Re: Custom music notation?

[Hans Åberg]

The staff system that LilyPond uses originally refers to the Pythagorean 
tuning, which has two generators. A symphony orchestra uses adaptive Just 
Intonation which requires an additional generator, but that is normally not 
notated, but microtonalist do, one can use for example Helmholtz-Ellis notation.

The original generators of the staff system are the perfect 8th P₈ = 2 rational 
interval and the perfect 5th P₅ = 3/2, but one can use others, for example, the 
major second M and the minor second m. LilyPond uses M and the sharp.

Any note can be expressed as an interval relative a tuning note, which 
traditionally is A4, though LilyPond uses C4, maybe a convention that comes 
from Helmholtz.

Here is how the notes position and its accidental are found on the staff 
system: The note relative the tuning note is an interval and can be expressed 
as a linear combination of the generators; say the generators are M and m, of 
relative scale degree 1. If the note's interval is k*m + l*M, where k, l are 
two integers, define the relative scale degree as k + l; this is also the 
distance from the tuning note on the staff system.

So to find the accidental: The staff system without accidentals refer to a 
scale, which traditionally is the pure minor relative A4. At the staff system 
position, subtract this interval of this note at the from that of the note one 
wants to typeset; this is an interval of relative scale degree 0, which is the 
accidental. For example, M - m is the sharp, and m - M the flat, and the other 
accidental intervals are multiples of these.

Written out in notes, using the sharps on the diagonal /, one gets this diagram:
      C#  D#  E#
    C   D   E   F#  G#  A#  B#
  Cb  Db  Eb  F   G   A   B
            Fb  Gb  Ab  Bb  C'
Transposing is the same as translation in this diagram.

One can get equal temperament approximations from the Pythagorean tuning by 
computing the continued fraction convergents of log₂ 3/2. The series starts 
7/12, 24/41, 31/53, 179/306, 389/665, 9126/15601, 18641/31867, 46408/79335, … 
The first in this series 7/12 is E12, the 12-equal temperament with P₅ = 7. But 
others are used, for example, 31/53, which is E53 with P₅ = 31, which has m = 
4, M = 9, is used in the description of Turkish music nowadays.