3. enharmonization of double diminished triad and double diminished seventh chord, and of the hard diminished seventh chord

The double diminished seventh chord is enharmonically equal to the dominant seventh chord: the notation (and thus, the inversion) varies, but the sound is identical.^ann This means that the context determines which chord is used, and that the context may change over this chord: it can be used as a pivot chord in an enharmonic modulation . For example, in example 17 below the chord at A can be understood as #IVdv6, Italian augmented, in B minor and B major; the chord at B on the other hand can be seen as V7 in C minor or C major.

Example 17 also shows how the enharmonization actually works: the augmented sixth is exchanged for a minor seventh, so that another note in the chord takes over the role of leading tone: at A the leading tone is: E# (leading to F#), at B the leading tone is: B natural (leading to C). We could therefore speak of 'leading tone change' or 'leading tone exchange.^ann This exchange is at the tritone.

The hard diminished seventh chord allows enharmonization as well: As the chord is symmetrical (what is easy visable when we add the octave of the root, see example 17 C and D). The 4/3 position of a hard diminished seventh chord can be understood as a root position (and logically: the 6/5-position as a 2-position) of another hard diminished seventh chord. In other words: Partial enharmonization of a hard diminished seventh chord results, unlike enharmonization of double diminished, in the same type of chord, in another inversion.^ann The context determines which inversion is at stake, and the context - as with double diminished - may change over the hard diminished seventh chord, as it can be used as pivot chord in an enharmonic modulation. For example, the chord at C in example 17 can be understood as IIhv4/3, French augmented, in B minor or B major; the chord at D as IIhv7 in C minor or C major. We observe again (like in double diminished/dominant seventh) that the augmented sixth is exchanged for a minor seventh (and vice versa!), so that another tone takes over the role of leading tone : at C in example 17, the leading tone is: E# (leading to F#); at D the leading tone is: B natural B (leading to C). We could therefore again speak of 'leading tone change' or 'leading tone exchange'. And the exchange is again at the tritone.. 

Obviously, enharmonization of one of these chords can be used to modulate; when we use one of the chords in this way, we make an enharmonic modulation.^ann
When we assume the very common enharmonization dominant seventh chord --> double diminished seventh chord the two keys involved often are a minor second apart, as double diminished often is used on #IV, and then normally resolves to V: 

In the sequential fragment in example 18 above we find two modulations. At first wemodulte from F major to E minor: In measure 4 the seventh of V7 of F major is enharmonized (Bb turns into A #), thus changing the chord to double diminished, which can be heard as German augmented in E minor. As a rule, composers use single notation in such situations - we could notate A# in the example instead of Bb. In the sequence, in measure 12, we find a modulation from E major to Eb minor. In this modulation three notes of V7 should be re-interpreted. But this difference with measure 4 is not really important: We could re-notate the chord as B - D# - F# - G .We would then modulate to D # minor instead of Eb minor - but that's of course the same key. The choice of Eb minor has a practical reason: It is easier to read than D# minor (no double sharps or double flats).

Please realize that such enharmonizations of chords always mean: partial enharmonization: At least one note of the chord remains unchanged. Complete enharmonization leaves the chord actually unchanged:We rename all notes, but the chord stays unchanged. For example, when we rewrite the chord Db - F - Ab - Cb , V7 in the key of Gb , it may turn into C# - E# - G# - B , V7 in the key of F# - but that is purely a notation issue. Which is not to say it does not exist: Sometimes it is handy to exchange sharps for flats somewhere in a piece...

As mentioned above: Enharmonisazion of a hard diminished seventh chord creates another hard diminished seventh chord. And although it can occur on V, FOOTNOTE PAR 5 hard diminished on II is much more common. If we imagine an enharmonic modulation with hard diminished in which this chord stands on II in both keys (albeit in different positions), then the two keys involved are in the distance of a diminished fifth:

But it is not absolutely necessary to really modulate.There is at least one possibilty to stay within the key, even when enharmonization is applied:  If we assume B minor, or possibly B major as the key of example 17, then the chords at B, F and H, and eventually at D as well can be understood as secondary dominant for the Neapolitan (bII). This way of using enharmonization within the key is used in practice mostly with the double diminished and dominant seventh chords: German augmented, # IVdv6/5,  is the enharmonic equivalent of (V7) ---> bII.  In both examples below the double diminished seventh chord F# -Ab - C - Eb is enharmonized to the dominant seventh chord Ab - C - Eb - Gb. The latter is V7 in Db, or - in the context of the key C minor: (V7) --> Neapolitan. We clearly see "leading tone exchange": In the double diminished chord F# is leading tone for G, in the dominant seventh chord C is leading tone for Db. The distance between these leading tones is a tritone, and indeed the resolution chords (G resp. Db) are tritone apart. Example 20b forms a somewhat extended variant of example 20a: 

The hard diminished seventh chord allows to play exactly the same trick: In that case IIhv7 is changed to (Vhv7) --> bII - but probably this occurs much less frequently. IN the following example IIhv6/5 in D minor: G # - Bb - D - E  is enharmonized to (Vhv2) --> bII6: Ab - Bb - D - Fb. Moments later, the same happens 'vice versa' (V7hv7) --> bII turns into IIhv4/3 (French augmented). In IIhv7 G# acts as leading tone for A, whereas  in (Vhv4/3) D is the leading tone for Eb - see measures 6 and 7 of the example. Like in the previous example, there is no modulation:

 

 

b.