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7700 Octatonic.Mus The Octatonic Collection The octatonic collection (or scale), as its name suggests, is an eight-member pitch-class set used as a scale or referential collection in much twentieth-century music. It was called by Olivier Messiaen a mode of limited transposition because it has only three discrete transpositions (by contrast to the diatonic scale, which has twelve). Here are the three forms of the octatonic: M2 m2 M2 m2 M2 m2 m2 M2 m2 OCT0,1 & bœ nœ #œ œ œ bœ ((nœ) œ bœ M2 m2 M2 m2 M2 m2 M2 m2 OCT1,2 & œ œ #œ #œ œ (#œ) #œ œ œ m2 M2 m2 M2 m2 M2 M2 m2 OCT2,3 œ œ (œ) & œ bœ œ #œ #œ œ Each form of the octatonic collection is named after the two lowest-numbered pitch integers that make up one of its component semitones. Note the regular alternation of half steps and whole steps in the collection. The octatonic collection shares certain features of the diatonic scale, but its extreme symmetry differentiates it from the major, minor or pentatonic scales.* Structure of the octatonic collection The octatonic collection can be considered in several different ways, all of which point to its symmetrical construction. In Forte's catalogue it is set class 3-28(0134679T); it is Messiaen's second mode of limited transposition. bœ œ œ œ & bœ bœ nœ #œ œ œ œ œ œ œ #œ #œ œ bœ œ #œ #œ œ œ {9,10} #œ {4,5} {7,8} {10,11} {2,3} {5,6} {8,9} {11,0} {0,1} {3,4} {6,7} {1,2} This is also true of each ic2 (whole tone) dyad, each ic4 (major third) dyad, and each ic5 dyad. It isn't true of ic3 or ic6 dyads, since these intervals are the ones that generate the collection itself (see below). Any given pitch class will occur in two of the three forms of the octatonic collection. Set class 3-10 (0369), the fully diminished seventh chord, divides the octave into four equal parts. Of the three distinct transpositions of 3-10, any two maybe combined to create a form of the octatonic collection. bœ œ & #œ œ OCT œ œ bœ œ bœ #œ 1,2œ œ OCT0,1 OCT2,3 *These orderings, with the lowest pc integers on the far left, are the ones usually given for easy identification and comparison purposes; compositionally, any note in the scale can be used as the focal point, or "do," of the scale. The Octatonic Collection -- p. 2 The octatonic collection also includes interlocking major triads whose roots are the first, third, fifth, and seventh notes of the collection, shown here ordered from C, a similar group of minor triads with the same roots, and a similar group of major/minor (dominant) seventh chords with the same roots: Minor triads on C, E , F , A Major/minor sevenths on C, E , F , A Major triads on C, Eb, F#, A b # b # M2 m2 œ bœ #œ#œnœ#œ œ bœ œ#œ œnœnœ bœ bœbœ #œ#œnœnœ#œ œ & œ œ œbœ œ #œ œ bœ œbœbœ #œ œ œ œ bœ œ #œ OCT0,1 OCT 0,1 OCT0,1 Major/minor triads (set class 4-17 (0347)) can also be built in the same places in the collection. The octatonic collection may also be considered as the union of a pair of symmetrical tetrachords, consisting of either a pair of {+1, +2, +1} tetrachords connected by a whole step or a pair of {+2, +1, +2} tetrachords connected by a semitone. {+1, +2, +1} {+2, +1, +2} {+1, +2, +1} {+2, +1, +2} bœ bœ œ & bœ bœ nœ #œ œ œ bœ bœ nœ #œ œ œ œ {+2} {+1} Appendix: Messiaen's Modes of Limited Transposition Olivier Messiaen (1908-92) made extensive use of scalar collections like the octatonic that have fewer than twelve distinct transpositions. In addition, there is a limited number of possible unique orderings of each mode; by contrast, there are seven unique ways to order the diatonic scale (corresponding to the six "church modes" plus the Locrian mode). Below is his system of "modes of limited transposition," taken from his Techniques of my Musical Language (1944). Note that in Messiaen's system the octatonic scale is the second mode. 1st mode (whole tone scale) 2nd mode (octatonic scale) 2 transpositions, 1 ordering 3 transpositions, 2 orderings œ bœ œ & œ œ #œ #œ #œ bœ bœ nœ #œ œ œ œ œ 5th mode 3rd mode 4th mode 6 transpositions, 3 orderings 4 transpositions, 3 orderings 6 transpositions, 4 orderings bœ bœ nœ œ bœ œ œ œ œ & œ œ bœ nœ #œ œ œ bœ nœ œ #œ œ œ bœ œ #œ œ 6th mode 7th mode 6 transpositions, 4 orderings 6 transpositions, 5 orderings #œ œ œ bœ nœ nœ œ & œ œ œ œ #œ #œ œ bœ nœ bœ œ #œ œ -2-.
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