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## Re: Custom music notation?

 From: Hans Åberg Subject: Re: Custom music notation? Date: Sat, 6 Feb 2021 15:23:07 +0100

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.