# (p.251) Appendix B: Non-Commutative Octatonic GIS Structures; More on Simply Transitive Groups

# (p.251) Appendix B: Non-Commutative Octatonic GIS Structures; More on Simply Transitive Groups

Let S be the octatonic family of pitch classes comprising C, C♯, D♯, E, F♯, G, A, and A♯. Eight of the standard “atonal” operations on the twelve pitch-classes transform S into itself; these operations are T_{0}, T_{3}, T_{6}, T_{9}, I^{C♯} _{C}, I^{E} _{C}, I^{G} _{C}, and I^{A♯} _{C}. The eight operations form a group on the twelve pitch-classes and therefore, mapping S into itself, induce a group of corresponding operations on S; we shall call those corresponding operations R0, R3, R6, R9, K, L, M, and N respectively.

It is not hard to verify that the latter group is simply transitive on S: Given members s and t of S, there is a unique OP, among the eight cited operations on S, satisfying OP(s) = t. (If t is in the same diminished-seventh chord as s, OP will be R0, R3, R6, or R9; if t is in the opposite diminished-seventh chord from s, OP will be K, L, M, or N.) We shall call this simply transitive group of operations STRANS1. The operations R0, R3, R6, and R9 may be thought of as “rotations,” to justify the use of the letter R in their names.

We can define another group of operations on S, STRANS2, as follows. R0 and R6 (as above) are members of the group; so are two “queer” operations Q3 and Q9. Q3 rotates each of the diminished-seventh chords within S, but in opposite directions; it maps C to D♯, D♯ to F♯, F♯ to A, A to C, and also C♯ to A♯ (not to E), A♯ to G, G to E, and E to C♯. Q9 is the inverse operation to Q3; it maps C to A,…, D♯ to C, and also C♯ to E,…, and A♯ to C♯.

Besides R0, Q3, R6, and Q9, STRANS2 also contains four “exchanging” operations X1, X2, X4, and X5. X1 exchanges pitch classes within S that lie one semitone apart; it thus maps C to C♯, C♯ to C, D♯ to E, E to D♯, F♯ to G, G to F♯, A to A♯, and A♯ to A. X2 exchanges pitch classes that lie two semitones apart; it maps C to A♯, A♯ to C, C♯ to D♯, F♯ to E, and so on. X4 (p.252) exchanges pitch classes that lie four semitones apart; it maps F♯ to A♯, E to C, G to D♯, and so on. X5 exchanges pitch classes that lie five semitones apart; it maps A to E, D♯ to A♯, F♯ to C♯, and so on. It can be verified that STRANS2 is a group of operations on S, and that it is simply transitive.

Using the method discussed in 7.1.1, we can develop a GIS structure for S in which the members of STRANS1 are exactly the formal transposition operations. We can call this structure GIS1 = (S, IVLS1, intl). In GIS1, then, applying any one of the operations R0, R3, R6, R9, K, L, M, or N to a member s of S amounts formally precisely to “transposing” the given s by a suitable corresponding interval of IVLS1. We must be careful to distinguish the operations K, L, M, and N, which are “GIS1-transpositions” under this formalism, from the operations I^{C♯} _{C} etc. that gave rise to them; I^{C♯} _{C} etc. are inversion-operations in a *different* GIS, a GIS involving a different family of (twelve not eight) objects, a different group of (twelve not eight) formal intervals, and a different function int. Likewise, and more subtly, we must distinguish the octatonic GIS1-transpositions R0, R3, R6, and R9 from the dodecaphonic atonal-GIS-transpositions T_{0}, T_{3}, T_{6}, and T_{9}.

As it turns out, the members of STRANS2 are exactly the interval-preserving operations for GIS1. Every member of STRANS2 commutes with every member of STRANS1. In fact, the members of STRANS2 are precisely those transformations on S that commute with every member of STRANS1.

Using the method of 7.1.1, we can develop another GIS involving the family S, a GIS for which the members of STRANS2 are exactly the formal transposition operations. We can call this structure GIS2 = (S, IVLS2, int2). In this GIS, applying any of the operations R0, Q3, R6, Q9, X1, X2, X4, or X5 to a member s of S amounts precisely to transposing s, formally, by a suitable corresponding interval of GIS2. The interval-preserving operations for GIS2 are exactly the members of STRANS1; those are in fact precisely the transformations on S that commute with every member of STRANS2.

Either GIS1, or GIS2, or both, might lead to results of interest in analyzing a variety of octatonic musics. STRANS2 and STRANS1, which figure as groups of interval-preserving operations in those respective GIS structures, are thereby also likely candidates for CANONical groups of operations in a variety of set-theoretical studies. The STRANS1-forms of a set within S are exactly the dodecaphonically transposed and inverted forms of the set that lie within S. The STRANS2-forms of a set within S are in general a more novel sort of family. Taking (C, E, G), for instance, we apply to it in turn the operations R0, Q3, R6, Q9, X1, X2, X4, and X5; its STRANS2-forms are thereby computed as R0(C, E, G) = (C, E, G), Q3(C, E, G) = (D♯, C♯, E), R6(C, E, G) = (F♯, A♯, C♯), Q9(C, E, G) = (A, G, A♯), X1(C, E, G) = (C♯, D♯, F♯), X2(C, E, G) = (A♯, F♯, A), X4(C, E, G) = (E, C, D♯), and X5(C, E, G) = (G, A, C). If Y is any one of those eight sets, and Y′ is any other one, and f is any one of the eight operations in STRANS1, then the number of (p.253) members of Y whose f-transforms lie within Y is the same as the number of members of Y′ whose f-transforms lie within Y′: INJ(Y, Y)(f) = INJ(Y′, Y′)(f). More generally, if f is any one of the eight operations in STRANS1, and A is any one of the eight operations in STRANS2, and Y and Z are any sets whatsoever within S, then INJ(Y, Z)(f) = INJ(A(Y), A(Z))(f): the number of members of Y whose f-transforms lie within Z is the same as the number of members of A(Y) whose f-transforms lie within A(Z).

As an exercise, the reader may consider a new family of pitch classes, S = (C, C♯, E, F, G♯, A), and develop on the new S two analogous simply transitive groups of operations.

More generally, suppose now that S is *any* family of objects and that STRANS is *any* simply transitive group of operations on S. Consider the family STRANS′ of transformations f on S such that f commutes with every member of the given group STRANS. It can be proved that STRANS′ is itself a simply transitive group of operations on S, and that every transformation A which commutes with every member of STRANS′ is (already) a member of the given group STRANS. When S is considered as a GIS whose formal transpositions are the members of STRANS, then the members of STRANS′ will be the interval-preserving operations. Dually, when S is considered as a GIS whose formal transpositions are the members of STRANS′, then the members of STRANS will be the interval-preserving operations. If STRANS is commutative, then STRANS′ will be precisely STRANS itself.
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