On Sun, 23 Dec 2007, Charles Henry wrote:
To split hairs, we want to constrain the total energy in mixing
signals, which means we have to expand the inner product.
I mentioned convex spaces possibly because you can deform your space so
that you don't have to do it with the inner product. If each of your
dimensions' values is an energy level instead of an amplitude, then
instead of forcing the inner product to be 1, you can force the simple sum
of all components to be 1, and that's a linear equation instead of a
quadratic equation. It changes the nature of how you cross-fade between
components, but that doesn't mean that it changes it in a bad way.
This ensures that we have a solid homotopy, where we're not
interpolating outside of our space (I stated this wrongly in the first
place!).
so, |x|^2=|y|^2=1
and |a*x+b*y|^2=a^2*|x|^2+b^2*|y|^2+a*b*<x,y>=1
What makes you think that? An inner product is only guaranteed to be
sesquilinear; in other words, it's "conjugate-commutative": some kind of
hybrid between commutative and anti-commutative.
You also forgot to multiply a*b*<x,y> by two because even in the
commutative case you have to count it twice.
|a*x+b*y|^2 = a^2*|x|^2 + b^2*|y|^2 + a*b*<x,y> + a*b*<y,x> = 1
If you don't use complex numbers you probably can say <x,y>=<y,x> and then
you can write it like:
|a*x+b*y|^2 = a^2*|x|^2 + b^2*|y|^2 + 2*a*b*<x,y> = 1
And then I don't know where you are getting to with your
simplifications, but really, if you use flatten it into a convex space,
it looks a lot friendlier for interpolation.
With a |x|=1 kind of space, the only nice stuff you can do on it is action
by an orthogonal matrix space. I wouldn't enjoy to have to mess with
square roots on that. OTOH it could be that the convex space thingie is
unusable in practice because one would want to work with amplitudes
instead, but I haven't really tried... It's not like I plan doing anything
with those structures any time soon.
I've always been fascinated (obsessed maybe? meh) with convolution
operators. I have often said some wrong things about these, but later
worked out proofs of general properties that are essential. L1 norms and
L2 norms are the most important. Convolution preserves L1 norms (proof
on request) in the following way (here | . | represents the L1 norm, |f|
= integral( -inf, inf, |f|dt)
and * is convolution
|x*y| = |x| |y|
In that case it might be easier to write slightly more verbose formulas
than having to explain the formula... e.g. L1(conv(x,y)) = L1(x)*L1(y),
where * is the ordinary product.
and in the L2 norm shown here with same notation
|f| = sqrt( integral( -inf, inf, f^2dt) )
|x*y| <= sqrt( |x| |y| )
BTW, note that the L2 norm in the spherical space is (isomorphic to) the
L1 norm in the convex space. (BTW, from now on, I will only use x,y to
talk about vectors in the spherical space, and will use different symbols
to talk about the convex space, e.g. convex(x) and convex(y))
To me, convolution makes a good operator for consideration in this
type of space. Maybe there's a modification to the definition we can
make to be sure that |x*y|^2=1 ?
Well, you could define the normalised convolution product as being
conv(x,y)/L2(conv(x,y)) ?
Let's say F(x),F(y) are Fourier transforms of the x,y vectors. Then the
convolution of x,y is a componentwise product (representable by diagonal
matrices if you prefer that, but i'll call it cp), according to the
Convolution Theorem, and F is energy-preserving, according to Parseval's
theorem. So F(conv(x,y)/L2(conv(x,y)) = F(cp(x,y))/L2(cp(x,y)). Does this
get you further in any way?
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?)
The values in the vector should be amplitudes of orthogonal components, right?
In the convex space, no, you deal directly with energy... but I suspect
that if you want to interpolate between timbres, it's better to linearly
interpolate energies instead of amplitudes, as it keeps total energy
constant.
Then, dissonance arises between pairs of frequencies by a nonlinear
function N(X) which takes the dissonance between each pair and creates
a vector of all possibilities.
diss(X)=N(X)*A*N(x)/2
where A=
[0 a1*a2 a1*a3 a1*a4 ....
[a1*a2 0 a2*a3 a2*a4 .....
[a1*a3 a2*a3 0 a3*a4 .....
where you see a1*a2, etc... I mean for it to be sqrt(a1*a2)
The elements are on the diagonal are zero because a single frequency
makes no dissonance with itself.
I don't believe this function. I'd expect the diagonal elements to follow
the same pattern as everything else. Then I'd expect the amplitudes to be
the elements of X and I'd expect the frequencies to be the indices of X.
I'm completely lost, but something like sqrt(a1*a2) definitely looks
wrong. It needs to be a formula such that when you combine a1 with itself,
a2 with itself, etc. it will give zero naturally, without having to make
an exception.
Plus, there's the added complication of non-linear critical band rate.
So, the dissonance function is different for different registers
(like the difference between a major third harmony in bass as opposed
to treble).
It's not "Plus"... I already mentioned this. I called it the window size.
You adjust it to match the boundary between tone and rhythm, and it is
this boundary that causes bass harmony to be touchier.
Attached is my resurrected/re-designed dissonance curves patch. The
last thing I was doing with it was to look at dissonace relations with
odd-harmonic series. Yes, it's very crudely coded--see for yourself--if
anyone wants to collaborate, I'm up for a fundamental overhaul.
Hmm... I haven't looked at it yet.
_ _ __ ___ _____ ________ _____________ _____________________ ...
| Mathieu Bouchard - tél:+1.514.383.3801, Montréal QC Canada
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