They are constructed symmetrically of minor 3rds. They are found diatonically in Harmonic minor and Harmonic major scales on VII (the raised 7th), and symmetrically in the "whole/half" diminished scale. This discussion is mainly about the symmetrical diminished scale and chord here.
This scale is "invertable" by minor thirds of course for example: Cdim inverts symmetrically to Ebdim, F#dim and Adim and the other remaining notes i.e. the 9, 11, b13, and Ma7 form ANOTHER dim7 chord and inversions that of course are a whole tone away from the original dim7 (Cdim). The chord used for this scale would be a form of Cdim7. The second of the 2 only modes would be "half/whole" and would be used for the roots of Dominant 7th chords with extensions b9, #9, #11, and 13 (which for in this case, Ddim7— i.e. Ddim7/Cdim7).
Functions of the Diminished 7th chord:
1) Leading tone function or dominant function popularly called VIIdim/ii.... where the leading tone diminished chord functions as if it were the 3rd of the dominant, for example: C#dim—Dmi7..... The C#dim acts or projects, as the 3rd of the dominant of Dmi7 i.e. A7(b9)—Dmi7. This is included as a secondary V7 within a given tonality. The scale used could be C# Sym-Dim but mode VII of D harmonic minor would be a probable first choice.
2) Passing chord function of dim7 are voice-led phenomena that uses 1/2 step voice-led motion to achieve a chromatic root motion. The common example (in C major) is Emi7—Ebdim7—Dmi7—G7 etc. Here the Ebdim7 is resolving by voice leading to Dmi7. If Ebdim7 can be interpreted as D7(b9) [and it can !!].... it is as if D7(b9) "resolves" to Dmi7 ... a slightly different chord quality on the same root !!
3) Auxlliary dim7 function is very useful and has many applications especially when the dim7 chord is assumed to be a part of the 4 dominant 7th chords it represents: i.d. Cdim/D—/F—/Ab—/B creates respectively: D7(b9), F7(b9), Ab7(b9) and B7(b9).
The Auxiliary dim7 CHORD FUNCTION acts like a "release" ( C6)—"tension" (Cdim7)—"release" (C6)....device.
C6 [or CMa7 or C7] — C dim (with extension possibilities) — C6 [or CMa7 or C7]....etc... can be and is used in many ways that one might not think come from Auxiliary dim.
In this example Cdim7 is the Auxiliary dim and can derive 4 dominant chords that could be called Auxiliary Dominants
or C6 (or C7)—F7—C7.....
or C6 etc.—Ab7—C6 etc... and even
C6 etc. — D7 —C6 has a milder tension...and release effect.
This becomes a lot of fun when one starts using some of the possible extension/slash/chord derivatives in passing chords. Why use them? because they help to harmonize melodic notes that aren't in the scale of the moment i.e Cmajor, for a very much denser and "active" harmonic sound.
The beauty of the Sym-Dim (whole —1/2) and Sym-Dom (1/2—whole) scales is the astonishing choices of extension color and slash chords available.
For example IN the Cdim scale (whole-1/2) on 9, 11, b13 and (ma)7 there resides 7th chords D7, F7, Ab7, and B7.....
Other chord qualities that are found in this scale also on the same 4 roots are: mi7(b5), 7(b5), mi7.
These are available as slash/chord components (don't forget inversions) of Cdim itself. Of course the C dim and inversions have extension possibilities which mix in the roots of D dim (F dim, Ab dim and B dim) to form for example: C dim9, C dim11, C dimb13 and popularly: CdimMa7 to name only a few. Ideally you can stack the two diminished chords in the make up of the symmetrical diminished scale to make a powerful if slightly crowded vertical chord as in Ddim7/Cdim7.
I have tried doing this on the fly and it works but to get to any sort of familiarity, these ideas should be written out in a tune context even if only with slash/chord symbols. A good tune— example of where this really come in is Benny Golson's Stablemates.... any tune will do if it has some "out-of-the-chord" tones the might need a reharminization. Reharminziation certainly is not restricted to these extended diminished concept chords and is a whole topic unto itself (endless really). The melody of Stablemates has some notes that are not very close to the harmony but Can be usually subsumed if played in a musical dynamically sensitive manner without any added passing changes. I'll try to get that idea of passing harmony both in the horizontal and vertical sense across in a graphic example the in the next blog—using extended dim 7 harmonies. These harmonies are used mostly as approach chords.
Cheers, everybody !!