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Re: Lilypond's internal pitch representation and microtonal notation

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 M-m and a flat m-M.

If I understand right, this is a key "trick" of your system, since such representations allow you to raise or lower the pitch without affecting
the degree.

So by extension, if we say that q is a quarter-tone, to raise or lower by a quarter-tone would be to add (m-q) or (q-m); and to raise or lower
by 3/4-tone would be to add (M-q) or (q-M).

.... but where/how in that system do we distinguish between for example
natural + 1/4 and sharp - 1/4 .... ?  Presumably the former is (m-q)
whereas the latter is (M-m)+(q-m) ... ?

It seems to me that the pitches natural+1/4 and sharp - 1/4 are the same
pitch (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: (M-m) + (m-n) = M-n and (M-m)+(n-M) = n-m.

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 key-signature that shows how to display a given pitch. I'm not sure exactly how to accomplish this, but it seems the proper logical
structure to me.

So one can do both. But the staff system will work on the algebraic level.

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