I am exploring different approaches to make sonifications or music of space-filling curves. This page contains a number of sound tracks of space-filling curves, mapped to sound using location mapping. For an introduction into space-filling curves mapped to sound using location mapping or other techniques, please check out the main page first. Below, I provide some additional examples that may be interesting, although probably less successful than those on the main page. In particular:
Here is an impression of the three-dimensional Meurthe curve as described here. It is 729 points of a 9 x 9 x 9 grid in 2 minutes, “raw” material, apart from some crescendo:
|1||C3 D3 E3 G3 A3 B♭3 C4 D4 E4|
|2||E4 F4 F♯4 G4 A♭4 A4 B♭4 B4 C5|
|3||C3 B♭2 A2 G2 F♯2 E2 D2 D♭2 C2|
Below are three different three-dimensional Hilbert curves as described in my article (“raw” material, 4096 sample points in 14 minutes, ending abruptly). These tracks are unpolished (and probably too long), but it may be interesting to check out the rhythms.
Cl00.cf.ff.dd is a so-called standing curve: the similarity transformations between the curves within the octants and the curve as a whole keep the vertical axis vertical. This is clearly audible: two voices (the horizontal axes) imitate each other's rhythms, while the other voice (the vertical axis) has a (slow) rhythm of its own.
The Beta curve is hyperorthogonal, which implies that no note in any voice is shorter than 2 time units.
Below is an attempt to create a “largo” from a space-filling curve. To get long notes, I used a Meander-like curve on a 4×4 grid, and I sampled it at a resolution of 16×16 (see figure). Each sample point sounds for 1.1 second. The melodic line of the second voice seemed to call for jumping down by a seventh instead of moving up by a second in “strategic” places, hence the non-standard selection of pitches. In the figure, the jumps are indicated by crossbars. Two hand-composed supporting voices are added below the two voices determined by the curve.
|1 / horizontal axis (“flute”)||D4 E4 F4 G4 A4 B4 C5 D5 E5 F5 G5 A5 B5 C6 D6 E6|
|2 / vertical axis (“trumpet”)||B4 C5 D5 E5 F4(!) G4 A4 B4 C5 D5 E5 F5 G4(!) A4 B4 C5|
Focusing on the technicalities of creating music from space-filling curves (and not on the obvious fact that I am not a trained composer), I am not quite satisfied with the result yet. Due to the space-filling character, all combinations of first-voice and second-voice pitch levels appear, and some of these are very dissonant (minor seconds). I did not manage to give all of these dissonances a proper place in the music. Possible solutions to explore:
Using barycentric coordinates, we can also make a sound track of a curve that is drawn on a triangular grid, using three voices. Here is the raw material of a first attempt at the Gosper “flowsnake” curve. Theory suggests that two-dimensional curves may be boring—personally I believe that the sound tracks on the main page show that this is not always true. For the Gosper curve, I am undecided. Maybe this curve has musical potential, maybe not.
Below are some examples of continuous-pitch renderings, 6 minutes each. This is unlike anything else on this page: there are no notes, the sonification is really a continuous mapping from time to pairs of pitch values. So, in a way, you could say that this is what space-filling curves really sound like:
Note: the first of these is a space-filling traversal, but not a space-filling curve, since it is not continuous.