders and nothing had put them side by side.

There are two routes to the distribution and this collection holds both. The stated route takes the metre ladder’s own weights for 4/4 — one on the downbeat, 0.85 on the third beat, 0.5 on the second and fourth, 0.15 on every offbeat quaver — and divides by their sum of 3.45. The measured route counts where the notes of the three tunes this collection carries actually fall: 28, 2, 26, 4, 28, 0, 16 and 0 across the eight positions, a hundred and four notes in twenty-eight bars.

They agree where it matters, and not everywhere. The stated weights come out of the preference-rule literature and are an account of what a listener infers; the counts are an account of what a composer wrote. The two put the eight positions in the same order — every position on a beat below every position off one, the downbeat and the third beat below the second and fourth — with one tie broken: the weights make the second and fourth beats equal where the tunes prefer the second, because a note on the third beat is often held through the fourth. And they price the tunes’ own rhythm at 2.23 bits a note and 2.41, a fifth of a bit apart.

Where they part company is a single position. The quaver after the downbeat is 4.52 bits under the stated weights and 5.70 under the counts, because only two notes in a hundred and four ever land there. The disagreement is more than a whole bit and it costs almost nothing, for exactly that reason: a position the tunes avoid enters the average in proportion to how rarely it is used. The routes are far apart where the evidence is thin, and that is the shape a disagreement should have.

The two positions with a count of zero are the interesting case. They are the offbeat quavers after the third and fourth beats, and no note in these tunes ever lands on either. The measured route has to charge them something finite, and it charges them the same floor the melodic measure gives an interval nobody plays. That floor is doing real work and it is named again below.

Seven rhythms, priced the way six lines were #

The twelfth rung prices six kinds of melodic line at eight notes each. The same shape of question is available here: seven rhythms, one bar each, priced against the distribution.

The ordering is the one a sight-reader would give, and the sizes are the informative part. Putting every note on a beat is the cheapest bar there is. Running quavers cost 4.35 a note, not because a quaver is hard but because half of them have to land somewhere weak. The tresillo — the pattern the metre ladder opened with, three onsets in the time of eight — costs 3.10.

Between the cheapest and the dearest is a factor of three, and every one of those rhythms is written with the same eight noteheads on the same eight positions of the same bar. Nothing about the ink distinguishes them.

The two routes can be run against each other on the same seven rhythms, and this is where they stop agreeing.

That reversal is the honest limit of the agreement. The two routes settle the tunes’ own cost within a fifth of a bit and settle which end of the list is which, and they disagree about the middle. An anticipation — a note pulled forward onto the quaver before a beat — is common enough in these tunes to be cheap under the stated weights and rare enough in them to be dear under the counts, and there is no third route here to break the tie.

So the ordering of two syncopations against each other is a claim about a repertoire, and a small one. The size of the gap between a bar on the beat and a bar off it is not. The counts make that gap 4.46 bits a note and the stated weights make it 2.18, and both are larger than the 1.32 bits that separate a scale from a chromatic run on the axis this ladder has been using for two rungs.

The rhythm is the larger half #

The number the debt asked for is the comparison, and it comes out on the wrong side of level.

These three tunes cost 1.89 bits a note in pitch and 2.23 in time. Where the note is, is 54 per cent of what a reader takes off the page, and which note it is, is 46.

The cleanest way to see that the two axes are genuinely commensurable is to compare the distributions rather than the tunes. The melodic distribution over interval sizes has an entropy of 2.09 bits; the distribution of onset positions has 2.23. A page with no theory in it — an interval drawn uniformly from the seven sizes that occur, a position drawn uniformly from the eight in the bar — would cost 2.81 and 3.00. So each axis saves its reader about three quarters of a bit, and the two are the same size to within six per cent.

The debt predicted that duration would be at least level with pitch, and the arithmetic puts it slightly above. That is a confirmation rather than a discovery, but the margin is worth stating: this is not an axis that has to be argued into relevance and then found to contribute a tenth of a bit. It is the bigger of the two.

It is worth being clear about why, because the reason is not that rhythm is intrinsically harder. It is that a bar of 4/4 read at the quaver has eight places a note can go and these tunes use six of them, while the melodic distribution is dominated by a single interval — a step of a tone is 43 per cent of everything these tunes do, and no position in the bar carries more than 27 per cent of their notes. The pitch axis is more predictable than the time axis in this repertoire, and a more predictable axis is a cheaper one.

What the trade costs, and what it does not #

A composer who wants a passage to be harder can widen an interval or displace a note, and the debt asked for that trade to be priced. It can be, and the units make the answer legible.

A syncopation and a leap of a fourth cost a reader the same. That is the exchange rate, and it holds under the other route too: the stated weights put a displacement at 2.74 bits, which is more than a semitone’s 1.84 and less than a third’s 3.10. Both routes put the move off the beat inside the range of melodic moves a composer already thinks of as costly, and neither puts it outside.

