Modulations by diminished seventh
The following steps explain how you work out which diminished seventh to use when modulating from one key to another. The most common way of modulating through a diminished seventh is to treat it as a preparation for the dominant of the new key.

1. take the dominant (chord V) of the key to which you want to modulate (the chord of Ab in the example below)
2. remember that diminished chords originate as chord vii7 so this is the chord you need. Remember: it is vii7 of the dominant of the new key that is neeeded not of the new key itself
3. in order to build this chord up you need to find the root of vii7 of the dominant chord. This will be a semitone below the root V in the new kwt (in the example below this will be a G natural)
4. build up your diminished chord in minor thirds above this root - the first two would be diatonic in the dominant of the new key while the last one would need flattening by a semitone
5. remember that the two tritones should resolve inwards (see diminished sevenths)

In the above example, the perfect cadence is not in root position so the modulation is not strongly confirmed. The music might move to a different key or confirm the present key at this point.

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