On â€œReverie in Prime Time Signaturesâ€ - Emory...
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On Reverie in Prime Time Signatures
ROBERT P. SCHNEIDER
The passage included below is the authors explanation of the following score, the main theme for the play Mathematical Science Investigation (MSI): The Anatomy of Integers and Permutations by number theorist Andrew Granville. MSI debuted at the Institute for Advanced Study in Princeton, New Jersey, on December 12, 2009. The author had the privilege of performing his score on analog synthesizer, alongside cellist Heather McIntosh and clarinetist Alex Kontorovich, at the showwhich consisted of seated actors reading from a script while three silent performers moved against a sparse, avant-garde backdrop of white paper. The explanation, excerpted from from the printed play program, reads as follows:
As the title indicates, the piece is written in prime-numbered time signatures; which is to say, there is a prime number of beats in each measure. The main theme plays in the time signature 7/4, which indicates 7 beats per measure, with an interlude that passes through the signatures 2/4, 3/4 and 5/4 as well. From the constraints imposed by these rhythmic patterns, melodies emerged naturally as I composed, special to each prime.
A second interlude happens in 29/4 time, occurring, by a pleasing coincidence, at the 29th measure of the composition: a musical rendition of the sieve of Eratosthenes, an ancient Greek method for identifying prime numbers.
Here, a high note pulses on every beat, rising in pitch at the perfect squares; while the cello plays a note on every other beat, the clarinet every third beat, and the keyboard plays a rich chord on every fifth beatthat every fifth beat is marked by a chord instead of a single note, is intended as a nod to the golden ratio, which is related to the square root of 5, and has historically been considered a model of aesthetic perfection by some writers.
Notice how the cello marks beats that are multiples of 2, the clarinet marks multiples of 3, and the chords mark multiples of 5. Clearly, the beats on which none of these instruments play must not be multiples of 2, 3 or 5, which is enough to identify them as primes among the integers relevant to the composition, each accompanied by only the high pulse; until the cello, clarinet, and keyboard chords sound together on the 30th beat (as 30 is a multiple of all three primes 2, 3 and 5), resolving before returning to the main theme.
In this tangled interlude, not quite random, our ears experience the formation of the sequence of the primes.
I have read that Leonardo Da Vinci may have hidden a musical composition in his painting The Last Supper, and that Roslyn Chapel in Scotland has musical notation encoded in the masonry. As a variation on this theme, I sought to encode a hint of real mathematics within the musical composition: Eratosthenes' first step toward understanding the primes.