Merkurius: the symmetry architecture of a piano piece from 1996

I composed Merkurius for piano in 1996, while I was studying composition with Peter Lyne at Kapellsberg in Härnösand, Sweden. It belongs to a series of pieces I called Planeterna ("The Planets"), together with Charon for piano, named after Pluto's moon, and Ganymedes for piano, piccolo and cello, named after one of Jupiter's moons. I wrote them before I knew Holst's orchestral suite The Planets. Of the three, Merkurius is perhaps the most mathematical.

Photograph of the first page of the printed score of Merkurius, with the title handwritten at the top: two piano staves, bars 1 to 10, with wide leaps between the hands and changing time signatures. — The first page of the original score of *Merkurius*, bars 1–10 (1996). © 1996 Christopher Yamba. All rights reserved.

Thirty years later I started going through my old compositions to archive them, and analysing them turned out to be more interesting than filing them away. Read as a mathematical object, the score of Merkurius is built from a small set of exact symmetries:

The time signatures of bars 5 to 48 form a palindrome: bar m has the same meter as bar 53 − m, across 44 bars.
A fast 20-note line on five pitch classes returns as its exact retrograde, in bars 6 and 34 and in bars 19 and 47. The inversion that links the two pairs differs in exactly one note, which the original score will have to settle.
A second line, on four pitch classes, appears exactly as original, inversion and retrograde, in bars 4, 17 and 36.
Inverting the E Lydian scale chromatically gives E Mixolydian.
In the first bar one hand falls by semitones while the other rises by major sevenths, so the interval between the hands stays a major sixth.
Bars 13, 26 and 40 split a whole-tone scale into two complementary augmented triads.

The article develops the group theory behind these observations. It uses Burnside's lemma to count how many genuinely different phrases such a system allows, for example 304 classes of marked five-note phrases, and it explains a 35-note cycle that appears when a five-note motif meets a repeated note every seven notes. The findings are checked by computer against the digital score, and the article keeps what the score literally contains apart from interpretation.

Next, I want to turn these structures into compositional techniques and write new pieces with them, drawing on what I have learnt since 1996 in mathematics, computer science, physics, AI, biology and bioinformatics.

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