On Sun, 25 Nov 2007, Charles Henry wrote:
On the signals level, we could have a non-linear manifold in a Hilbert space. Sets of functions with constant total energy and identical pitch, for example. Then, psychoacoustics represents the map of this space into timbre space (a psychological space).
Functions with constant total energy are a convex space. This is like a linear space except it changes one rule: in a vector space, if a,b are scalars and x,y are vectors, then ax+by is a vector. In a convex space, there's the additional restrictions that a+b=1 and a>=0 and b>=0, so that you can only blend vectors together by various ponderations, without adding any gain. In 2-dimensional space, any base of a convex space generates a convex polygon (polyhedron if 3-dimensional space instead).
If you are using the affine space, you can't simply add and you can't simply multiply by a scalar: instead, the fundamental operation is the convex sum of "vectors": as a single operation, you add together any number of vectors, weighted, where the total weight has to be 1, so that the amplitude of the fundamental sticks to 1.This makes good sense to me. The operators on this space are still a little fuzzy, though.
Actually, note the difference with convex space: in an affine space, you are not restricted to a>=0 and b>=0. I can only call the latter a convex sum because energy is nonnegative. (Btw, are the values in the vector supposed to be energy values or amplitude values?)
suppose f(t) is a complex tone with frequencies of 100, 200, 300, 400, 500 and g(t) has frequencies of 100, 200, 330, 400, 500 Then, when we mix the two tones together there is dissonance between the frequencies of 300 and 330, which wasn't present in either of the two tones.
Dissonance is a somewhat complicated operation, imho. How you compute it? It's definitely non-linear. It could be a quadratic form, perhaps. Think of it as a matrix sandwiched between twice the same vector so that the result is a scalar. e.g. diss(x) = x'*A*x, where apostrophe means transpose. What would be a good A ?
Forget the matrix syntax, because this vector space is R^R... but matrix ideas can be mapped to functions. What's the dissonance function A(i,j) for two frequencies i,j? Or maybe A(i,j,w) where w is the window size that the dissonance is relative to. I guess that there are many valid and useful dissonance functions, depending on taste.
diss(x) = integral of integral of A(i,j)*x(i)*x(j) di dj
By infinity, I mean, can we take a harmonic complex tone and change the amplitudes of the partials, to achieve any given sharpness/dullness of the tone? Essentially being able to increase the central moment of spectral denisty without bound.
This looks like a job for equalisers... but it requires a signal that has infinitely many partials.
Bounded in terms of the dimensions of timbre. For example, dissonance. Can we have a tone which is maximally dissonant? Are there boundaries on the other dimensions of timbre?
Apart from A(i,j)=A(j,i) and A(i,i)=0, I don't have much knowledge of what would make a good dissonance function. I can't tell what's maximally dissonant without having a dissonance function first.
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