Multiplication (music)

The mathematical operations of multiplication have several applications to music. Other than its application to the frequency ratios of intervals (for example, 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. Multiplication originated intuitively in interval expansion, including tone row order number rotation, for example in the music of Béla Bartók and Alban Berg. Pitch number rotation, Fünferreihe or "five-series" and Siebenerreihe or "seven-series", was first described by Ernst Krenek in Über neue Musik. Princeton-based theorists, including James K. Randall, Godfrey Winham, and Hubert S. Howe "were the first to discuss and adopt them, not only with regards to twelve-tone series". 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 Mx, x being the multiplicator: Mx(y) ≡ xy mod 12 The following table lists all possible multiplications of a chromatic twelve-tone row: Note that only M1, M5, M7, and M11 give a one-to-one mapping (a complete set of 12 unique tones). This is because each of these numbers is relatively prime to 12. Also interesting is that the chromatic scale is mapped to the circle of fourths with M5, or fifths with M7, and more generally under M7 all even numbers stay the same while odd numbers are transposed by a tritone. This kind of multiplication is frequently combined with a transposition operation. It was first described in print by Herbert Eimert, under the terms "Quartverwandlung" (fourth transformation) and "Quintverwandlung" (fifth transformation),, and has been used by the composers Milton Babbitt, Robert Morris, and Charles Wuorinen.