Multiplication (music) - Pitch Class Multiplication Modulo 12

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, being the multiplicator:

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 by Herbert Eimert, under the terms "Quartverwandlung" (fourth transformation) and "Quintverwandlung" (fifth transformation) (Eimert 1950, 29–33), and has been used by the composers Milton Babbitt (Morris 1997, 238 & 242–43; Winham 1970, 65–66), Robert Morris (Morris 1997, 238–39 & 243), and Charles Wuorinen (Hibbard 1969, 157–58). This operation also accounts for certain harmonic transformations in jazz (Morris 1982, 153–54).

Thus multiplication by the two meaningful operations (5 & 7) may be designated with M5(a) and M7(a) or M and IM (Schuijer 2008, 77–78).

  • M1 = Identity
  • M5 = Cycle of fourths transform
  • M7 = Cycle of fifths transform
  • M11 = Inversion
  • M11M5 = M7
  • M7M5 = M11
  • M5M5 = M1
  • M7M11M5 = M1
  • ...

Read more about this topic:  Multiplication (music)

Famous quotes containing the words pitch and/or class:

    “It is beyond a doubt that during the sixteenth century, and the years immediately preceding and following it, poisoning had been brought to a pitch of perfection which remains unknown to modern chemistry, but which is indisputably proved by history. Italy, the cradle of modern science, was at that time, the inventor and mistress of these secrets, many of which are lost.”
    —Honoré De Balzac (1799–1850)

    “I never feel so conscious of my race as I do when I stand before a class of twenty-five young men and women eager to learn about what it is to be black in America.”
    —Claire Oberon Garcia, African American college professor. As quoted in the Chronicle of Higher Education, p. B3 (July 27, 1994)