Why is the vii dim 7 important in minor keys?
Question: Why is the vii dim 7 important in minor keys, like the V7 in majors? I've noticed in my scale reference book they give the tonic and V7 arpeggios for the major scales, but for the minors they give the tonic and the vii dim 7th arpeggios. - K.N.

Answer: What I like about this question is that it has no practical relevance; it stems from pure curiosity. And like many explorations founded on curiosity, there's stuff here to be discovered. I've never really thought about this one myself, but I can explore it along with you.

The resolution by movement of a half-step is an important factor in driving harmonic changes. In the familiar V7 - I of a major key we are used to hearing a diminished 5th or augmented 4th resolve in such a way that two of its pitches move by half step to notes of the following chord:

That kind of movement is also what makes the augmented 6th chords work: the dissonant 6+ expands by half steps at each end to make an octave. The normal resolution of the "French" aug 6 chord contains three half-step moves:

In short, movement by half-step adds to the logic of a harmonic change, and if it's a case of a dissonant interval resolving by half step the feeling is even stronger.

When playing tonal music the minor scale usually appears with its 7th degree raised, to allow a half-step leading tone that can resolve to the tonic. With that 7th raised one can play a V7 chord in minor, just as in major, allowing the strong V7-i chord change. But the V7-i in minor is not quite as strong as V7-I in major, because the 7 in the V7 has to move down by a whole step rather than a half step:

How can the minor key make up for this failing and regain its self-esteem? It can turn to its own advantage something not found in the major scale: the minor key has a half step above its fifth, which means that without changing any notes of the harmonic minor, a diminished seventh chord on the last note of the scale has two half-step resolutions, restoring a lot of that voice-leading power.

The diminished triad built on vii is just a substitute for the V7; it's the same notes as V7, but lacking the root. Adding a diminished seventh to that dim 7 triad makes it work even better, but in a major key one would have to alter another chord tone to get a dim 7. In harmonic minor it's right there, ready to use. So it makes sense in minor that the vii dim 7 chord is a useful substitute for the V7 - especially in a scale practice book whose goal is to teach you the notes of the harmonic minor without introducing pitches outside the scale. They could have given you an ordinary V7 - i, but this has more color and displays more of the nature of the harmonic minor.

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