On 09/21/2010 02:16 PM, Hans Aberg 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 ... :-) Anyway, I will take the time to go over your answers properly and surely be back with more questions. >> ... 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 _______________________________________________ lilypond-devel mailing list lilypond-devel@gnu.org http://lists.gnu.org/mailman/listinfo/lilypond-devel