The page is read by an eye

How much music a page holds multiplied the two axes — the horizontal spacing rule and the vertical capacity — and got the one design constraint on notation that is not about legibility. Its last paragraph named what every quantity in the ladder leaves out.

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.

Two facts about that reader are well measured, they are in different units, and the spacing rule is what converts between them.

Two facts in different units #

The eye–hand span is measured in notes. A sight-reader’s eye sits ahead of the note being played, and how far ahead is one of the better-studied quantities in music psychology. It is about four notes for a competent reader and rather more for an expert, and it is a note count rather than a distance — a reader confronted with wider spacing does not read fewer notes ahead, they look further along the page.

A fixation is measured in millimetres. The useful field of one fixation is a couple of degrees of visual angle, which at a music stand’s distance of half a metre is about nineteen millimetres. That is a property of the fovea and it does not care what is printed in it.

The spacing rule converts. The axis that is not a time axis is the rung about horizontal spacing: an engraver’s rule gives each note a width, in staff spaces, that is a sublinear function of its duration. Multiply by the rastral size — the physical height of one staff space, which is the one real length notation has — and a note occupies a definite number of millimetres.

At a common orchestral rastral of 1.75 millimetres and the standard spacing rule, a note occupies 5.4 millimetres, so a span of four notes is 21.5 — against a useful field of 19.2.

The span does not fit inside one fixation. Reading music is necessarily a sequence of fixations rather than a single wide view, and that is true at every part-sized rastral.

Which means the eye has to move, and easily can #

The next question is whether the eye can move fast enough, and this is the bound everybody expects to bind.

One fixation covers 3.6 notes at this layout. At a hundred beats a minute with two notes to the beat, that is 3.3 notes a second, which needs 0.93 saccades a second.

The eye’s ceiling is around four a second.

So there is a factor of four in hand. This layout runs out of eye movements at 428 beats a minute with two notes to the beat, which is 14 notes a second and is not a tempo anybody plays.

The saccade rate has enormous headroom and never binds. That is the negative result of the rung and it is worth stating plainly, because the eye–hand span is the quantity the literature is about and it turns out to describe the strategy rather than the limit.

What does bind is the fovea #

The other bound is much simpler and it is the one that matters.

A notehead has to subtend enough visual angle to be identified. Its width is about 1.18 staff spaces, so at a rastral of r millimetres it is 1.18r millimetres wide, and at half a metre it subtends an angle proportional to that. Taking twelve minutes of arc as the angle needed to identify a glyph as complex as a notehead in context gives a minimum rastral of 1.63 millimetres.

The standard rastral sizes are a series that engravers have used for two centuries: rastral 0 at 2.4 millimetres down to rastral 8 at 1.2. Against that bound:

Five of the nine clear the bound, and they are the five used for parts. The four that do not are the ones used for study scores.

That is a division the model did not know about. The rastral series is a printers’ convention with no acoustics in it, and the point at which it crosses the acuity bound is the point at which its own conventional descriptions change from part to score.

Why the score sizes are allowed to fail #

The four small rastrals are not unreadable and the model says why they are not.

The bound depends on the viewing distance. Half a metre is a music stand: a player sitting back, arms free, instrument in the way. A study score is read at a desk, at perhaps thirty-five centimetres, which is a factor of one and a half nearer — and the acuity bound scales with the distance, so it falls to about 1.1 millimetres.

At that distance every one of the nine clears.

So the rastral series is not two arbitrary halves; it is one series read at two distances, and where it divides is where the distance changes. A part is read from a stand and a score is read from a desk, and the sizes follow.

There is a second reason the small sizes are allowed to fail, and it is that a study score is generally not being sight-read. A conductor studying a score is reading at their own pace with unlimited time; a player reading a part is reading at the tempo. The saccade bound is irrelevant to both and the acuity bound is much softer when there is no time pressure.

