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".

Joseph Schillinger described an operation of "polynomial time multiplication" (polynomial refers to any rhythm consisting of more than one duration) corresponding roughly to that of Pitch multiplication above. A theme, reduced to a consistent series of integers representing the quarter, 8th , or 16th note duration of each of the notes of the theme, could be multiplied by itself or the series of another theme to produce a coherent and related variation. Especially, a theme's series could be squared or cubed or taken to higher powers to produce a saturation of related material.

Herbert Eimert described what he called the "eight modes" of the twelve tone series, all mirror forms of one another. The inverse is obtained through a horizontal mirror, the retrograde through a vertical mirror, the retrograde inverse through both a horizontal and a vertical mirror, and the "cycle of fourths transform" or Quartverwandlung and "cycle of fifths transform" or Quintverwandlung obtained through a slanting mirror. With the retrogrades of these transforms and the prime, there are eight permutations. Joseph Schillinger embraced not only contrapuntal inverse, retrograde, and retrograde inverse—operations of matrix multiplication in Euclidean vector space—but also their rhythmic counterparts as well. Thus he could describe a variation of theme using the same pitches in same order, but employing its original time values in retrograde order. He saw the scope of this multiplicatory universe beyond simple reflection, to include transposition and rotation (possibly with projection back to source), as well as dilation which had formerly been limited in use to the time dimension (via augmentation and diminution). Thus he could describe another variation of theme, or even of a basic scale, by multiplying the halfstep counts between each successive pair of notes by some factor, possibly normalizing to the octave via Modulo 12 operation,(

Some Z related chords are connected by M or IM (multiplication by 5 or multiplication by 7), due to identical entries for 1 and 5 on the APIC vector.