Re: Diatonic/modal transposition function

[Hans Aberg]

On 30 Dec 2008, at 03:08, Graham Breed wrote:

> Right, that's how it looked to me.  If it doesn't assume equal
> temperament it should work generally.  But I haven't tested it.
>
> Note that I discovered it is possible to get more than 7 scale steps
> to the octave (see the development list).  So this would probably
> generalize accordingly.  But I don't know if anybody's ever going to
> make the (AFAIK global) change of scale size and then want to do modal
> transpositions.  If they do they can hack your code.

I think one in general just add a degree-pitch, but the amount depends  
on the scale degree of the pitch itself. So it should generalize to  
microtones and arbitrary scale degrees, but also to other diatonic  
permutations (inversions, etc). A more formal description:

LilyPond uses essentially (if I have understood Graham right) pairs in  
the set Z x Q, where the first integer is the scale degree, and the  
second the (logarithmic) pitch. Then choose a transition function f: Z  
-> Z x Q; if the process should be invertible, the restriction Z -> Z  
should be invertible. Then any p in Z x Q is mapped onto p + f(d(p)),  
where d(p) is the scale degree of p.

It might be good to have such a general function, since various pitch  
permutations are used by composers.

  Hans