
From:  Hans Aberg 
Subject:  Re: Lilypond's internal pitch representation and microtonal notation 
Date:  Tue, 21 Sep 2010 15:19:38 +0200 
On 21 Sep 2010, at 14:16, Carl Sorensen wrote:
A sharp is Mm and a flat mM.If I understand right, this is a key "trick" of your system, since such representations allow you to raise or lower the pitch without affectingthe degree.So by extension, if we say that q is a quartertone, to raise or lower by a quartertone would be to add (mq) or (qm); and to raise or lowerby 3/4tone would be to add (Mq) or (qM)..... but where/how in that system do we distinguish between for examplenatural + 1/4 and sharp  1/4 .... ? Presumably the former is (mq) whereas the latter is (Mm)+(qm) ... ?It seems to me that the pitches natural+1/4 and sharp  1/4 are the samepitch (i.e. enharmonic equivalents) and that it is appropriate to have either one represent the same pitch.
They will be the same in a particular tuning, but they may transpose differently in the staff system. Change the amount so that one raises with 1/3, then that will be paired with one lowering 2/3, and similarly, one lowering with 1/3 will be paired with one raising 2/3. The sum of those distances is that of a sharp.
In algebraic terms, choose a neutral n between m and M. The total pitch system will be i m + j M + k n, where i, j, k are integers. But the staff system only has the pitches i' m + j' M. When taking the difference with the staff note, reducing the degree to 0, and taking away the sharps/flat (a multiple of M  m), there will result a multiple n  m or n  M.
Now choose n so that n  m is neutral plus 1/3. Then n  M lowers 2/3, and m  n with 1/3. In with an exact quartertone, both will work. A sharp is M  m, so there are two candidates for sharp  1/4: (Mm) + (mn) = Mn and (Mm)+(nM) = nm.
With one choice, they are actually equal in any tuning, but with the other choice, they will become separate.
The display of the accidentals leading to that pitch should likely be a property of a keysignature that shows how to display a given pitch. I'm not sure exactly how to accomplish this, but it seems the proper logicalstructure to me.
So one can do both. But the staff system will work on the algebraic level.
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