Charles Henry czhenry at gmail.com
Sun Nov 25 17:19:15 CET 2007

```The problem with my examples, which I thought were bad was that
sometimes, I was using x(t) and y(t) as if they were signals, which
can be added and subtracted, and sometimes as vectors as functions of
time in an abstract timbre space.
Some of the presumed dimensions of timbre are things like
consonance/dissonance, formant location, which seem to be consistent
with the loose definition of timbre.  I have to be more careful, so I
don't bullshit myself out of existence or make some other bad argument
:)

> I think that you are right that it can't be all done within a linear
> framework: there needs some slight mangling of a linear space to do the
> work.

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).

If we start with the assumtion that timbre space is a linear vector
space, the psychoacoustical mapping function has to be pretty wild.
On the other hand, we can map a sphere onto a plane with fixed
boundaries (not a vector space), without any trouble.

> Oh, well, you could do most of the work using a linear space, and then at
> the last possible moment, divide the space by products by a positive real,
> so that there is one element per possible direction (or by any real, so
> that there is one element per possible axis). This could be called R[x]/R+
> or R[x]/R respectively. Also, this could be called "spherical space" and
> "projective spherical space", respectively.
>
> You could also suppose that the fundamental's amplitude is always 1, which
> is another way to give you exactly one element for all possible loudnesses
> of a sound that is otherwise the same. This is also better because then it
> ensures that it's a unique timbre, as you can't set all odd harmonics to 0
> in such a situation (this would have allowed you to pretend a 440 Hz sound
> is also a 220 Hz sound and such). Also, sometimes affine spaces are easier
> to work with than quotient spaces even if you use those quotient spaces as
> little as possible.
>
> A neato aspect of R[x]/R+ is that even though vector addition doesn't work
> on it, vector multiplication by matrix works quite well, and for example
> R^42/R+ can be acted upon by SO(42,R) and most any other matrix group...
> although SO(42,R) is the most tightly fitting matrix group in this
> case: SO matrices preserve the L2-norm of vectors, so what nicer thing can
> there be for a set of pseudo-vectors in which L2-norm has been made
> irrelevant?

> 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.

> > OK, so how about linearity?  If we take two timbres x(t) and y(t),
> > then we can construct a timbre z(t,a)=ax(t)+(1-a)y(t)   (0<=a<=1)
> > which interpolates between x and y.
>
> This is a special case of the "convexity" requirement.
>
> > And let's take a particularly bad example.  We'll take x(t) to be a
> > harmonic series.  Then, we'll let y(t) be the same harmonic series,
> > with a single mis-tuned partial, while keeping pitch constant.  Then
> > z(t) becomes dissonant moving between x(t) and y(t), even though
> > dissonance was not significant in x(t) or y(t).
>
> I don't quite understand how this works. Can you make a version of this
> example with actual figures?

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.

> > Can we move the central moment of spectral density all the way to
> > infinity while keeping pitch constant?
>
> What do you mean "to infinity" ?
>
> Anyway, it depends on how "perceptual" you are trying to be, supposing
> that we don't argue on the meaning of "to infinity".

One kind of dimension of timbre is the sharpness/dullness of a tone,
based on the distribution of spectral energy among the components.  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.

> > If the space of timbres is bounded, then it cannot be a vector space
> > (because it fails to be closed under scalar multiplication).
>
> You mean bounded how? bounded in amplitude or in frequency? if it's
> bounded in frequencies, it's still linear. but you sound like you mean
> it'd be bounded in amplitude, which wouldn't be as much linear, but the
> spherical space above would make this issue moot anyway.

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?

I don't think I did a very good job of laying out my ideas.  But this
was a good discussion (for me) to examine them in detail and try to be
specific enough.

Chuck

```