Felce

A fractal looks like itself at every scale: the fern’s frond, the coastline, the tree. Benoît Mandelbrot gave them the name in 1975. The same year Richard Voss and John Clarke, physicists, measured how the loudness and pitch of music and speech fluctuate, from Bach to the Beatles to radio talk, and found the same law everywhere: a power spectrum falling as 1/f, the fingerprint of something self-similar in time, between white noise (no memory) and brownian motion (nothing but memory). Composers had been writing self-similar music long before anyone measured it.

In a prolation, or mensuration, canon every voice sings the same melody at once, each at its own speed. Johannes Ockeghem’sMissa prolationum is built of them; in the second Agnus Dei of Josquin’s Missa L’homme armé super voces musicalesthree voices sing one line at three speeds. It is a fractal in time: the slow voice is the fast one, magnified.

The Danish composer Per Nørgård found, in 1959, a melody that is such a canon with itself, his infinity series: start on 0; the note at 2n is the one at n turned upside down, the note at 2n + 1 is the one at n plus a step. Every fourth note of it is the series again, so played against itself at a quarter of the speed it keeps landing on the same notes. He builtVoyage into the Golden Screen (1968) and his second symphony (1970) on it.

Take a motif, and replace each of its notes by the whole motif, starting on that note; do it again. The tune at every level has the outline of the tune at the level above, as each branch of a fern has the outline of the frond. Tom Johnson explored such melodies in Self-Similar Melodies (1996); Robert Walker’s program Fractal Tune Smithy makes them from any seed. Kept in time, the levels are a prolation canon of each other: the felce plays them together if you ask.

The logistic map, x → r·x(1 − x), was Robert May’s example, in Nature (1976), of how simple rules make complicated behaviour: as r grows, one note, then two alternating, then four, eight, sixteen, faster and faster (Mitchell Feigenbaum found the constant ratio, 4.669…), then chaos, with windows of order inside it. The Mandelbrot set is the set of points c where z → z² + c, started at 0, never escapes; click a point and hear its orbit: in the big cardioid it settles on one note, in the bulbs on loops of 2, 3, 4…, outside it escapes. Voss’s dice, which Martin Gardner described in Scientific American (April 1978), make 1/f melodies with a handful of dice thrown at different rates.

The formulas include what dedicated programs have used: MusiNum, Lars Kindermann’s “music in the numbers”, counts and plays how many 1s each number has in binary, a melody that contains itself at every power of two; the Thue–Morse sequence; the ruler sequence; the steps of the 3n + 1 problem. Bytebeat, found by Ville-Matias Heikkilä (viznut) and others in 2011, plays a formula of the sample counter as the waveform itself. And the Weierstrass timbre is a fractal as a sound: the function Karl Weierstrass presented in 1872, continuous everywhere and smooth nowhere, whose harmonics sit at the powers of b, falling as the powers of a.

Aristid Lindenmayer, a biologist, described in 1968 how algae grow with a grammar: every symbol rewritten at once, A into AB and B into A, generation after generation. Drawn by a turtle, such grammars make the classic fractals and lifelike plants. Przemysław Prusinkiewicz, in “Score generation with L-systems” (1986), read the turtle’s curve as a score: its height the pitch, a horizontal run a held note. Felce reads it that way, and another: the branches of a plant as voices, each starting where it grows from the trunk.

A language model writes one word at a time, choosing by probability, and its patterns show: phrases that return, favourite words, and, when it goes wrong, loops. Made into music, a word to a note, they can be heard, and seen in the self-similarity matrix, the picture Colin Morris’s SongSim made of song lyrics (2017), where a chorus is a diagonal.

Then the measures. Zipf’s law: in what people write, the second commonest word is about half as common as the first, the third a third. Repetition: Sean Welleck and his colleagues (2019) counted the four-word phrases a model repeats, near none in people’s writing, many in a model that degenerates. Compressibility: Jürgen Schmidhuber argued (1997) that beauty is in what a mind can compress, a pattern found; a loop compresses to nothing, noise not at all.Entropy, Claude Shannon’s measure of information (1948). And Birkhoff’s aesthetic measure (1933), order divided by complexity. Each has a range where people’s texts and music usually sit, between order and surprise. None of them knows what the words mean.