Re: Lilypond's internal pitch representation and microtonal notation

[Hans Aberg]

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.