On Nov 19, 2007 11:06 PM, Mathieu Bouchard matju@artengine.ca wrote:
On Fri, 16 Nov 2007, Charles Henry wrote: I don't mean frequencies of sine waves, I mean frequency of any kind of periodicity that is found.
Yes, I was sure you knew what you were talking about. I just had to jump on it, and add some parts that I felt you could have included. Also, you got some good info here:
Notes of many instruments have a percussive/click sound in the attack, which tend to have a wider spectrum than the main part of the note, so there could be a 4 Hz component anyway, but this is not what I mean anyway. What I mean could include the fact that the comb effect is at 4 Hz, but it's more abstract than that: playing any melody, you can simply count the time between attacks or between changes of note, and see it as a set of periodic patterns. It could get as far as taking any interval and turning it into the corresponding frequency, even though there may be a complete absence of actual periodicity.
On FidoNet in the mid-nineties, I was getting acquainted with the theory that rational intervals between notes (e.g. 5/4, 3/2) could correspond to rhythmic patterns as far as they could be expressed with a similar notation: thus you can see a major chord as being similar to a pattern involving a superposition of 4/4, 4/5 and 4/6 beats. Needless to say that in "normal" music, "normal" chords don't match the ratios of "normal" beats, except in extremely simple cases such as "power chords".
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. I'm planning to apply for grad school at FAU this month. My plans are not sure now, but I will eventually work on this.
I don't have the impression that we need topology in order to access all that we need to do what we have to do. So far, I think that the interest of using topology in music is just so that we have topology and music together... just an alternate way of expressing the already expressible. (Please convince me that some things in music are easier to think about using topology...)
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. I have started working on a patch lately to simulate the trajectory of a particle as it flies across the surface of a torus (it's remarkable simple, so far--a couple of phasors and boom, there it is). Next thing is to add functions that will map the particle's trajectory onto sounds (the tough part).
Some current rhythm perception research focuses on dynamical systems, which can have those long-range correlation properties. (again the action of perception is still a function of 1-D time) The dynamical system can have a non-integer dimension (a fractal), so you might be on to something to speculate additional dimensions in sound.
The Hausdorff dimension of a set that is a subset of some space can't be bigger than that of that space. If anything, you get above the 1-D of the time dimension, but never above the number of dimensions of the space that the trajectory lives in. Even then, you are approximating a phenomenon using a fractal, which does not mean that the phenomenon is fractal any more than real numbers are real and that infinity is infinite: there's a lot of theoretical gimmickry there. Many phenomena look fractal only within a precise range of orders of magnitude.
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). But then, there's the psychological space, which has drastically fewer dimensions, and they're not linear.
I conjecture that timbre perception may be better explained through topology. A common figure in analyzing instrumental timbre is a multi-dimensional scaling technique. Similarity between timbres is visualized in a linear space with a metric, corresponding to the straight-line distance. If it were possible for timbre space to be a non-linear manifold, similarities would correspond to distances along a path in the presumably curved space. 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. However, getting from A to B, and showing this is true would take an exquisitely designed experiment, a real work of art :P
Chuck