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.
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