From Chapter 40 of the Gauldin:
Enharmonic spellings as part of modulations
New keys with enharmonic spellings. We’ve already seen this some with modulations to bII and bVI. If you start in Ab major and modulate to bII, it’s easier to read as A major rather than Bbb major.
Pivot chords with enharmonic spellings. In C major, bVI is an Ab major triad (Ab – C – Eb). Changing the spelling of the triad to G# (G# – B# – D#) would give you V of C# major or minor.
Enharmonic spellings of fully-diminished seventh chords. Because fully-diminished seventh chords divide the octave equally into minor thirds, any note of the chord can serve as the root. This leads to an enharmonic spelling of the chord and a new target resolution. c# – e – g – Bb would resolve to D, but if respelled e – g – Bb – Db then the chord would resolve to F. (Note the minor third difference.) You usually find these enharmonic spellings creating a vii°7/V in the new key, but this is not always the case.
V7 enharmonically spelled as a Ger+6 (Ger65). We already know that a Ger+6 chord sounds the same as a V7, with the minor seventh spelled as an augmented sixth. This relationship allows for either chord to be enharmonically spelled and resolved accordingly. In C major, bVI is Ab – C – Eb – F#. If the F# is changed to Gb, the result is an Ab – C – Eb – Gb chord, or V7 in Db. Spelling a Ger+6 in the original key as a V7 will lead to a modulation one half step up. Changing V7 in the key to a Ger+6 takes you to a key one half step down. Also possible are enharmonic changes of secondary dominants to Ger+6 chords. The resulting modulation is still one half step down from where the secondary dominant chord would have resolved. For example, V7/IV in major changed to a Ger+6 will lead to a modulation to iii. (or more likely, III)
Altering fully diminished seventh chords to make dominant seventh chords. Lowering the root of a fully diminished seventh chord by chromatic half step will create a dominant seventh chord. Example: the fully diminished chord C# – E – G -Bb becomes the dominant seventh chord C – E – G – Bb (C# -> C) when its root is lowered. Instead of resolving to D, the chord now resolves to F. Since we know that fully diminished seventh chords can be enharmonically spelled with any chord tone serving as the root, then we can lower any chord tone by chromatic half step and end up with a collection of pitches that can be enharmonically spelled as a V7.
Modulation by strict harmonic sequence.
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