Multiplication (music)
The mathematical operations of multiplication have several applications to music. Other than its application to the frequency ratios of intervals (e.g., Just intonation, and the twelfth root of two in equal temperament), it has been used in other ways for twelve-tone technique, and musical set theory. Additionally ring modulation is an electrical audio process involving multiplication that has been used for musical effect.
A multiplicative operation is a mapping in which the argument is multiplied (Rahn 1980, 53). Multiplication originated intuitively in interval expansion, including tone row order number rotation, for example in the music of Béla Bartók and Alban Berg (Schuijer 2008, 77–78). Pitch number rotation, Fünferreihe or "five-series" and Siebenerreihe or "seven-series", was first described by Ernst Krenek in Über neue Musik (Krenek 1937; Schuijer 2008, 77–78). Princeton-based theorists, including James K. Randall (1962), Godfrey Winham (1970), and Hubert S. Howe (1967) "were the first to discuss and adopt them, not only with regards [sic] to twelve-tone series" (Schuijer 2008, 81).
When dealing with pitch-class sets, multiplication modulo 12 is a common operation. Dealing with all twelve tones, or a tone row, there are only a few numbers which one may multiply a row by and still end up with a set of twelve distinct tones. Taking the prime or unaltered form as P0, multiplication is indicated by , being the multiplicator:
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