Free Inquiry Topic #3: Musical Fractals

What are Musical Fractals?

Musical Fractals are hard to define in specific terms, and in a technically mathematical sense, they are extremely rare. A more accurate way to describe the musical fractal is as a self-similar musical pattern that is heard performed at various speeds. This playing of a single musical or rhythmic pattern at different speeds imitates the effect of magnifying a spatial or visual fractal at varying degrees of depth.

As John McDonough and Andrzej Herczynski explain in their article, Fractal patterns in music (found in Chaos, Solitons and Fractals, Vol. 170, May 2023, 113315):

There is currently no broadly accepted consensus on what constitutes a fractal pattern in music. Statistical analyses, as discussed above, provide various measures of structural complexity for a sequence of notes, such as the self-affine index � or the autocorrelation (Hurst) exponent H, but these can be numerically computed for any time series without requiring self-similarity, however loosely understood. Yet self-similarity across multiple scales, whether exact or approximate, is the hallmark of all fractals, their sine qua non. Furthermore, the exponents � and H can be understood to characterize asymptotically increasing periods, or long-time memory of the piece (whether perceived by a listener or not), whereas fractal dimension D characterizes asymptotically decreasing periods, or instantaneous coincidence.

For a strictly self-similar time series, D=2−H, but in music that is highly improbable, except in very short passages. For whole multipart symphonic movements, whose character, motifs, instrumentation, scoring, and tempi, vary greatly, even approximate overall self–similarity is simply unachievable; indeed, it would hardly be desirable. Difficulties arise even with a single line of music because it is not a simple time series; notes have values (durations) and pitch (frequency), and both must be taken into account when defining self-similarity. Compositions made entirely from notes of the same pitch, or from notes of equal value, are rare, although such passages sometimes do occur in longer works.

The approach proposed here aims to avoid treating music as a simple time series and to discern scaling regularity akin to algorithmically defined fractals, that is, reoccurrence of the same pattern at finer and finer scales, with sensitivity to the melody itself. It is modeled on a close analogy with infinitely fine mathematical fractals (such as the Cantor Set or the Koch Curve) and their quasi-fractal graphical representations, wherein only a finite number of subscales can be manifest. In musical pieces, which are finite in length and (in the Western tradition) have a limited range of note values, only a few scaling orders can be expected.

The key insight of this analogy is that whereas in a spatial quasi-fractal structure (such as an image), the same pattern recurs simultaneously at multiple spatial scales at a given location, in music, the same melodic line must occur at multiple time scales at a given instant. A musical fractal pattern, therefore, requires that a motif be performed simultaneously at a few different tempos, creating an intricate interplay of the theme with its faster or slower versions – musical self-similarity.

Two specific compositional techniques for realizing such complexities in music can be identified. The primary music genre, which depends on the simultaneous rendition of the same theme at various tempos, is the prolation canon, also called the mensuration canon (Brothers [48] calls it “motivic scaling”). The technique was almost certainly invented by Johannes Ockeghem, who might have also given it the name. Every movement of his remarkable Missa Prolationum, which survives in a manuscript completed in 1503, offers a different variant of this form. Ockeghem’s most gifted student, Josquin des Prez, developed it further in his masses with an unmatched finesse and inventiveness.

The second fractal technique, described formally in its simplest version by Henderson-Sellers and Cooper [47], consists of the sequential refinement of a given motif wherein each note is replaced by a faster rendition of the entire theme but transposed, so that it starts with that very note. The melody is thus constructed by layering nested sequences and encompasses the simultaneity of multiple scales in a figurative sense, in a single line of music, which becomes faster and faster at each order, but retains the outline of the slower version in the leading notes of each scaled copy. This algorithmic technique has been deployed in creative ways by many composers and thus merits consideration here.