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So, I'm reading Allen Forte's Structure of Atonal Music and in the sub-chapter 1.4 Inversionally equivalent pc sets he affirms:

It is important to observe that whereas transposition does not imply prior inversion, inversion always implies subsequent transposition, even if it is the trivial case t = 0

So, I don't get why inversion always implies subsequent transposition. Any explanations?

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This is kind of a weird artifact of Forte’s original formulation of the operation, and is often ignored nowadays. The basic idea for him is that the transposition operation Tn is a single operation wherein all of the notes are increased by n (mod 12). The inversion operation TnI is actual a dual function wherein first the notes are inverted around C (PC 0) and then they are transposed to be the specific notes the composer wants. He could have defined an operation I that cuts out the middle man and gets the same results, and many indeed think of it that way now, but it does lead to a more complex, less intuitive idea of inversion.

Let’s say we were analyzing a composition that makes use of two alternating harmonies: one is the notes D, Eb and F# and the other is F, G#, A. Both of these collections are members of the (014) set class, but we might be interested in the specific relationship between the two. In Forte’s way of thinking, two things have happened. The composer didn’t merely transpose the first set to get the second, the composer inverted as well. For simplicity, Forte defines inversion as mirroring around C, so the I operation of D–Eb–F# creates Bb–A–F# by definition. That’s the right shape of the set for the second chord, but not the right notes, we still have to transpose the notes by 11. So the T11 is happening in addition to I, and we call it the dual operation T11I.

Again, there’s no reason we could just call it I11 and define inversion slightly differently than Forte did, as many contemporary theorists do. For example, earlier editions of Straus’ Introduction to Post-Tonal Theory emphasize the dual TnI operation, but the most recent edition veers much more heavily toward In. Still, Forte has a point that an inversion operation which is completely separated from transposition clarifies other aspects of the nature of the two operations.

  • I was hoping you'd answer! – Richard Aug 20 '18 at 19:58

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