Leaps do not fall where offbeats do

Every load on this anchor is a sum. The fourteenth rung adds 1.89 bits for which note it is to 2.23 for where the note is; the fifteenth adds 0.67 for how long it lasts; the sixteenth adds a tie term on top. Each addition assumes the terms are independent, and the fourteenth rung said so and called the interaction the thing it could not reach.

It is reachable, and on the corpus this ladder already uses. A tune does not put its leaps and its offbeats at random with respect to each other.

The mutual information between the two axes is 0.31 bits a note — a fifth of the smaller of them, and a sixth of a note’s total load.

What the shared bits are #

The picture is a table of counts and the pattern in it is one every musician would have predicted and nobody here has measured before.

Large moves land on strong beats. Small moves and repeats land everywhere, and they are what fills the offbeats. So a reader who has read a note’s position and found it between two beats has already learned that the interval is probably small — and the pitch term charges them for learning it again.

The size of the effect is the part that had to be computed. The pitch axis alone carries 1.55 bits over the corpus and the position axis 2.23; if they were independent the pair would cost 3.79. Measured jointly they cost 3.47, and the difference is the 0.31 the two share.

That is a fifth of the pitch axis, which is a large fraction of the smaller quantity and a small fraction of the total. Every reading load this anchor has published is high by about six per cent, uniformly, which is enough to matter for a ratio and not enough to change a ranking.

It is worth being clear about what “shared” means here, because the phrase invites a wrong picture. The two axes do not each contain a copy of the same 0.31 bits that a reader could take from either. What the number says is that the joint distribution is more concentrated than the product of the two marginals — knowing one narrows the other — and a reader who reads both pays for the narrowing once instead of twice. There is no bit anywhere on the page that is duplicated; there is a page whose two descriptions are not independent.

The check that it is measuring music #

A mutual information computed over a grid can be an artefact of the grid: divide the bar finely enough and every note lands somewhere unique, at which point the position axis records the note’s identity and the two quantities share everything by construction.

At two positions in the bar the interaction is 0.08 bits, at four it is 0.27, at eight it is 0.31 — and at sixteen it is still 0.31, to every digit.

That flat top is the check. Every onset in these three tunes lands on the quaver grid, so a division finer than eight has nothing new to sort them by and can reveal no more shared structure. A quantity that kept climbing as the grid refined would be counting the grid; this one stops exactly where the notation does.

It also says the eight-position grid the whole anchor uses is the right one for this corpus rather than a convenient one, which is a thing every previous rung assumed and none of them checked.

Which is the same shape as the thirteenth rung’s result #

This anchor has met a correlation between two axes once before and drew the opposite conclusion from it, which is worth putting side by side.

The thirteenth rung found that four parts are easier to read than two independent ones, because the parts are related and a reader who has read one has partly read the others. Its correlation is between voices and it makes a score cheaper than the sum of its lines.

This one is within a note and it makes a line cheaper than the sum of its axes. The two are the same arithmetic — a joint entropy below the sum of its marginals — applied at two different scales, and the anchor has now found it at both.

That suggests the shape of what is left. If a reading load is a joint entropy over everything on the page, then every structure a reader can exploit reduces it, and the sum of independent terms is a ceiling that a real page never reaches. The measure this ladder has been building is an upper bound with corrections subtracted from it one at a time, and the corrections have all pointed the same way.

Whether there are more of them is the honest open question. The tie term and the position term are surely correlated too — a tie crosses a beat and therefore lands somewhere specific — and nothing here has measured that.

What the corpus cannot say about the interaction #

Three tunes are three tunes, and the reading above is one that a wider corpus could easily overturn — or, more likely, sharpen in a direction this one cannot see.

All three are in four, all three are diatonic, all three are simple enough to be universally known. Music that leaps on offbeats deliberately — a syncopated line, a Baroque hemiola, a jazz head — would have a smaller interaction, because the association this measures is precisely the one such music sets out to break. So 0.31 bits is a figure for very regular music, and the honest statement is that it is an upper bound rather than a value.

Which produces the prediction that is most worth testing. A syncopated repertoire should have a smaller interaction and therefore a higher reading load at the same two axis values — that is, two lines whose pitch and position terms are identical should differ in how hard they are to read according to how much the two are correlated. No account of sight-reading difficulty this collection has found contains such a term, and it is the only prediction on this anchor that could distinguish the joint measure from the sum.

The three tunes do not agree, and that is informative #

The measurement above pools three tunes, which hides whether they behave alike.

Taken separately the association is strongest in the Beethoven, whose leaps sit almost exclusively on beats, and weakest in the children’s song, which is nearly all repeated notes and steps and therefore has very little interval variety for the position to be correlated with. That is not a difference in habit so much as a difference in what there is to correlate: a tune with no leaps in it cannot put its leaps anywhere.

Which means the interaction is bounded above by the pitch axis’s own entropy, and a tune with a narrow interval vocabulary has a small interaction for an uninteresting reason. The share reported — a fifth of the smaller axis — is the right way to quote it precisely because it normalises that away.

It also says where a larger effect would be found. A melody with a wide interval vocabulary and a strict placement habit would have both a large pitch entropy and a large interaction, and the obvious candidate is a Baroque instrumental line: wide leaps, strong metre, and a great deal of both. That is one measurement away and it is the same measurement, run on different notes.

Where the same object already is on this site #

The correlation this rung measures is not a new fact about music, and it is worth saying which other anchors have it and in what form.

The metrical anchor asserts that positions have weights and that music prefers the heavy ones. The melody ladder finds that a tune is nearly all small steps and that the large ones are rare. Neither of them puts the two together, because each is a statement about one axis.

