Mathieu Bouchard matju at artengine.ca
Sun Dec 23 19:50:04 CET 2007

```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.

_ _ __ ___ _____ ________ _____________ _____________________ ...
| Mathieu Bouchard - tél:+1.514.383.3801, Montréal QC Canada
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