matju at artengine.ca
Tue Nov 20 06:06:43 CET 2007
On Fri, 16 Nov 2007, Charles Henry wrote:
> You won't be able to find those low frequencies like 4 Hz, unless one of
> your instruments contains them, like drums for example.
I don't mean frequencies of sine waves, I mean frequency of any kind of
periodicity that is found.
> Percussion instruments can have those low frequencies. And the result
> of adding up the fourier contributions from periodic sequences has an
> effect like a comb filter on the spectrum of the orignal instrument,
> which makes the peaks. If you have an instrument in a higher frequency
> range, you probably won't find those low e.g. 4 Hz frequencies, but you
> could find them in the envelope following signal of the original.
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.
> but that's just nitpicking..haha I find it interesting to consider how a
> song structure could have more than one dimension...
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".
> But a loop is a path. So, we start from point A, we go to point B and
> come back. So, if we have a measure of 8/8, we can represent it as the
> path in the plane which follows e^(2*pi*i*t/8) or many other paths.
> Still we have a clearly defined topology (btw, I'm just learning
> topology, so I'm feeling my way through this).
> A function maps points in time onto the loop. Again we have just one
> dimension. We can extend our loop into a sphere. or a torus or any
> other surface in more than two dimensions with holes in it.
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
> but still it breaks down... we can only have the paths as functions of
> time. So, no matter how complicated the song structure gets, you can
> flatten it into a single function. Any thoughts?
And yet, to express this function, you'll probably want to break it down
into several functions, for modularity. The advantage of putting
everything in one big function is somewhat overrated. Already, any
abstraction mechanism in math is a way to modularise and outsource meaning
so that it doesn't have to be specified in the main function(s)
> 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.
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| Mathieu Bouchard - tél:+1.514.383.3801, Montréal QC Canada
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