Alex, you wrote
|as I was perusing your sight, came across your page investigating showing periodic data on a circular graph and it reminded me of a style of spectrogram I was interested in exploring, one which makes the representation of harmonics more accessible (for me anyway). ... The idea is that the spiral represents the non linear frequency relationships where, as shown in this awfully crude picture, each harmonic of any series of frequencies exists at the same angular position (which can be colour coded). I would love to see this if it already exists.|
Before I can say whether it exists, I need a better idea of what it is. I suspect that what you're imagining is a chimera, but from a single picture, I can't tell. Up/down refers to frequency, but what are the units? If you measure frequency in cycles per second (linear frequency), then the harmonics of any given pitch are equally spaced (since they are 1, 2, 3, 4, 5, 6 ... times the fundamental frequency). If you measure frequency in pitch space (log frequency), then the harmonics get closer together as you go up (since the intervals between successive harmonics are P8, P5, P4, M3, m3, ...). On this page is a chart of frequencies and a picture of them in linear and log space:
(Hmm, for some reason, Excel picked colors for some of the harmonics that are so close to white that the lines are invisible. You'll have to use your imagination on those.)
In your diagram, the spacing between harmonics (both vertically and as an absolute distance around the spiral) gets larger as you go up, so the two ways of representing frequency that I'm familiar with don't map to it in an obvious way.
Maybe I'm making some incorrect assumptions about how to interpret the picture. You've shown a single pitch; if there were two pitches, would the second one appear on the same spiral. or would it have its own spiral? If it's a different spiral, what is the spatial relationship between one spiral and another? If it's the same spiral, where would the harmonics of the second one appear? For example, if the two pitches were C and G a P5 higher, what would the picture look like?
A related question: why a spiral and not (for example) a helix?
You've seen some of the visualizations I've done that relate to this idea (in the History series); the enclosed file contains a few more variants I explored (V17, V21, V22, V23, V24, V26). The problem with picking a single space (dimension) for frequency is that there is no space in which both equal frequency distances (the spacing between harmonics) and equal pitch distances (the distance spanned by a particular musical interval) are represented by equal visual distances. In V21, I tried to address this by having the x-axis (horizontal) be linear frequency and the y-axis (vertical) be log frequency. I didn't do a very good job of it, but I could tell from this first attempt that it wasn't an idea that would "just work," like some do. Might be worth revisiting ...