The ladders overlap along their whole length, which is what it means for two axes to be on one scale. The dearest metrical move — onto one of the two positions these tunes never touch — is 6.07 bits, above a minor third and just below an interval nobody plays. The cheapest non-trivial melodic move, a repeated note, is 0.47, above the second beat’s 0.11 and below the fourth beat’s 0.81.

And here the debt asked for something the arithmetic will not give. It hoped that the trade could be priced, with the implication that a composer faces a rate that varies and therefore a choice with a best answer somewhere in the middle. It does not. The two terms are added, so a contour of equal difficulty is a straight line at forty-five degrees, the exchange rate is one bit for one bit everywhere along it, and there is no interior optimum to find. A bar is cheaper for having one fewer note in it, always, and the model’s favourite bar is the empty one.

That is a real result rather than a failure to find one, and it says where the interesting question actually is. The rate being flat means the only thing left to ask is where on the contour a repertoire chooses to sit, and these tunes sit at 1.89 and 2.23 — not balanced, and not at either extreme.

One number rather than two #

If a page’s difficulty is two numbers, a reader has to be told how to combine them. If the two are in the same units, they add, and a page has one number.

Fifteen per cent of the pairs invert, and the clearest instance is a page a teacher would recognise. A chromatic run written entirely off the beat costs 9.66 bits a note; a wide-leaping line written on the beat costs 9.14. Measured by their pitches alone the leaping line is more than twice the chromatic run, and that ordering is the wrong one.

That is the argument for quoting one number. Two numbers, reported side by side, invite a reader to use whichever is larger, and on this plane the larger one is the wrong one on one comparison in seven.

The joint number also does something to the twelfth rung’s headline figure that is worth stating plainly. That rung reports a factor of nearly four between a tune and a wide-leaping line. Add the second term and the same comparison, both lines on the beat, is a factor of 2.3 — because the rhythmic cost is shared and a shared term dilutes a ratio. The pitch axis alone overstates how different two pages are, and here it puts them 63 per cent further apart than the joint number does.

The span is the quantity the eleventh rung built its reading distance on and the one the twelfth rung converted from notes into bits, so it is the natural place for a second term to show up. It does: a tune as written gives four notes, the same tune in running quavers gives 2.64, and the same tune displaced entirely off the beat gives 1.94.

Half the span, and not one notehead has moved up or down. A teacher who says a student loses their place in syncopated music is describing that number.

Which computation produced the numbers #

The metrical weights are the metre ladder’s own, unchanged: one, 0.15, 0.5, 0.15, 0.85, 0.15, 0.5, 0.15 across the eight quaver positions of a bar of 4/4. They are stated rather than fitted, they are the ratio the preference-rule literature uses, and every figure above that names the stated route uses exactly them.

A position’s probability under that route is its weight divided by the sum of all eight, and its cost is minus the log to base two of that. Under the measured route the probability is the count of notes these tunes put there divided by a hundred and four, and the cost is the same logarithm, with a floor of 0.004 for a position no tune ever uses — the same floor, and the same value, the melodic measure gives an interval that does not occur.

A rhythm’s cost per note is the mean of its notes’ position costs. A line’s cost per note is the twelfth rung’s readingLoad, unchanged. A page’s cost is the sum of the two, which is the whole of the joining and is the reason the contours in the plane are straight.

The exchange ladders are both quoted as bits above the cheapest move of their own kind, because an interval has to be some interval and a note has to land somewhere: the absolute levels carry an offset that neither decision can avoid, and subtracting it is what makes the two columns readable across. Positions of equal cost are drawn as one rung, because two positions the distribution scores alike are one choice to a reader.

The joint span divides a fixed budget by each page’s cost. The budget is four times what these tunes cost as written, so the reference page returns four notes by construction and every other page is a ratio against it.

Where the model stops #

Each onset is priced on its own. That is the same independence assumption the melodic measure makes and it fails in the same way. A run of syncopations is a pattern, and the eighth displaced note in a row is not a surprise to anybody; here it costs exactly what the first one did. The chromatic run is the melodic version of that defect and this is its exact twin, arriving from the other axis.

The corpus is three tunes. Twenty-eight bars, a hundred and four notes, all in four, all diatonic, all European. Two of the eight positions have a count of zero and are held up entirely by the floor, and the 6.07-bit rung in the exchange figure is the floor’s number rather than a measurement.

The same sample distorts the melodic ladder in a way that shows up here. A minor third costs 6.66 bits on this distribution and a fifth costs 4.66, because these three tunes take fifths four times as often as they take minor thirds. Nobody would defend that as a general fact about melody, and it propagates directly into the exchange table.

