Multiplication (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.

[3] Multiplication originated intuitively in interval expansion, including tone row order number rotation, for example in the music of Béla Bartók and Alban Berg.

[4] Pitch number rotation, Fünferreihe or "five-series" and Siebenerreihe or "seven-series", was first described by Ernst Krenek in Über neue Musik.

[5][4] Princeton-based theorists, including James K. Randall,[6] Godfrey Winham,[7] and Hubert S. Howe[8] "were the first to discuss and adopt them, not only with regards [sic] to twelve-tone series".

[4] Pierre Boulez[16][dubious – discuss] described an operation he called pitch multiplication, which is somewhat akin [clarification needed] to the Cartesian product of pitch-class sets.

This technique was used most famously in Boulez's 1955 Le Marteau sans maître, as well as in his Third Piano Sonata, Structures II, "Don" and "Tombeau" from Pli selon pli, Éclat (and Éclat/Multiples), Figures—Doubles—Prismes, Domaines, and Cummings ist der Dichter, as well as the withdrawn choral work, Oubli signal lapidé (1952).

[23] The combination of interpolation, infrapolation, and ultrapolation, forming obliquely infra-interpolation, infra-ultrapolation, and infra-inter-ultrapolation, additively sums to what is effectively a second sonority.

A different result is obtained by starting with the "3 perfect fifths stacked", and from these non-beating notes tuning a tempered major third above and below; this is Euler-Fokker genus [333].

[29]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.

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).

[30] 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.