Herman Haverkort > Recursive tilings and space-filling curves > Sound of space-filling curves > Axis mapping
This approach to sonifying plane-filling curves is one of several approaches I have explored. For the other approaches, please click here.
This web page contains a number of sound tracks based on axis mapping. Admittedly, this page is rather rudimentary, lacking figures in particular. Moreover, so far, I have tried axis mapping only for one particular space-filling curve. I might work on it later, but for now: if you do not understand what is going on here, it might help to have a look at at my location-mapping approach first.
For d = 1 up to 6, the following sound track plays, one after another, a sketch of how a d-dimensional Harmonious Hilbert curve (as described here) traverses a grid of 4d subcubes. The track uses six different pitch classes (A, G, F, E, D, C) that correspond to steps in the 1st, 2nd, 3rd, 4th, 5th and 6th dimension, respectively (whether the step goes up or down in that dimension is not encoded). On top of that, there are 2d−1 voices that each sketch one of the 2d−1 facets of the d-dimensional cube that are each traversed in the order of the (d−1)-dimensional Harmonious Hilbert curve. In these sketches, more emphasis is given (by means of slower decay of the sound burst) to each 2nd, each 4th, each 8th, each 16th step etc. The sketches for the facets are aligned with the sketch of the d-dimensional curve, that is, the “steps” of the sketch of each (d−1)-dimensional curve are played when the d-dimensional curve makes the corresponding steps. Thus, the entire sound track is unisono, but the voices for the 2d−1 facets sound in different registers (the axis of the outward normal determines the octave, and the sign of the outward normal determines the timbre).
Note that in the six-dimensional section, the sketches for the facets play 11 times 1023 notes, while the six-dimensional sketch is only 4095 notes. Thus, on average, each note is hit by almost three facet voices. It is mostly the variations in how many (and which) facet voices are playing that create the dynamics of the sound track (there are no hand-composed notes or dynamics on this track!). One may listen to the salient features of the unexpectedly rich sound track and think about how those features arise from the structure of the curve and the six-dimensional cube—or one can just listen to the “music”.
Below is a track with an alternative pitch set (C, D, E♭, F♯, G), playing the Harmonious Hilbert curves up to five dimensions:
The following track plays the curve up to four dimensions (on C, B, A, G), but at a higher resolution (8 × 8 × 8 × 8), resulting in a track that is less dense: there are 7 facet voices of 511 notes each on a track of 4095 notes, so there is lots of “quiet time” in which one hears only the four-dimensional curve.