15. The diminshed 7th (Rdim7) and augmented (Raug) chords: As balander-chords and their harmonic role for resolutions. The smalest modulation cycles, of order 2,3 and 4

Each such chord has all its notes as roots, that is why are called balander-chords. 

The role of diminished and augmented chords creating the emotion of anxiety  and resolving to a major (or minor) with two (or one)  notes close by one semitone to two notes of them. 

About the stanrd dominant 7th  resolution e.g. G7->C 

Interval explanation by notes in one semitone distance

This is called the standard Dominant resolution e.g. G7->C  , D7->G, C7->F etc The key to explain of this anxiety-serenity resolutions is that the dominant 7th chord e.g. G7 is the superposition of the Gmajor+Bdiminished, and the diminished chord is by itself a bit sad with anxiety, which is correlated with the Cmajor chord which has serenity because it contains at least one notes of one semitone distance of notes of the C major. These are the notes B, with one semitone distance from the C of Cmajor. The  same is the explanation why the  Cdim7 below resolves to G, because it has the notes D# and F# with one semitone distance from the notes D, G of Gmajor. Sometimes even Cm will be considered resolving to G, because of the D# note at one semitne distance from the D note of G, and the major chord sounds with more serenity than the minor chord!

And the same is the explanation why in the Andalusian Cadenza  Dm, C, Bb A. , the Bb resolves to A, because it has 3 notes one semitone distance apart from corresponding notes of A.

Similarly as the E7 resolves to A so the Em7b5 resolves to A too because it has the 3 notes D, F , G# one semitone apart from the C#, E , A of A chord. And similarly the Edim7 resolves to A as it has the notes D, F one semitone apart from the C#, E  of A

For the same reason if D7 resolves to G, then also Daug resolves to G because Daug has the notes  F#, C# at one semitone distance from the notes G, D of G major chord and Daug sounds in anxiety.


See also 

https://www.youtube.com/watch?v=sDEicex433w
SCALES THAT HAVE DIM7 AND AUG CHORDS

All the 7-notes scales that have steps made from 1,2,3 semitones have more or less chords of the type diminished  and augmented, except of major and minor


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