Dmitry Kormann, a composer from São Paulo, Brazil, explains how he uses fractal-like patterns to structure his music in “Fractal Music” at + Plus Magazine (12 July 2010):
I found that in order to get that really nice feeling of balance between randomness and predictability that one gets from contemplating fractals on a computer screen — what I like to call the fractal effect — a minimum of three iterations is necessary. Coincidentally, Harlan Brothers also specifies . . . that in order for a structure to earn the title of fractal, a minimum of 3 levels of scalar affinity must be present, as with less than that, mathematical power-law relationships will not exist (meaning we cannot see if the transformation is constant at larger scales).
- “3 levels of scalar affinity” = 3 repeats (“iterations”) of the transformation, e.g., the same motif has to be repeated using three different rhythms;
- “power-law relationships” = scale-invariant transformations (a.k.a. dilatations)—or, strictly speaking, self-similar transformations—i.e., some element of the motif, e.g., the pitch, has to stay the same.
Here’s a visual fractal: the Sierpinski Gasket . . .
Kormann gives the example of how in The Rite of Spring Stravinsky “created forward momentum by making motifs come out from inside one another”:
Stravinsky alternates progressively larger instances of a new motif with progressively smaller instances of an ongoing one.
This is a kind of two steps forward, one step back—or telescoping—pattern that allows new motifs to be gradually introduced.
Musical fractal effects aren’t new: what Harlan Brothers calls motivic scaling was present in the Baroque mensuration or prolation canon, characterized by a melody or rhythmic motif that is repeated in different voices simultaneously at different tempos; see the Agnus Dei (midi) of Josquin Desprez’s first Missa l’Homme Armé Super Voces Musicales. (Prolation is Latin for “lengthening.”)
Here each voice sings the same music, but at different speeds. In the original score only one part is given: a notation over the single line of music indicates the three prolations to be used, and a second notation over the line indicates where each voice should end if sung correctly. A modern example is Arvo Pärt’s Cantus in Memoriam Benjamin Britten: