Generator

Seven positions for twelve pitches

The twelve semitones of an octave against the seven places a staff has to put them. Five of the twelve have no position of their own and are written as an alteration of a neighbour, which is what an accidental is: not an extra symbol on a complete axis but the repair for an axis with five values missing. The vertical distance on the page therefore counts letters, not pitch, and two notes the same distance apart on the staff are not the same distance apart in the air.

Drawn above with its standard settings, which is almost never how an essay draws it: an essay states the numbers it is arguing about, so the figure a reader meets there is about that argument rather than about the drawing in general. Every option a placement passes is checked against the ones this function actually reads, because an option it does not read is silently ignored and the figure quietly draws what is above instead.

It makes a noise. 17 of its 17 placements carry sound, built from the same numbers as the drawing, offering these buttons: 11 positions of the stave, read in the treble clef, 5 positions of the stave, read in the treble clef, A third — the step between two clefs, The eleventh a staff spans, The same positions read in the bass clef: one drawing, another set of notes.

Called by 6 essays

the blast radius of changing it

The chain the page counts on. Every way of writing a pitch with up to one accidental, laid out along the chain of fifths and grouped by the pitch class each one sounds as. The vertical axis is what a keyboard has and the horizontal axis is what the page has: the spellings of one pitch class are seven steps of a fifth apart, which is exactly the distance a comma is measured over. Notation did not choose an arbitrary redundancy; it kept the coordinate a tuning system is built in, and equal temperament is the projection that loses it.

The stave is not a ruler

A hundred and eighty essays here draw pitch against an axis somebody computed. The one axis every reader already owns is the five lines, and it is not a pitch axis at all: it counts letters. Seven positions carry twelve pitches, so the same vertical distance is two intervals before an accidental is allowed and six after — and the accidental is not an extra symbol on a complete scale but the repair for a scale with five values missing.

scales · Notation

The staff holds 11 positions and nothing fits in it. Each clef's eleven staff positions — five lines, four spaces and the space either side — as a bar on an axis that counts letters, with eight ranges laid underneath. The clefs step through the axis in thirds and cover fifteen positions of offset between them. Every range drawn is wider than eleven positions: the four voices span 13, 12, 13, 13 and the four instruments 24, 24, 25, 23, so the best clef for each still leaves 1 to 7 positions off the staff. A clef is a choice of which end sticks out.

The clef is an integer

The first essay on notation found that the staff's vertical axis counts letters rather than pitch, and named the clef as a question it was leaving open. Paid, it is arithmetic: a staff holds eleven letters, no voice or instrument is that narrow, and the eight clefs of European practice step through the axis in thirds — a spacing that buys everything a set of fifteen would buy on a wide range, for eight.

scales · Notation

What eleven positions cover, at seven to the octave and at twelve. A clef's 11 positions, read as a range, in three systems. At seven positions to the octave they cover 1.57 octaves — an octave and a fourth, which is the seventh rung's own number. At twelve they cover 0.92. Writing a 44-semitone range then takes 3 staves and about 15 ledger positions on the staff, and 4 staves and 33 on a chromatic one. The accidentals a chromatic staff removes are paid for in vertical space, at a rate the two integers fix.

The notations invented for the overflow

Every proposal to replace the staff since the seventeenth century is a response to a specific overflow, and the commonest one — a chromatic staff with twelve positions per octave instead of seven — makes a trade computable from the same two integers the clef essay counted. It buys the accidentals outright and pays 40 per cent of the range, three ledger positions for one, and twenty-three enharmonic distinctions that are not merely absent from the page but unrecoverable.

scales · Notation

How much music each notation fits on a page. The two axes multiplied. Vertically, a system is as tall as the staves the range needs; horizontally, a system holds as many notes as fit once the shortest is wide enough to read. the staff, seven to the octave: 3 staves to the system, 4 systems and 38 notes to a system, 152 notes to the page — 46 seconds at 100 beats a minute, so a page turn every 46 seconds; a chromatic staff, twelve to the octave: 4 staves to the system, 3 systems and 38 notes to a system, 114 notes to the page — 34 seconds at 100 beats a minute, so a page turn every 34 seconds; a whole-tone staff, six to the octave: 2 staves to the system, 7 systems and 38 notes to a system, 266 notes to the page — 80 seconds at 100 beats a minute, so a page turn every 80 seconds. a whole-tone staff, six to the octave holds 2.33 times what a chromatic staff, twelve to the octave does, which is a difference of 1.00 page turns a minute — and a page turn is a thing a player with two hands occupied cannot do.

How much music a page holds

Nine earlier essays have measured notations, and every one of them is a page — a two-dimensional object read in a fixed order by a reader who has to turn it. One measured the vertical axis and another the horizontal, and multiplying them gives the one design constraint on notation that is not about legibility at all: a chromatic staff turns pages a third more often than an ordinary one, and a proportional spacing rule turns them nearly twice as often as a columnar one.

scales · Notation

Every standard rastral size against the two bounds a reader imposes. Print the notes larger and the eye-hand span stops fitting inside one fixation, so the reader has to saccade ahead faster than the eye can move. Print them smaller and a notehead stops subtending enough angle to be identified. Both bounds come from the reader and neither from the music. At 100 beats a minute with 2 notes to the beat, the acuity bound sits at 1.63 millimetres and the saccade bound at 7.5 — so the saccade rate is nowhere near binding and acuity is doing all the work, which is the opposite of what the eye-hand span suggests. 5 of the 9 standard rastrals clear the acuity bound: rastral 4 and larger. Those are exactly the sizes used for parts, and the ones below are used for study scores — which are read at a desk rather than played from at a stand, and a shorter viewing distance moves the bound with them.

The page is read by an eye

A sight-reader's eye sits a fixed number of notes ahead of the sounding one and a fixation takes in a fixed number of millimetres, and the spacing rule converts between them. Two bounds follow, from the reader rather than from the music — and the one everybody would expect to bind does not. The saccade rate has enormous headroom at any playable tempo, and what decides is acuity.

scales · Notation

The same eight notes are four times as much to read. How many bits each note of a line carries, taken as minus the log of the probability of the interval that reached it, under the distribution of melodic steps measured over the tunes used throughout. A scale costs 1.76 bits a note and a wide leaps costs 7.02 — a factor of 4.0 at the same number of notes on the page. Every quantity computed until now counts notes, and the page cannot tell these apart: eight quavers are eight quavers of horizontal space whichever line they spell.

A reader does not read notes

Eleven earlier essays count notes, and the page cannot tell one line of eight quavers from another. A reader can: a scale of eight is one object where eight leaps are eight. Measured against the melodic interval distribution, the same eight notes are four times as much to read — and the eye–hand span, the best-measured quantity in the reading literature, is four notes of a tune and one of a leaping line.

scales · Notation

All figures · What can be heard