I am exploring different approaches to make sonifications or music of space-filling curves. One of these approaches is proximity mapping. For an introduction and examples of proximity mapping, please check out the main page first. Below, I provide more details of the mapping, and some additional examples that may be interesting.

For ease of description, consider a d-dimensional self-similar space-filling curve f: [0,1] → [0,1]^d (however, the approach below can also be applied, mutatis mutandis, to non-continuous traversals). We traverse the curve, that is, the values f(t), from t=0 to t=1, sampling it at a set T⊂[0,1] of n uniformly spaced values of t. When we are at a specific value of t, we generate a tone for each other point f(t′) with t′∈T, generating louder tones for points that, together with f(t), define pairs for which the space-filling curve does not preserve locality well. So in total, we generate roughly |T|² tones: there are |T| time steps t, and in each step, we superimpose |T|−1 tones, one for each pair (t,t′) with t≠t′.

Frequency: with a typical mapping, the generated tone for the pair (t,t′) would be 24·sin(½π(t−t′)) semitones above C♯4 (so it will be between C♯2 and C♯6). However, for many sound tracks, I use a different mapping—as long as it is a non-increasing or non-decreasing function of t-t′.

Amplitude: let r be the Euclidean distance between f(t) and f(t′), and let V be the volume of a d-dimensional sphere with radius r. One could argue that in a hypothetical ideal locality-preserving space-filling curve, all points of that sphere would be covered by the curve section from f(t−½V) to f(t+½V). Therefore, we can say that f(t′) is “too far away” along the curve by a factor |t−t′| ⁄ (½V); we denote this factor by s. We now map this factor to an amplitude of, in principle, 0.01·cap(αs²), where α is a parameter that controls the sensitivity of the mapping (in practice, I used α values between 0.001 and 0.05), and cap(x) = x ⁄ (1+x) is a function that dampens higher values so that the amplitude stays below 0.01. However, within distance r, there will be a number of points proportional to r^d that generate waves of similar frequency and amplitude, and, unless they are somehow in phase, they will add up to a sound with an effective amplitude that is a sublinear but still rising function of r^d. If we would not compensate for that, points farther away would completely dominate the sound track due to their sheer number. Therefore, for each point f(t′), we divide the amplitude by r^βd, for a suitable parameter β. So far, for the three- to seven-dimensional cube grids, I got best results with β values between 1/2 and 1/3. (A value of 1/2 makes sense theoretically, since the standard deviation of r^d independent signals with standard deviation 1 is equal to r^d/2, but sometimes lower values work better in practice; I believe it might have something to do with how we perceive the loudness of mixtures of different frequencies.). For the permutation traversal orders, I used β=4.

Phase: to prevent unintended interferences, a random phase shift is applied to each tone.

A showcase of nice tracks can be found here. Below is an additional set of five six-dimensional traversals (4⁶ points at 60 points per second, α=0.005). These do not sound nice, but they might be interesting to compare. They are roughly ordered from smoothly (and quite boring) rising sound at the top to less-smoothly rising and melodically more interesting sound at the bottom. Does this correspond to an ordering by locality-preserving properties? All space-filling curves and traversals are described in my manuscript on higher-dimensional curves.