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Section 33.8 Inversion (T\(\text{}_{n}\)I)

Inverting a set using T\(\text{}_{n}\)I is a compound operation. The first step is to invert each note below C using C as an axis. For example, E is a major 3rd above C, so E would invert to A♭, a major third below C.

The second step of inversion is to apply the T\(\text{}_{n}\) interval. So, to calculate T\(\text{}_{3}\)I for the note E, one would first invert E to A♭ (this is T\(\text{}_{0}\)I), then transpose the A♭ up 3 semitones to B. (Theorist Joseph Straus simplifies the nomenclature to I\(\text{}_{n}\) instead of T\(\text{}_{n}\)I, but the outcome remains the same.)

Let’s try inverting a pitch–class set, applying T\(\text{}_{7}\)I to [2, 4, 5] (or D, E, and F). Inverting the notes to the opposite side of C using C as an axis yields pitch numbers 10, 8, and 7 (or B♭, A♭, and G), which in ascending order is 7, 8, and 10. Then transposing [7, 8, 10] at T\(\text{}_{7}\) raises each note 7 semitones, resulting in [2, 3, 5] (or D, E♭, and F).

Subsection 33.8.1 Identifying T\(\text{}_{n}\)I for Inversionally–Related Sets

To determine n of T\(\text{}_{n}\)I for two inversionally–related sets, write the second set backward and add the notes of the two sets together. Each sum will equal n. Let’s use our two sets from the previous example above: [2, 4, 5] and [2, 3, 5].

First set in order: 2 4 5
Second set backward: + 5 3 2
n of T\(\text{}_{n}\)I: 7 7 7
Table 33.8.1

This confirms the sets are related at T\(\text{}_{7}\)I.