: How to negatively harmonize a major key into different modes other than Aeolian (minor) mode? (T7i inversion) The T7i inversion transforms a major key into its parallel minor (Aeolian mode). How do I transform a major key

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The TnI designation indicates pitch-class inversion. In short, you invert the given pitch around C, and then transpose it by n semitones. (That may seem backwards; why not transpose it first, since "T" is written on the left? That has to do with left-right and right-left orthography, which we don't really need to get into here.)

So let's say we have a C triad, C E G, and we want it to undergo T7I. Let's invert C around C to get C, then transpose by 7 to get to G. Thus, T7I (C) = G.

Now let's invert E around C to get A♭, then transpose by 7 to get E♭. T7I (E) = E♭.

Following this same process, T7I (G) = C.

Thus the C E G triad, when transformed by T7I, results in C E? G.

Some of you may also know a quicker way: TnI (x) = n - x.

In any event, if we do this with the entire C-major scale, we get:

C = G
D = F
E = E?
F = D
G = C
A = B?
B = A?
C = G

Meaning that T7I of the C-major scale really gets us to G Phrygian, assuming a one-to-one mapping of the pitch classes that are inverted.

And ultimately, any n in TnI will result in a Phrygian collection, assuming this same one-to-one mapping. If n is 1, C major maps to C? Phrygian; if n is 2, it maps to D Phrygian, and so on. The only way to get Aeolian is if you decide that you don't need to have this one-to-one mapping. And at that point, since the resulting collection is the major scale (and its modes), you could just decide to use whatever mode you want.

But perhaps you're thinking of keeping C as tonic throughout. In that case, here's the result according to each n number:

0 = C Phrygian
1 = C? Phrygian (no C, so you cannot keep tonic)
2 = D Phrygian, the same collection as C Dorian
3 = D? Phrygian (no C, so you cannot keep tonic)
4 = E Phrygian, the same collection as C major
5 = F Phrygian, the same collection as C Locrian
6 = F? Phrygian (no C, so you cannot keep tonic)
7 = G Phrygian, the same collection as C Aeolian
8 = G? Phrygian (no C, so you cannot keep tonic)
9 = A Phrygian, the same collection as C Mixolydian
10 = A? Phrygian (no C, so you cannot keep tonic)
11 = B Phrygian, the same collection as C Lydian