The grid is a free parameter and it costs less than it looks. Everything above reads the bar at the quaver, and reading it at the semiquaver inserts eight weaker positions and raises every cost. Under the stated route the tunes’ rhythm goes from 2.41 bits a note to 2.55 at the semiquaver, 2.63 at the demisemiquaver, and converges on about 2.74 however far the subdivision is taken, because each new level is worth 0.3 of the one above it. Under the measured route it does not move at all: every note of these tunes is on the quaver grid, so refining it adds positions with a count of zero and changes no probability. The parameter is real and it is bounded at a third of a bit.

The two terms are added, which assumes they are independent. A leap onto a weak beat is very probably harder than the sum of a leap and a weak beat, and nothing here has a term for the interaction. It is the largest thing missing and it is not decidable by arithmetic.

And bits are not effort. Converting information into reading difficulty assumes a reader whose capacity is a channel, and nothing on this page measures a reader at all. The numbers are good for ratios between two things measured the same way, which is how every one of them is used.

What the picture cannot show #

It cannot show duration. Every quantity here is about where a note starts. A minim and two tied crotchets have the same onsets and the same cost, and they are not the same thing to read.

Nor can it show a rest. A position with no note on it contributes nothing, so a bar whose second beat is a rest is charged exactly as a bar whose second beat was never written. A rest is a symbol with a position and a length, a reader has to take it in like any other, and on a page with much rest on it that is most of what there is to take in.

It cannot show the beaming. A bar of seven is 2+2+3 or 3+2+2 depending on how the quavers are grouped, and the grouping is in the beams rather than in the signature. The weights used here are read off a metre that has already been decided.

It cannot show a metre the reader has not been given. The whole measure is relative to a fixed bar with a fixed beginning, and rotating where the bar starts changes the number without moving a note — the same four onsets that cost 2.12 bits read from one place cost 6.58 read from the next. On a cycle with no beginning at all the question has no answer.

It cannot show tempo. The same bar at half the speed is the same set of positions and a different reading problem, and the preferred beat rate is a fact about a listener that the page addresses with a word in Italian.

Nor can it show a performance. Everything here quantises to a grid, and the deviations from that grid are the substance of a groove rather than noise around it.

And it still cannot show a page. How much music a page holds is a capacity in notes, and a bar that is dear to read is not a bar that takes more room; the layout bound and the reading bound remain two different constraints that happen to be multiplied together.

Whose notation, and whose tunes #

The staff, the bar line and the eight quaver positions of a bar of 4/4 are the European engraving tradition’s. The weights are the preference-rule tradition’s — Lerdahl and Jackendoff’s metrical hierarchy, and the syncopation measure Longuet-Higgins and Lee built on a hierarchy of the same shape — with the particular numbers stated by this collection rather than fitted by anybody to anything. The counts are three tunes: the Ode to Joy theme as Beethoven first states it in 1824, Ah! vous dirai-je, maman as printed in 1761, and Frère Jacques.

Those are simple, diatonic, strongly metrical and all in four, and every number above inherits that. The clearest sign of it is the tresillo, which this measure prices at 3.10 bits a note — a syncopation, dearer than four crotchets by half again. In the repertoires the tresillo belongs to it is not a syncopation at all; it is the metre, and a reader raised on it would find its positions the cheap ones and the four square crotchets the surprise. The rhythmic axis is a claim about a repertoire in a way the pitch axis is not, because a bar has no natural strong positions the way a semitone has a natural size, and this collection has a whole essay on the notation forcing that choice.

The historical observation is about teaching rather than about repertoire, and it is unusually direct. The Galin–Paris–Chevé method, taught widely in France from the 1830s, separated the two axes deliberately: rhythms were drilled on a single repeated pitch, with spoken time-names and no melody at all, before a pupil was allowed to read the two together. That practice is an assertion that the two are separable and that neither is a rounding error beside the other, and it was made without a number for a century and a half. The number is 1.89 against 2.23 — and the half the method drilled as preparation for the real business of reading pitches turns out to be the larger of the two.

Where this ladder goes next #

Fourteen 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; the two axes multiplied; the eye that has to cross them; the fact that what is being crossed is not uniform; the fact that what is above and below it is not independent; and now the fact that when is at least as much to read as what.

What the ladder owes now is duration. Everything on this rung prices where a note starts, and a notation records how long it lasts as well — a minim and two tied crotchets are one cost here and are not one thing to read. The measurement is available from the same three tunes and needs nothing this collection does not already hold: take the duration of every note, condition it on the gap to the next onset, and price it by the same logarithm. The prediction is that the term will be small, well under half a bit against 1.89 and 2.23, because a duration is nearly determined by the onsets on either side of it — a note lasts until the next one starts, and the information is only in the exceptions, which are rests and ties. If that comes out right, then a single line’s reading load is fully accounted for by two axes and a small correction, and what is left over is the interaction term this rung cannot reach. If it comes out large, the accounting is wrong somewhere and the place to look is the tie, which is the one mark on the staff that makes a note out of two noteheads.

It is arithmetic, and it is the last term available without a corpus or a reader.