czhenry at gmail.com
Fri Nov 23 20:16:43 CET 2007
On Nov 22, 2007 11:55 PM, Mathieu Bouchard <matju at artengine.ca> wrote:
> 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...
Actually what I'm referring to is the dynamical systems perspective on
pitch perception that I keep harping on about (work by Julyan
Cartwright and colleagues, and articles from Chialvo). It's the only
analysis I've seen that gives some kind of outside support for the
perception of a fundamental bass. For example, Schenker analysis is a
well-developed music analysis technique, but I haven't personally read
any support for it, outside of music theory.
> 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 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
> > 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.
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.
> > 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 ?
I would take two variables to parameterize the surface a1 on [0,1) and
a2 on [0,1)
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)
or using cylindrical coordinates
theta=2*pi*a1, r=2+cos(2*pi*a2), z=sin(2*pi*a2)
> > 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).
My reasoning was that we can create 1-1 functions on a subset of the
continuous functions to R^N.
If we have a function on a finite interval which is also band-limited,
we can map this space onto the coefficients of a finite fourier
series. All of the non-integer frequencies on the continuum still
exist, but the spectrum can be sampled. When we have a real-valued
continuous function on the interval [0,T), we can sample the spectrum
on 1/T without losing any information. Just like when we have a
complex function (a spectrum) on the interval [-fs/2,fs/2) or any
other half-open inteval, we can sample in the time domain on 1/fs
seconds, without losing information, as long as we know the interval
of the spectrum.
> > But then, there's the psychological space, which has drastically fewer
> > dimensions, and they're not linear.
> Did you get into algebraic psychology yet?
That's the first time I've ever read those words put together. That
sounds interesting. I can see that "A Functional Theory of Cognition"
by Norman H. Anderson deals with this topic..
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