33.8: Inversion (TnI)
33.8 Inversion (TnI)
Inverting a set using TnI 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 Tn interval. So, to calculate T3I for the note E, one would first invert E to A♭ (this is T0I), then transpose the A♭ up 3 semitones to B. (Theorist Joseph Straus simplifies the nomenclature to In instead of TnI, but the outcome remains the same.)
Let’s try inverting a pitch-class set, applying T7I 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 T7 raises each note 7 semitones, resulting in [2, 3, 5] (or D, E♭, and F).
33.8.1 Identifying TnI for Inversionally-Related Sets
To determine n of TnI 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].
This confirms the sets are related at T7I.