Re: Microtonality

[Hans Aberg]

On 27 Sep 2013, at 08:45, David Kastrup <address@hidden> wrote:

> Well, today's xkcd, at the surface more being about LilyPond's choice of
> extension language, still seems somewhat on-topic here:
> 
> <http://xkcd.com/1270/> (mark the mouse-over text)

Perhaps some mathematical abstractions can help:

The set of intervals I one wants to typeset, i.e., the set of pitches expressed 
relative a tuning note, is a free abelian group [1]. So it looks like a finite 
product of copies of Z, the set of integers.

There is scale degree function deg: I -> Z, setting the tuning note to 0, which 
is a group homomorphism: one chooses the scale degree for each generator, and 
extend by linearity. This function is used to find the position of the notehead 
in the staff system.

The staff system requires there to be a section s: Z -> I, that is, a function 
s with the property that the function composition  deg o s  is the identity on 
Z, i.e., deg(s(k)) = k for all k in Z. This just expresses that each position k 
on the staff, expressed as an integer relative the tuning note, has a note 
value in I, denoted by s(k).

When typesetting a note x in I, first find deg(x) to get its position on the 
staff relative the tuning note; so there should be a notehead in that position. 
Then y = s(deg(x)) is the note that the staff expresses in that position.

The note z = x - y has degree 0: deg(z) = deg(x - y) = deg(x) - deg(y), since 
deg is a group homomorphism; and deg(x) - deg(y) = deg(x) - deg(s(deg(x))) = 
deg(x) - deg(x) since s is a section.

So z should be typeset using an accidental. There can be various algorithms for 
that.

There can be different generators for I. Traditionally, one uses the perfect 
fifth P5 and the octave P8. One can choose a different set of generators, for 
example the minor second m and the major second M.

When doing microtonality, just add more generators. When computing interval 
values to make the music playable, different choices of free bases of I lead to 
a matrix equation.

Hans


1. Abelian because of transposition, finitely generated as we only want to 
express a finite set of pitches, and torsion free, as we cannot realize finite 
order elements as intervals when given values. Then a finitely generated 
torsion free abelian group is known to be free.