Re: Towards a new pitch representation

[Hans Aberg]

On 30 Dec 2010, at 23:16, Felipe Gonçalves Assis wrote:

> 1. How should we represent alterations?
>
> It is clear to me that the most general representation would be as
> a list of integers of arbitrary length (see sections 1 and 2 of the
> attachment).

I made a proposal for a representation, and there is Haskell code  
available if you are interested.

In it, the staff positions are expressed by a linear combinations of a  
sequence of seconds m, M, n_0, ..., n_k (whatever names you want to  
give them). If one only has a minor second m and a major second M,  
then sharps raise with the interval M - m and the flats lower with  
that interval.

So I generalized this as to compute for any set of accidentals. The  
algorithm will depend on the typesetting preferences, but it is  
possible to compute them without any reference to pitches at all.

> 1.1. How do we implement the mapping from alterations to tones?
>
> A simple idea is to replace the list of integers by a list of  
> rationals,
> corresponding to the value of the alteration in tones. This is the
> current approach, except for the fact that the list has length 1.

Rational numbers are not sufficient for the theoretically given: one  
must be able to take roots, for example when computing meantone values.

So, for the exact given values, I added those roots. This suffices, as  
one is never going to add the numbers of the intervals - interval  
addition correspond to multiplication of numbers. The implementation  
representation I chose was as a prime factorization - a map from prime  
numbers to rational numbers. This seems to work well, as one is  
typically just working with a few small prime numbers.

But when working with inexact intervals, like in pitch bends or given  
s cents, etc., this proved inconvenient.

So I added an inexact part, a floating point number (double). The  
representation is thus (r, x), where r is the exact number above, and  
x a floating point number. When x = 1.0, the representation is  
considered exact.