PLAN FOR A MICROTONAL COMPOSITION IN ℤđ?‘?# ROBERT PETER SCHNEIDER We sketch a plan for a musical piece in a microtonal scale, in which the sequence of prime numbers creates melodic patterns arising from the algebraic structure of the scale, following up on a conversation with Neil Sloane of The On-Line Encyclopedia of Integer Sequences at the ninth Gathering for Gardner conference. We may define a microtonal sequence of tones by dividing the octave into đ?‘š distinct pitches in equal-temperament, using the well-known formula for the đ?‘›th tone đ?‘‡(đ?‘›) in the sequence đ?‘‡(đ?‘›) = đ??š ∙ 2
đ?‘› −1 đ?‘š
,
where đ??š denotes some fundamental pitch, which we choose to set equal to middle C on the piano. For our composition, we divide the octave into đ?‘?# tones, where đ?‘? is prime and đ?‘?# denotes the product of all primes less than or equal to đ?‘?. Then we have the formula for the đ?‘›th tone in this sequence đ?‘› −1
đ?‘‡(đ?‘›) = đ??š ∙ 2 đ?‘? # . From this sequence we define a scale, or subset of tones, to which we will restrict the choice of pitches used in the composition. We choose to include a tone đ?‘‡(đ?‘›â€˛ ) in the scale if đ?‘›â€˛ and đ?‘?# are relatively prime, i.e. gcd đ?‘›â€˛, đ?‘?# = 1, beginning the scale with the fundamental tone đ?‘‡(1); which is to say, we exploit the units in ℤđ?‘?# . There are đ?œ‘(đ?‘?#) such numbers less than or equal to đ?‘?#, by definition of the Euler totient function đ?œ‘, so there are đ?œ‘(đ?‘?#) tones in the scale within a single octave. For example, choosing đ?‘? = 5 will select an eight-tone scale from 5# = 2 ∙ 3 ∙ 5 = 30 distinct tones in the octave, as đ?œ‘ 30 = 8. Obviously, all primes greater than đ?‘? are relatively prime to đ?‘?#; then the prime-numbered tones associated with prime values of đ?‘›â€˛ > đ?‘? are among the members of our musical scale. Our composition is carried out by playing a melody in this scale consisting of đ?‘‡(1) followed by the aforementioned prime-numbered tones, sounded sequentially in a steady rhythm. The performers may choose to restrict the resulting melody to a one-octave range based on aesthetic and practical preferences. We anticipate from our research in the theory of numbers that a recurring pattern will seem to be established initially, but will decay as the composition progresses; eventually degenerating to random-sounding melodies, then to silence. The composition ends upon occurrence of the first “silentâ€? octave, containing no prime-numbered tones. If we extend our conception of music to include tone patterns selected from a predetermined scale according to a definite rule, then the planned composition provides one way to hear what mathematician Marcus Du Sautoy poetically refers to as the “music of the primes.â€?