The reason I thought of this community to post it in, is that I'm going to phrase the idea in terms of a math problem:
What functions, f(x) (reals to reals), have the property that
for all real-valued k,
the sum, with n going from negative infinity to infinity, of f(n+k)
does not depend on k?
In particular, I'm looking for functions f(x) with this property that take a global maximum at 0, are even, are increasing when x is negative and decreasing when x is positive, are always positive-valued, and take a limit of 0 at infinity and negative infinity....
The context: I want to find a mechanism (deterministic or aleatory) to create pieces of music out of works of visual art. In order to do this, I'd like to have a musical representation first of color.
First off: In color theory, each color is assigned a place in the "Color Solid" according to three dimensions: Hue, Saturation and Intensity.
The Intensity of a color is how dark or light it is.
The Saturation of a color is how bright or dull it is.
The Hue of a color is its position on the color wheel: yellow and blue-green are examples of hues.
The "North Pole" and "South Pole" of the Color Solid are white and black: just as the longitudes of earth's North and South Poles are undefined, black and white have undefined saturation and hue.
My plan is for each hue to represent a certain abstract pitch, not dependent on octave, e.g. F# or Bb, but also including the pitches between the pitches in the normal twelve-tone scale. I also want the notes to get higher as the colors get lighter. A combination of these two axioms means that a set of certain colors, as a certain hue progresses continuously in intensity from dark to light (leaving saturation aside for the moment), should correspond to a continuous progression of pitches, all of which are (say) Bb.
The way I plan to do this is similar to (what I believe is) how the Rising Tone works, that is, a tone that gives the illusion of constantly getting higher and higher, while in reality staying exactly the same. (An arpeggiated version of the Rising Tone can be heard in Super Mario 64 for the Nintendo 64, when one tries to ascend the endless stairway to the final Bowser level with less than 70 stars.) Instead of having a single pitch representing each color, I'll have an octave chord: instead of one Bb, I'll have a whole chord of all the Bb's in the tone generator's range, each one at a different volume level. The intensity of the color will then correspond to the central pitch of the chord, which will probably not actually be in the chord; however, it is the central pitch in that the farther a particular one of the chord tones (all of which are a whole number of octaves apart from each other) lies from the central pitch, the quieter it will be.
You've probably already guessed where this is going: the function f(x), above, represents the volume for each tone in the chord. Presumably, the volume will be the highest when the chord tone actually *is* the "central pitch"; therefore we can represent the range of all possible pitches by the infinite real line, an increase or decrease of 1 on the line being equivalent to raising or lowering the pitch an octave (so we can take the base-2 logarithm of the frequency, plus or minus a constant perhaps, to get the corresponding position on the number line). Whatever the central pitch is will correspond to 0; then k will correspond to the distance from this central pitch the actual tones present in the chord are, and the criterion I've italicized above is then (I hope...!) seen to correspond to the condition that this chord will exist at the same total volume no matter where the chord tones are relative to the central pitch....
... phew! And no, I haven't even dealt with saturation yet. (Maybe different saturations could correspond to different possible values of the solution function f, if there exist more than one? Or maybe they could correspond to different source sounds that the chords are constructed from in the first place?) And how is actual dynamic music to be made out of this? Well, that's far more loose and floaty a concept than this is, so I think I'll type that up in my regular livejournal... coming soon!
I don't even know who to give this full idea to. Maybe someone knows someone who has a connection....
UPDATE: The Big Picture is now at http://abangaku.livejournal.com/52433.h