Re: Lilypond's internal pitch representation and microtonal notation

[Hans Aberg]

On 21 Sep 2010, at 16:05, Joseph Wakeling wrote:

> > Yes - accidentals do not affect the degree: they are of degree  
> > zero. One
> > can add notes and intervals on this abstract level, and the degrees  
> > add
> > as well. In mathematics, a function f is called a homomorphism (of
> > abelian groups) when f(0) = 0, f(x + y) = f(x) + f(y), f(-x) = -f(x).
> > The degree, which is the sum of the coefficients is a homomorphism.
>
> Ahhh, NOW you are bringing the discussion into terms that I can
> appreciate.  I have studied group theory, but it was 10+ years ago
> without using it since, and hence my recollection is extremely flaky.
> So it was difficult to work out what you were getting at with your
> earlier explanations which tried to limit the use of precise
> mathematical terminology.  In my case I need _more_ maths, not less,
> because it helps me re-familiarize myself with the exact concepts I  
> need
> to understand your system ... :-)

Good. I tried to keep low on such terminology.

> Anyway, I will take the time to go over your answers properly and  
> surely
> be back with more questions.

Sure.

> > > ... so if we extend this vectorial representation to a 3D case for
> > > quarter-tones, (x_M, x_m, x_q), (NB my q is different from yours:-)
> >
> > Yes, some mathematicians do that error, too, though in print, q as a
> > variable might be typeset in italic, whereas as constant in upright  
> > type.
>
> I don't understand what you consider an error here.  I understood very
> well that you were using the letter q to represent a coefficient.  I
> just wanted to use q to represent a group element, so re-labelled the
> coefficients of the elements M, m and q by x_M, x_m and x_q.
>
> It may be an error to think of "vectorial representation", but ... :-P

Just the common typo, like in
  sum_{i=1}^n a_i + i b_i.
Here, one "i" is an index variable - it should be typeset in italic.  
The "i" before b_i is the imaginary unit, which strictly speaking,  
since it is a constant, should be typeset upright, though rarely it is.

So the situation is similar: I used "q" as an index, and you  
introduced another for a quarter-tone.