On Nov 22, 2007 11:55 PM, Mathieu Bouchard matju@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 under detuning.
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 ?
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) 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..
Chuck