Multiplication (music)

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This article is about multiplication in music; for multiplication in mathematics see multiplication.

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.

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[edit] Pitch class multiplication modulo 12

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:

  • M_x(y) \equiv xy \pmod{12}

The following table lists all possible multiplications of a chromatic twelve-tone row:

M M × (0,1,2,3,4,5,6,7,8,9,10,11) mod 12
0 0 0 0 0 0 0 0 0 0 0 0 0
1 0 1 2 3 4 5 6 7 8 9 10 11
2 0 2 4 6 8 10 0 2 4 6 8 10
3 0 3 6 9 0 3 6 9 0 3 6 9
4 0 4 8 0 4 8 0 4 8 0 4 8
5 0 5 10 3 8 1 6 11 4 9 2 7
6 0 6 0 6 0 6 0 6 0 6 0 6
7 0 7 2 9 4 11 6 1 8 3 10 5
8 0 8 4 0 8 4 0 8 4 0 8 4
9 0 9 6 3 0 9 6 3 0 9 6 3
10 0 10 8 6 4 2 0 10 8 6 4 2
11 0 11 10 9 8 7 6 5 4 3 2 1

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 in Eimert 1950, and has been used by the composers Milton Babbitt, Robert Morris, and Charles Wuorinen.

[edit] Pitch multiplication

Pierre Boulez (1971) described an operation he called pitch multiplication, which is somewhat akin to the Cartesian product of pitch class sets. Given two sets, the result of pitch multiplication will be the set of sums (modulo 12) of all possible pairings of elements between the original two sets. Its definition:

X \times Y = \{ (x+y)\bmod 12 | x\in X, y\in Y\}

For example, if multiplying a C major chord {0,4,7} with a dyad containing C,D {0,2}, the result is:

\{ 0,4,7 \} \times \{ 0,2 \} = \{ 0,2,4,6,7,9 \}

In this example, a set of 3 pitches multiplied with a set of 2 pitches gives a new set of 3 × 2 pitches. Given the limited space of modulo 12 arithmetic, when using this procedure very often duplicate tones are produced, which are generally omitted. This technique was used most famously in Boulez's 1955 masterpiece Le marteau sans maître, as well as in his Third Piano Sonata, Pli selon pli, Eclat (and Eclat multiples), Figures-Doubles-Prisms, Domaines, and Cummings ist der Dichter, as well as the withdrawn choral work, Oubli signal lapidé (1952) (Koblyakov 1990; Heinemann 1993 and 1998).

[edit] References

  • Boulez, Pierre. 1971. Boulez on Music Today. Translated by Susan Bradshaw and Richard Rodney Bennett. Cambridge, Mass.: Harvard University Press. ISBN 0-674-08006-8.
  • Eimert, Herbert. 1950. Lehrbuch der Zwöfltontechnik. Wiesbaden: Breitkopf & Härtel.
  • Heinemann, Stephen. 1993. "Pitch-Class Set Multiplication in Boulez's Le Marteau sans maître. D.M.A. diss., University of Washington.
  • Heinemann, Stephen. 1998. "Pitch-Class Set Multiplication in Theory and Practice." Music Theory Spectrum 20, no. 1 (Spring): 72-96.
  • Koblyakov, Lev . 1990. Pierre Boulez: A World of Harmony. Chur: Harwood Academic Publishers. ISBN 3-7186-0422-1.

[edit] Further reading

  • Howe, Hubert S. 1965. “Some Combinational Properties of Pitch Structures.” Perspectives of New Music 4, no. 1 (Fall-Winter): 45–61.
  • Rahn, John. 1980. Basic Atonal Theory. Longman Music Series. New York and London: Longman. Reprinted, New York: Schirmer Books; London: Collier Macmillan, 1987.
  • Winham, Godfrey.1970, “Composition with Arrays”. Perspectives of New Music 9, no. 1(Fall-Winter): 43–67.
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