This paper presents an approach for representing space-filling curves by sound, aiming to add a new way of perceiving their beautiful properties. In contrast to previous approaches, the representation is such that geometric similarity transformations between parts of the curve carry over to auditory similarity transformations between parts of the sound track. This allows us to sonify space-filling curves, in some cases in up to at least five dimensions, in such a way that some of their geometric properties can be heard. The results direct attention to the question whether space-filling curves exhibit a structure that is similar to music. I show how previous findings on the power spectrum of pitch fluctuations in music suggest that the answer depends on the number of dimensions of the space-filling curve.