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location_mapping

The sound of space-filling curves by location mapping: additional examples

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:

Three-dimensional

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:

voice pitch selection
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.

curve name nickname audio
Cl00.cf.ff.dd (unnamed)
Ca00.gs Beta
Se33.c7T Mosquito

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.

More two-dimensional curves

Meander 4x4

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.

voice pitch selection
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:

  • try other transpositions (maybe there is one where the dissonances are in less awkward places);
  • try a hexatonic scale;
  • try a curve that fills a triangle in a square grid (such as the Sierpinski curve) and position the triangle in the two-dimensional pitch space such that the second voice always remains at least a major second below the first;
  • convince a trained and/or talented composer to have a go at it.

Gosper flowsnake curve

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.

The well-known Gosper curve (344 vertices in 5 minutes)
raw sound track

Continuous pitch

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:

generator and sketch audio

Note: the first of these is a space-filling traversal, but not a space-filling curve, since it is not continuous.

location_mapping.txt · Last modified: 2018/04/03 14:23 by administrator