The M5 and M7 transforms are multiplicative transformations of twelve-tone sets. These multiplicative transformations involve the multiplication of each pitch-class number by either 7 (modulo 12) or 5 (modulo 12). The M7 transform maps the chromatic scale onto the circle of fifths. In other words, the succession 0, 1, 2, 3,...11 times 7 produces the succession 0, 7, 2, 9,...5. This has the effect of allowing all even-numbered pitch-classes to remain unaltered while transposing all odd-numbered pitch-classes by a tritone. The M5 transform of the same succession, (0, 1, 2, 3,...11), which maps the chromatic scale onto the circle of fourths, yields 0, 5, 10, 3,...7. The attentive observer will notice that the M5 transform of this succession thus produces the inversion of that produced by the M7 transform.
Further investigation will show that M11 produces the inversion of the original succession, which can be thought of as representing the M1 "transform." These two pairs, [5 and 7] and [1 and 11], include all of the prime numbers between 0 and 11. Furthermore, the M5, M7, M11 (which we don't need, since inversion is already included within the commonly-used twelve-tone operations), and (trivially) M1 ("trivially" because it leaves the starting set unchanged) are the only multiplicative operations which, when performed upon a complete twelve-tone set, produce all twelve pitch-classes. Multiplication of a set by any other number produces redundancies. M6 yields only the same 2 pitch-classes (two notes a tritone apart), M4 (and M8) only the same three (the three notes of the same augmented triad), M3 (and M9) only the same four (the four notes of the same diminished-seventh chord), and M2 (and M10) only the same six (two occurrences of the six notes of the same whole-tone scale). More information can be found in the documentation from Charles Wuorinen's Simple Composition cited below.
Rotation is an already well-known process which is further explained in the documentation from Charles Wuorinen's Simple Composition cited below. Matrix maker 3.0.2 allows not only the traditional form of rotation which is applied to the entire set, but also, in the case of a complete twelve-tone set (as opposed to those of eleven or fewer members) gives the user the option of instead specifying intra-hexachordal rotation.
Stravinsky verticals represent a very special form of rotation given its name by the fact that it represented one of the principal compositional techniques used by Igor Stravinsky during the final (so-called "serial") period of his compositional career. This procedure, which has since been adopted by a wide variety of composers, is explained in the documentation found in Charles Wuorinen's Simple Composition mentioned below.
Wuorinen, Charles. Simple Composition. New York: C. F. Peters Corporation, 1979. Originally published by Longman, Inc. (Copies of the original Longman edition can occasionally be found in stores specializing in used books.)
The M5 and M7 transforms are discussed in pages 98 through 101.
Rotation is discussed on pages 101 (last paragraph) through 108; this discussion, as noted below, also includes the information about a specialized form of rotation called "Stravinsky verticals."
The information concerning Stravinsky verticals is found on pages 105 (last paragraph) through 108 (down to and including Example 74).
Further information about all of these topics is available, much of it in articles in the periodical Perspectives of New Music, and especially in those by Milton Babbitt, Hubert S. Howe, Jr., Godfrey Winham, and Claudio Spies.
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