matju at artengine.ca
Fri Nov 23 06:55:17 CET 2007
On Tue, 20 Nov 2007, Charles Henry wrote:
> Yes, but there is evidence for the fundamental bass that occurs between
> pairs of notes, with a strength dependent on those ratios. Complex
> harmonies could have multiple fundamentals. It's a mystery to me how
> harmony/rhythm work at a fundamental level.
Well, so far, most of the time you see "fundamental", there's only one at
a time, for each block of music you want to figure out the fundamental of.
But different fundamentals can be extracted for any given interval, and
those intervals can be a window sliding through time, looking at any
"dinote" (pair of notes), and there can be multiple windows of different
sizes that account for different levels of human memory and of musical
understanding... (?) I think that we could analyse music using whole
networks of fundamentals...
and also, a theory of musical understanding should be resistant to
"detuning", because many forms of detuning are used in music and yet
humans can automatically figure out what the fundamental is _intended_ to
be (rather than what it is physically).
> The topology bullshit was plainly bullshit. But I was trying to
> stretch what I know, and try to see a way for song-structure and
> rhythm to take on more than one dimension.
There are many discrete or semi-discrete phenomena in construction of
music, so using the Reals, an uncountable noncompact continuum, is pretty
counterproductive. Instead of trying to use cartesian powers of the Reals
in some form, try cartesian products of different algebraic structures
that you will not use as (math) vectors.
> I have started working on a patch lately to simulate the trajectory of a
> particle as it flies across the surface of a torus
Are you doing it in terms of a particular embedding with a particular
curvature of the space, or do you use a modulo-Euclidean space in the
style of PacMan ?
> That's just the thing I was getting at. We have music as a function
> from 1-D into the space of all possible sounds. Assuming the space of
> sounds is band-limited and compact in time, it is actually a finite
> dimension (a gigantically huge finite dimension).
Not necessarily... if you fit all sounds in one master period, yes, but if
you are using a continuum of frequencies, you have a continuum of possible
dimensions. The finite dimensions of the FFT (and of other discrete
interval transforms) are because there is a master fundamental frequency
(that is not zero).
> But then, there's the psychological space, which has drastically fewer
> dimensions, and they're not linear.
Did you get into algebraic psychology yet?
> I feel absolutely certain that I can convince you that timbre is *not* a
> vector space, using only the defining properties of a vector space.
Ok, let's do that. How do you prove it?
> However, getting from A to B, and showing this is true would take an
> exquisitely designed experiment, a real work of art :P
That's a detail :-P
Especially as I believe that timbre is a vector space. This is as long as
we agree that timbre is a reduced form of the spectrum of a periodic
sound, and not the more complicated things that happen with attacks, nor
the whole range of an instrument.
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| Mathieu Bouchard - tél:+1.514.383.3801, Montréal QC Canada
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