The correlation is what happens when the two preferences are applied to the same note. A composer choosing a leap has, on both accounts, chosen something rare and conspicuous, and putting a conspicuous interval in a conspicuous place is a stylistic habit rather than a constraint — which is exactly why a repertoire could break it and this one does not.

There is a third place the same object appears and it is not on this anchor at all. The metre-induction ladder asks how a listener infers a bar from a stream of onsets, and what makes that possible is exactly the unevenness this rung measures: if every position in the bar were equally likely, no metre could be inferred from onsets alone. So the interaction is the readable form of the thing metre induction is the audible form of, and a music with no interaction would be a music whose metre a listener could not find.

That is a stronger statement than the reading-load correction and it is worth separating from it. The correlation is not a convenience for a reader; it is the same statistical structure that makes the notation interpretable in the first place, seen from the page instead of from the ear.

So the 0.31 bits is a measurement of a style, and this anchor has been quietly measuring style since its ninth rung without saying so. The pitch axis is the interval distribution of three particular tunes; the position axis is where three particular tunes put their notes. Both of them are stylistic and only the interaction makes that visible, because it is the only one of the three that would be zero for a music that had no habits.

What a correction of six per cent is worth arguing about #

A sixth of a note’s load sounds substantial and six per cent of it sounds negligible, and both descriptions are of the same number. Which one is right depends on what the number is for.

For ranking pages it is negligible. The correction is the same for every cell of the anchor’s grid, so subtracting it moves every page by the same amount and changes no ordering at all. A reader wanting to know whether one line is harder than another can ignore it entirely.

For quoting a rate it is not. The eleventh rung’s eye-hand span is a constant load divided by a per-note load, so a six per cent error in the denominator is a six per cent error in how many notes a reader holds — and the span is about four notes, so the correction is a quarter of a note. That is inside the measurement’s own uncertainty and it is the kind of thing that accumulates when a chain of results is quoted through each other.

For comparing repertoires it is the whole thing. Two repertoires with identical axis values and different interactions differ only in this term, and the difference between a strict style and a syncopated one is exactly that comparison. The correction is invisible where the anchor has been looking and decisive where it has not, which is the usual position of a term that took seventeen rungs to reach.

What the pictures cannot show #

The interval axis is bucketed into five sizes and the bucketing is chosen. Finer buckets would raise both the pitch entropy and the interaction, coarser ones would lower both, and the share — a fifth — is the quantity least sensitive to the choice. Nothing here swept it.

A hundred and one moves is a small sample for a two-dimensional table. Forty cells and a hundred observations means most cells hold one or two notes, and a mutual information estimated from sparse counts is biased upward: the estimator finds structure in noise. The bias for a table this shape is of order a few hundredths of a bit, which is a tenth of the number reported, and correcting it properly needs more data rather than more arithmetic.

So 0.31 should be read as “about a fifth of the smaller axis” and not as three digits. The saturation check survives that entirely, because a bias that is constant across grids cannot produce a flat top.

There is also a distinction the arithmetic runs straight past. Mutual information is a property of the distribution, and a reader who has not learned that distribution gets none of the saving. A child reading their first page has no idea that leaps prefer beats; a professional has internalised it so thoroughly that they will misread an offbeat leap as an onbeat one, which is a documented sight-reading error and is what an over-strong prior does. So the 0.31 bits is the saving available to somebody who knows the style, and the cost of getting it wrong to somebody who knows it too well is not in the accounting anywhere.

And the measure is symmetric while reading is not. Mutual information says the two axes share information; it does not say which one a reader uses to predict the other, and a reader’s eye takes in a notehead’s vertical position and its horizontal position at the same fixation. Whether the saving is available in the order a reader needs it is a question about eye movements, and the eleventh rung is the only thing on this ladder that touches them.

What would have to be true for the interaction to be zero #

It is worth asking what a music with no interaction would look like, because the answer is not “random” and it is not obviously bad.

A repertoire in which interval size and metrical position are independent puts leaps as readily between beats as on them. That is not disorder: it is what a great deal of music does deliberately, and the devices have names. A syncopation is a strong event in a weak place. A hemiola is a whole passage of them. A jazz melody’s characteristic anticipation puts the note the harmony wants an eighth before the beat that wants it.

So an interaction of zero is a style with a great many of those, and an interaction of 0.31 bits is a style with almost none — which is what three regular tunes should give. The measure is a syncopation meter, read backwards, and that is a more useful thing to say about it than the reading-load correction is.

It also gives the quantity a sanity check nothing else on this anchor has. If the interaction is a syncopation meter, then measuring it on a ragtime melody should give something near zero, and measuring it on a chorale should give more than 0.31. Neither is done here and either would take an hour.

Whose music, and what a page costs after the correction #

The corpus is a Beethoven theme, a French children’s song and one more tune of the same kind: eighteenth- and nineteenth-century European melody at its most regular, chosen because a reader is likely to know them and not because they represent anything.

For that music the accounting is now complete in its terms. Which note it is, where the note is, how long it lasts, whether it is a note at all — four terms — less what the first two share. A single line’s reading load is 4.47 bits a note rather than the 4.79 the sum gives, and no further term is available without either a corpus this collection does not have or a reader it cannot recruit.

The next rung puts that number to the use it was always for.

Where this ladder goes next #

Seventeen rungs, and the accounting closes with this one. The terms are measured, the interaction is measured, and the total is a bit rate per note that a page can be characterised by.

What has not been done is the thing every rung since the twelfth has been pointing at: a composer chooses between a hard rhythm and a hard interval, and until now there was no exchange rate between them. There is one now, because both are in bits: the six kinds of line this ladder prices span 5.26 bits and the seven kinds of rhythm span 4.46, so one axis’s whole range buys 1.18 of the other’s. What that says about which of the two a composer should spend, and about which pages are equally hard for reasons that look nothing alike, is one drawing and the end of this ladder.