...when is a perfect fifth interval a diminished 6th and why?
This mixes up distance with interval names.
A perfect fifth
P5 and a diminished 6th
d6 are the same distance - 7 semi-tones.
But the interval name is a combination of an interval number - like 5 - and then a quality - like perfect.
D is six diatonic steps so its number is 6. But when we account for the accidentals
Db the distance gets smaller by two semi-tones and so it is called diminished. It's a diminished sixth.
If it were renamed to
D is a 5 and with 7 semi-tones distance it is perfect, it's a perfect fifth.
Db are enharmonic equivalent spellings.
You could say a
P5 and a
d6 are enharmonically equal intervals, but technically they are not the same interval.
That particular interval -
d6 - is a bit odd. Two intervals non-diatonic intervals that are relatively common and could be mis-identified are:
- augmented second
A2 which is enharmonically a minor third
m3, it is found in the harmonic minor scale between scale steps
- augmented sixth
A6 which is enharmonically a minor seventh
m7, it is found usually in minor keys as one of several augemented sixth chords. In
C minor the basic form of this chord would be
Ab C F# with
F# being the
Those two example can be a good way to become familiar with these enharmonic spellings with practical examples that you will encounter fairly commonly.