Oriental microtonal music

[Hans Åberg]

Graham Breed wrote a file regular.ly <http://regular.ly/> that allows for 
retuning into any ET (around C4, not A4), but Adam Good is expert on Turkish 
music specifically.

There are some issues with Turkish, Persian and Arabic music that must be dealt 
with when adapting for LilyPond: Notation system limitations, transposability, 
and MIDI output.

Because of those issues, I would prefer to use Helmholtz-Ellis notation. There, 
LilyPond does not have the double arrowhead accidentals.

Though Persian music is transposed, in Turkish makam and Arabic maqam, one may 
rename. In LilyPond, transposability is hard wired, so some care must be taken 
here: Turkish music has different notation systems, but the common AEU 
(Arel–Ezgi–Uzdilek) is not transposable, as its sharp ♯ and and doublesharp 𝄪 
represents microtonal accidentals, not the ordinary ones. AEU also writes the 
pitch a 4th above the sounding pitch.

As for MIDI, the traditional tuning is Pythagorean tuning with Pythagorean 
commas for microtonal alterations in Turkish music, though with departures in 
actual performance. In Arabic and Persian music, these have drifted 
considerably, the double is a possibility.

LilyPond does not support exact Pythagorean tuning, but it could be replaced by 
E₅₃ (53 ET) by the use of regular.ly <http://regular.ly/> if included in the 
distribution. An alternative that does not require this file is E₇₂ (72 ET), 
which relative E₅₃ and Pythagorean tuning changes the diatonic scale into E₁₂ 
(12 ET), but alters the microtonal neutral seconds only with a few cents.

I believe that Adam Good used E₇₂ for the rewritten Turkish makam file.

The original LilyPond makam file had an error making it not transposable:

E₅₃ has minor second m = 4, major second M = 9 commas (E₅₃ tonesteps), so that 
the sharp ♯ = 9 - 4 = 5 commas, and the Pythagorean comma is approximated by 1 
E₅₃ tonestep, also called a comma. Therefore, in E₅₃, one arrives at the 
microtonal alteration of one comma by dividing the E₅₃ M by 9 or the E₅₃ ♯ by 5.

A type of error one sees though, is that these last rules are not applied to 
E₅₃ but to E₁₂:

In the original LilyPond file, they reasoned to divide the M of E₁₂ further in 
form of the LilyPond accidental, which is internally represented as a rational 
number where ♯ is 1/2. So the doublesharp 𝄪, which has a value 1, is divided by 
9, but unfortunately the ♯  of 1/2 is not then an integral multiple of that. 
The new one by Adam Good corrects this issue.

If the ♯ of E₁₂ is divided to 5 parts, one arrives at E₆₀, which is done in a 
suggested Persian file. The Arabic files written so far use E₂₄, possibly 
because LilyPond did not have the rational accidentals originally, but only 
this 24 ET.