The band never closes #

Plotting the two bounds against tempo says how much of a real constraint this is.

At forty beats a minute the band between them is a factor of eleven — any rastral at all works. At three hundred it is a factor of 1.54, which is still four of the nine standard sizes.

The band does not close at any playable tempo. So the answer to whether a page can be laid out too densely to sight-read is: not by the eye’s speed. The eye is not the bottleneck in music reading, and every difficulty a sight-reader has is somewhere else — in recognising the patterns, in the hands, in the memory.

That is a useful negative and it disposes of an intuition. The impression that a dense page is hard to read is real and it is not about the fovea’s throughput; it is about how much a reader has to decode, which is a different quantity entirely and one this ladder has no measure of.

The vertical axis has its own version of the same bound #

Everything above is horizontal, and the eighth and ninth rungs measured two axes rather than one.

The vertical question is not about spacing but about discrimination: a reader has to tell a line from a space, which is a judgement about a position to within half a staff space. Half a staff space at rastral 4 is 0.87 millimetres, which at half a metre subtends about six minutes of arc — comfortably above the limit for judging whether a small object sits on a line or between two, which is a vernier acuity task and one the eye is remarkably good at.

So the vertical axis has more headroom than the horizontal, which is a small vindication of a notation that has five lines rather than more. The clef is an integer and the notations invented for the overflow are the rungs about what happens when a range exceeds what five lines carry, and the answer is always a symbol — a clef change, an octave sign, a ledger line — rather than a finer grid.

That is the right design given these numbers. Making the staff finer would trade a task the eye is very good at, judging a position against a line, for one it is worse at, resolving two nearby marks. Adding a symbol costs nothing visually and costs the reader an inference instead.

The stave is not a ruler is the ladder’s first rung and it makes the same point from the other end: the staff’s vertical axis is not linear in pitch, and it does not need to be, because what it is doing is naming positions rather than measuring distances.

What a page turn is, in these terms #

The tenth rung’s own product was a page turn, and the reader adds one thing to it.

A page holds a number of notes, from the vertical capacity times the horizontal spacing, and at a stated tempo that converts to a number of seconds. Making the rastral smaller puts more notes on a page and lengthens the interval between turns.

The acuity bound is therefore a bound on how far that can be pushed: a part cannot be printed smaller than about 1.6 millimetres to the staff space, so a page cannot hold more than a certain number of notes, so a turn cannot be delayed beyond a certain time.

That is the one place in this ladder where the reader and the page’s arithmetic produce a single number together, and it is what the tenth rung asked for.

What the spacing rule is really trading #

There is a reading of the horizontal rule that this essay makes available and the ninth rung could not.

The axis that is not a time axis found that every engraving rule puts a note somewhere other than where its moment is: proportional spacing is the only rule that is true to time and it wastes an enormous amount of paper on long notes, so every practical rule compresses the long ones and displaces the short ones from their moments.

That trade was presented as being between fidelity and paper. It is also a trade against the reader, and in a direction the rules get right.

A rule that compresses long notes puts more notes in a fixed number of millimetres, which raises the notes per fixation and lowers the saccade rate. Since the saccade bound has a factor of four in hand, that is buying something the reader does not need — but it also lowers the distance a reader’s span covers, which means a reader looking four notes ahead is looking a shorter way and is more likely to have the span inside a single fixation.

At the tightest practical setting the span does fit. So the rules that engravers arrived at are, among other things, rules that bring the eye–hand span toward the size of one fixation, and the sublinearity is what does it.

That is offered as a consequence rather than as a motive. Nobody was measuring fixations in the eighteenth century, and the rules were arrived at by looking at pages and judging them handsome.

It has one testable corollary. The rules differ from each other in how sublinear they are, so they differ in how many notes a fixation covers — and a reader given the same music under two rules should saccade at measurably different rates. That is a measurement on an eye-tracker rather than an arithmetic, and it is the only prediction in this rung that could distinguish the rules on the reader’s behalf rather than on the engraver’s.

