Re: Lilypond's internal pitch representation and microtonal notation

[Hans Aberg]

On 20 Sep 2010, at 14:48, Joseph Wakeling wrote:

> > > I saw the post but was not sure quite how to interpret it.
> >
> > I expected someone to ask for details. In the past, I discussed  
> > part of
> > it with Graham Breed, who did some LilyPond microtonal  
> > implementation,
> > but perhaps he is not working on it anymore.
>
> I get the feeling activity is rather focused right now on getting 2.14
> released ... :-)

So the problem is finding someone willing to have a look at it from  
the LilyPond side.

> I also discussed microtonal stuff with Graham Breed a while back,  
> but we
> weren't really able to bring anything to a satisfactory conclusion.

Graham's code is within LilyPond's current model. We discussed a bit  
this extension, but at that time, I had not made a full generalization.

> > If you want, I can explain it - the algorithm itself is very simple.
> > Writing up it in math style will probably not make it more  
> > accessible.
>
> It would make it clearer to me, surely.  What I'd like to see is for  
> it
> to be written up in a structured way along the lines of

Sure, just get back if you want more details or examples.

> (i) this is the
> problem that needs solving,

The staff can be an arbitrary scale (see below), and one has music in  
another. The problem is to compute, for any note, the accidental to  
use, and the key signature. The latter follows from the former, though.

The scales of the staff and the music choose their pitches as integer  
linear combinations from an abstract set of seconds. In the  
traditional typesetting, one has a minor second m and a major second  
M, so it is all combinations p m + q M, where p, q are integers, which  
can be identified with all pairs (p, q). Mathematically, I am looking  
the free abelian group generated by the intervals one wants to use.

The traditional staff scale is the minor scale: 0, M, m+M, ...,  
corresponding to a, b, c, .... This can be extended to other types of  
accidentals by adding neutral seconds n_1, ... For Persian, Arab,  
Turkish music, and Just Intonation relative Pythagorean, in each, just  
one neutral suffices, but it has different values when giving it a  
value.

> (ii) this is the approach the algorithm
> takes to solve the problem,

One needs an order between the seconds, which is a choice of  
preference by the one typesetting the music.

> (iii) this is the algorithm.

Call the sum of the coefficients of a linear combination the degree of  
the note. A requirement is that the staff provides a note for every  
degree (also having that degree) one wants to typeset a note for.

In order to typeset a note, first compute its degree, and then  
subtract the staff note of that degree. The result is a note of degree  
zero, which should be typeset using an accidental. An accidental is  
represented by a linear combination with positive coefficients of  
pairs of distinct seconds.

Compute its accidental by successively reducing the first (relative  
the order of the seconds) positive and negative coefficients until it  
stops (the note is 0).

As you can see, there are really no requirements of what scales to  
use. So it pushes the staff notation to its utmost capabilities.

> > > I did wonder if the fact it was Haskell code was part of the  
> > > reason for
> > > the lack of response.  I have a lot of admiration for Haskell but  
> > > I can
> > > see there being problems extending Lilypond with yet another  
> > > language.
> > ,
> > It should not be difficult to translate into Scheme - no specific
> > Haskell features are used, only better syntax and type system to help
> > structuring the code. It is just a page.
> >
> > The difficulty is to figure out to put it into LilyPond.
>
> Indeed, it sounds like a pretty fundamental
> major-version-number-changing kind of modification.

It might be. But it is a generalization, that is, one builds an  
interface on top which expresses the current LilyPond model. Then the  
rest can be changed when one has time.

> As a related issue, have you considered how (different kinds of)
> transposition would be handled in your pitch scheme?


This is much simpler: the linear combinations are vectors that you  
just add. For example, if a, b, c, ... are represented by 0, M, m 
+M, ..., and you want to transpose from b to c, just add m. A sharp is  
M-m and a flat m-M. If you want transpose from a note x to a note y,  
just add y - x.