33.8: Inversion (TnI)

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33.8 Inversion (TnI)

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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.

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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.)

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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).

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33.8.1 Identifying TnI for Inversionally-Related Sets

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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].

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�� Table 33.8.1.

First set in order:		2	4	5
Second set backward:	+	5	3	2
n of TnI:		7	7	7

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This confirms the sets are related at T7I.