Mirror Form of Multiplication
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 (Eimert 1950, 28–29). With the retrogrades of these transforms and the prime, there are eight permutations.
Furthermore, one can sort of move the mirror at an angle, that is the 'angle' of a fourth or fifth, so that the chromatic row is reflected in both cycles. . . . In this way, one obtains the cycle-of-fourths transform and the cycle-of-fifths transform of the row. (Eimert 1950, 29; trans. Schuijer 2008, 81)Read more about this topic: Multiplication (music)
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