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 “whole/half” scale/chord.
The symmetrical diminished scale is also "invertable" by minor thirds. For example, C symmetrical diminished (referred to as "Sym Dim") scale “inverts” symmetrically to Eb Sym Dim, F# Sym Dim, and A Sym Dim. The remaining notes in this scale i.e. the 9, 11, b13, and Ma7 themselves form a diminished seventh chord (D dim) and inversions that are a whole tone away from the original.
The second of the only two true modes of this symmetrical diminished scale would be "half/whole" (referred to as "Sym Dom") and would be used for the roots of dominant seventh chords with the potential extensions of b9, #9, #11, and 13 (C Sym Dom i.e. Ebdim7/Dbdim7) e.g. C13(b9#9#11) and chords with the same extension potential a minor third away (Eb7, Gb7. and A7).
Note that C Sym Dim: CD EbF GbAb AB C …. is radically different than C Sym Dom: C C#D# EF# GA BbC. C Sym Dom is related to and a mode of C# Sym Dim.
Functions of the diminished seventh chord:
- The leading tone function or dominant function of the diminished seventh is popularly called VIIdim/ii. Here the leading tone (half-tone below the root) diminished chord functions as if it were the third of the dominant of the chord it is leading to. For example in C#dim—Dmi7, the C#dim acts as the third of the dominant of Dmi7 i.e. A7(b9). This is acts like a secondary V7 within a given tonality. The scale used could be C# Sym-Dim but mode VII of D harmonic minor could also be used.
- The passing chord function of the diminished seventh features voice leading that is used in a descending half step motion. The common example (in C major) is Emi7—Ebdim7—Dmi7—G7 etc. Here the Ebdim7 is creates an urgency to resolve by voice leading to Dmi7. If Ebdim7 can be interpreted as D7(b9), it is as if D7(b9) "resolves" to Dmi7 ... a different chord quality on the same root.
- The auxiliary function of the diminished seventh is very useful and has many applications especially when the dim7 chord is assumed to be a part of the four dominant seventh chords it potentially projects: i.e. Cdim/D—/F—/Ab—/B creates respectively: D7(b9), F7(b9), Ab7(b9) and B7(b9).
The auxiliary diminished seventh chord function acts like a "release" from a chord that remains stationary but is treated rhythmically with its root diminished chord i.e. C6—Cdim—C6 or: C6 at rest — Cdim7 — in tension—C6 at rest. This has many uses.
The auxiliary function can be applied to other chord qualities such as Ma7 or C7 etc., and can be used as approach chords to emphasize the arrival of another chord.
In this example Cdim7 is the auxiliary diminished and can derive four 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 which has a milder tension...and release effect.They can be utilized with extension/slash/chord derivatives in passing chords. They can be used to harmonize melodic notes that aren't in the scale of the moment for a very much denser and "active" harmonic sound and still sound effective because of the voice-leading that is available with this diminished chord function.
Many effective choices of extension color and slash chords are inherent in the Sym-Dim (whole/half) and Sym-Dom (half/whole) scales.
For example, in the Cdim scale (whole/half) on 9, 11, b13 and (ma)7 there resides sevenths chords D7, F7, Ab7, and B7. These dominant 7ths chords would use D (half/whole), F (half/whole), Ab (half/whole), and B (half/whole) respectively.
Other chord qualities that are found in this scale also on the same four roots are: mi7(b5), 7(b5), mi7 but they are treated as extensions of those dominant seventh chords: i.e. Fmi7(b5)/D7 = D13(b9#9+11) and also as slash chord components.
The C diminished seventh chord and inversions have extension possibilities which when mixed in with 9, 11, b13 and ma7 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.
These ideas should be worked out and written out in a tune context even if only with slash chord symbols. For example: Benny Golson's Stablemates. Any tune will do if it has some "out-of-the-chord" tones that might need reharmonization. Reharmonziation certainly is not restricted to these extended diminished concept chords and is a whole topic unto itself.