The time signature is a claim is the rung about the other thing a page asserts that is not a measurement, and it is the same shape of argument: a convention that looks like a description of the music and is really an instruction to the reader.

Which computation produced the numbers #

The horizontal spacing is the engraver’s rule from the ninth rung, giving each note a width in staff spaces as a function of its duration; the vertical capacity is the eighth rung’s, giving a system’s height from the staff’s own positions plus the ledger lines a range needs.

The reader’s constants are all published figures with ranges around them, and every one is named rather than buried: a span of four notes, a useful field of two degrees, a viewing distance of 55 centimetres, a saccade ceiling of four a second, and a notehead 1.18 staff spaces wide.

The acuity criterion is twelve minutes of arc, which is generous compared with the limit for detecting a fine line and appropriate for identifying a glyph in a cluttered field.

Where the model stops #

Acuity is one number for one task. Reading a notehead’s position on the staff is a much finer discrimination than detecting the notehead, and reading an accidental or an articulation is finer still. A model with one angle for everything is a caricature, and the true bound is probably set by the fine marks rather than by the noteheads.

The span is a mean. Eye–hand span varies enormously with skill, from about two notes for a beginner to eight or more for an expert sight-reader, and it varies within a performance with how predictable the music is.

Reading is not note by note. A skilled reader takes in a chord, a scale fragment or a familiar figure as one unit, so the note count that matters is a count of chunks. That makes the effective span longer in notes and does not change the millimetres.

And nothing here is about music. The whole model treats a page as a field of glyphs at a spacing, and the actual difficulty of sight-reading is about pattern recognition — which is why a hard piece and an easy piece printed identically are not equally readable. A melody is a walk, not a set is the collection’s account of what makes a sequence of pitches predictable, and predictability is what a chunk is made of.

Nor is the notation the only one. Three notations, one progression and what a tablature keeps are the rungs about systems that encode different things, and a tablature’s glyphs are numerals rather than positions — a different identification task with different acuity demands, which this model has no way to compare.

What the picture cannot show #

It cannot show the page turn as an event. A turn is a hand leaving the instrument, and the constraint is about when a hand is free rather than about how much is on the page. How much music a page holds says the same and neither rung has a model of the player’s hands.

Nor can it show the second staff. A pianist reads two staves at once, which is a vertical span as well as a horizontal one, and nothing in a one-dimensional model of fixations addresses it.

And it cannot show the screen. A great deal of music is now read from a tablet at a fixed size and a fixed distance, with no page turns and no rastral series. The bounds still apply and the conventions the series encodes do not.

Whose notation, and when #

The rastral series is a European engraving convention, standardised in the nineteenth century and still in use — the numbers and their conventional descriptions are what a publisher’s specification sheet contains. The spacing rules are the same tradition’s.

The reading measurements are twentieth-century experimental work, mostly on Western staff notation and mostly on pianists, and the eye–hand span in particular has been measured many times with fairly consistent results.

The historical claim the figure supports is narrow and rather satisfying. The rastral series was arrived at by printers balancing cost, legibility and the size of a sheet of paper, over a long period, with no measurement of anybody’s eye. It divides exactly where a modern account of visual acuity says it should, and the division is between the sizes read at a stand and the sizes read at a desk. Nobody designed that boundary and everybody has been using it.

Where this ladder goes next #

Eleven 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; and now the eye that has to cross them.

What the ladder owes now is the chunk. Every quantity above counts notes, and a reader does not read notes — they read figures, and a scale fragment of eight notes is one object where eight unrelated pitches are eight. That is the difference between a sight-readable page and an unreadable one, and it is not a property of the layout at all. This collection has an account of what makes a sequence of pitches predictable, on the melody ladder, and applying it to a page would give the first quantity in this anchor that depends on what the music is rather than on how it is printed.