czhenry at gmail.com
Sat Nov 17 02:16:41 CET 2007
> "reprise", "beat" and such, are just larger scale splittings of the time
> dimension in the same way that frequency separates from time. Reprises and
> beats and rhythms are full of periodic patterns, just like the sound waves
> themselves, but at a different scale, which doesn't make the physical ear
> resonate anymore, but appeal to the brain's taste for sequencing. Thus a
> beat may have frequencies like 4 Hz and 2 Hz and 0.333 Hz in it (whatever
> is roughly in that range), whereas larger-scale song structures may have
> frequencies like 0.1 and 0.01 Hz. You could call rhythm and song structure
> a third dimension of music.
You won't be able to find those low frequencies like 4 Hz, unless one
of your instruments contains them, like drums for example. 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.
but that's just nitpicking..haha
I find it interesting to consider how a song structure could have more
than one dimension...
Consider the familiar time-points analysis. We structure events in
music as a function of time.
f: R (time) -> (set of possible sound events)
The topology in this case is clear. It's a line, and music is a
function mapping 1-D into the space of all possible sounds.
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
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
Then, you could have an infinite variety of paths, representing
different ways of looping, different periods, etc...
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?
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.
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