Mathieu Bouchard matju at artengine.ca
Tue Nov 27 08:06:53 CET 2007

```On Fri, 23 Nov 2007, Charles Henry wrote:

> The theory (dynamical systems/pitch) is actually good for this too.
> There is a slight pitch shift when the frequency ratios become slightly
> detuned, but the overall fundamental produced is reliable under
> detuning.

In a nutshell, how does it work? E.g. I have a tone at 200 Hz and another
at 301 Hz and we want it to believe that it's a fifth, that the
fundamental is 100.16 Hz or so, and perhaps also that the fifth interval
is good within 0.33% and that the fundamental is good within 0.16%.

> Like using the mod 12 arithmetic, or other groups?  Or making loops
> (using finite groups)?  I think I can see how it would be useful.  The
> whole idea was confusing to me in the first place... it still is.

Yes, but it can also be various things other things: for example it could
lack closure, such as the [0;1] interval, or the half line (unless you
restrict it to multiplication).

You may also decide to work with Q (and in a computer implementation, you
may also decide to work with true numerators and denominators instead of
floats)

Because Q is a field, it can be used to construct true vector spaces, and
because it's a suitable enough ring, it can be used to construct some kind
of complex numbers: Q[x]/(x^2+1) sounds perfectly reasonable, and btw, so
does Z[x]/(x^2+1).

Beside the complex numbers themselves, modulos of various kinds can be
useful, and not just discrete ones. An especially important example is
R/(2*pi*Z), the ring of all distinct angles in radians. Likewise for R/Z.
It's useful to make Z act upon both of these too (Group Actions). It may
be more convenient to just use the unit circle of the complex plane

Overall, I want to say that what's important is to imitate the hearing of
sounds, not to imitate what individual neurons do nor what we think they
do. Let's say the brain might work only with sigmoid-clipped matrix
products with approximate real numbers (they don't, but computer neural
networks do). Then it's possible that you are better off using a
structure that does more directly what you want (relative to what you
know and how you think), rather than whatever contorsion of the same thing
a billion years of random mutations has come up with.

> PacMan :) I would take two variables to parameterize the surface a1 on
> [0,1) and a2 on [0,1) and use x=cos(2*pi*a1)*(2+cos(2*pi*a2),
> y=sin(2*pi*a1)*(2+cos(2*pi*a2), z=sin(2*pi*a2)

You don't have to embed it in a non-modulo space, especially if computing
things in terms of x,y,z is more complicated than in terms of a1,a2. If
you already plan to compute directly in the a1,a2 square, then I don't
know what x,y,z are for.

> My reasoning was that we can create 1-1 functions on a subset of the
> continuous functions to R^N.

So, if your subset of continuous functions is isomorphic to R^N, what's

And more importantly, when you say band-limited, what kind of spectral
analysis is it relative to? Does "compact in time" mean that your function
is zero everywhere outside of an interval, or that it is periodic? If it's
periodic, then you end up with a finite number of frequencies, but else,
you still have an infinite number of possibilities in a compact set of
frequencies, because the frequencies in-between the supposed harmonics are
not aliased to a specific weighting of harmonics: e.g. with a function
that is sin(1.5*x) over [-pi;+pi) and 0 elsewhere, the spectrum has a
dirac at frequency 1.5/2pi and is 0 elsewhere, right?

>> Did you get into algebraic psychology yet?
> That's the first time I've ever read those words put together.

I haven't read anything on the topic, only stumbled upon the name at one
point, and just the combination of the two words impressed me. I should
look it up, in case it can tell me what are the eigenvectors of my thought
patterns.

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