Generator

What one chord symbol leaves open

Four-part arrangements of C major inside the ordinary vocal ranges, counted under each of three notations. A chord symbol admits 480 of them; a Roman numeral, which adds the inversion, 152; a figured bass, which puts the bass note itself on a staff, 76. Each notation is a quotient, and the number is the size of the class it maps a chord onto.

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. 8 of its 8 placements carry sound, built from the same numbers as the drawing, offering these buttons: I – IV – V – I, close together, I – vi – ii – V – I – IV – V – I, close together, The same four symbols with the bass two octaves under: one chart, two pieces of music.

Called by 3 essays

the blast radius of changing it

What one progression leaves open. Realisations of I–IV–V–I in four parts with no parallel fifths or parallel octaves, counted exactly by a dynamic programme over the voicings rather than sampled. The chord symbols admit 16,100,352,296; the Roman numerals 59,418,496; the figured bass 2,042,672. The three notations differ by four orders of magnitude, and every one of them was in daily professional use.

Three notations, one progression

A figured bass, a Roman numeral and a chord symbol are three professional notations for the same four chords, and the number of four-part realisations each of them admits can be counted exactly rather than argued about. With no parallel fifths or octaves the counts are sixteen billion, fifty-nine million and two million: a factor of eight thousand between the loosest and the tightest. What each one collapses is what its tradition thought a chord was, and the three do not agree.

harmony · 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

A notehead in four parts costs 1.30 bits and one in two parts costs 1.89. What one notehead asks of a reader, against how many parts are on the page, for a progression realised by the voice-leading solver used here at 2 semitones of motion a voice a chord. The horizontal rule is a note of a single melody under the measure established earlier, 1.89 bits, which is what a texture costs when its parts have to be read one at a time. The bars are the harmonic reading: the chord, charged at the worst case of 2.81 bits for one of seven diatonic degrees, plus the logarithm of how many voicings of it the previous chord could legally have moved to. At two parts there is no bar, because a duet has no complete voicing of any triad — it cannot state the harmony and has to be read as 3.79 bits of two independent lines. Every thicker texture is cheaper a notehead than the thin one, and the four-part figure is an upper bound.

Four parts are easier to read than two

Twelve earlier essays read one line, and a score is several at once. Measured through the voice-leading model, a notehead of a four-part chorale asks a reader for 1.30 bits and a note of an independent line asks 1.89 — so twice the ink is less than three quarters of the load. The reason is a boundary those essays already established: a duet has no complete voicing of any triad at all, so two parts cannot be read from their harmony and have to be read as two melodies.

harmony · Notation

All figures · What can be heard