Could someone please describe how and when a diminished-major chord would be used. How would one lead up to it, resolve it, and are there any rules about inversions with it? An explained example usage in common-practice music would be helpful.
Usage in common practice harmony? Maybe, but the spelling may vary, and, in most cases, the major seventh in the chord is going to be non-harmonic. I've experimented with some examples that make the assumption that the resolving diminished chord or diminished seventh acts as a truncated dominant minor ninth. There are four possible implied roots as a result. These examples aren't exhaustive by any stretch.
To simplify things, I'm using the same lead-in in examples a-d: a minor ninth chord on G becoming a first inversion diminished chord on B, with B in the descant being held as a suspension. B/C♭ could be handled equally well as an appoggiatura or accented passing note.
There is a certain amount of parallelism in these four examples: (a) and (b) run parallel first inversion chords in the top three voices over a contrary bass; (c) implies parallel minor ninths, which is fine, given that the roots are missing and there are no perfect fifths or octaves; and (d) has the top voices moving down in parallel forming a direct fifth. The implied roots are parenthesised in blue, omitted notes are in red, and applicable enharmonic spellings are just placed beside the original note in parentheses. The last case only really applies to example b, where the resulting melodic diminished third would not be unusual as centering motion in the minor mode.
You'll note that the spellings do change, depending on the direction of motion of the individual notes of the chords and the implied tonal area - this chord is an artifact of voice leading. In examples c & d, where the implied root is A♭, it seemed wise to respell B as C♭ moving to B♭♭ (and thence to A♭). Note that C and C♭ thus appear together. Jazz theory doesn't particularly like this, but classical theory tends to consider something like this as combining functions in the voice leading - the English cadence works in very similar fashion.
The last example, (e), uses the heretofore missing implied root, but implied no longer - it considers B in the descant as the root, so that the chord is actually a gapped 4th inversion B minor ninth, with the rather unusual omission of the seventh. An inverted ninth, especially the fourth inversion, would have been considered taboo through most of the common practice period, but Schoenberg broke the ice in Verklärte Nacht by using a major ninth chord in this inversion. This would be very much Late Romantic chromaticism.
Under the circumstances, inversions aren't really material: the chord is either going to act like a diminished chord with a B/C♭ non-harmonic tone, or it is going to act like a gapped inversion of a dominant minor ninth, and if you invert the voices so that the B is in the bass, it is a gapped dominant minor ninth (and you would probably be wiser dropping the chord's fifth than its seventh).
An enharmonic equivalent of the diminished major seventh chord (in second inversion) arises naturally from a first inversion of the diminished seventh chord (with the upper part inverted once more), if you replace the degree that was originally the dominant's seventh again with the dominant-fundamental:
X:1 L:1/2 M: K:C %%score (T1 T2) (B1 B2) V:T1 clef=treble V:T2 clef=treble V:B1 clef=bass V:B2 clef=bass % 1 [V:T1] _A, B, (F G) [V:T2] F, _A, B, B, [V:B1] D, F, _A, _A, [V:B2] B,, D, D, D,
One way this could happen is when the main voice ends a cadence by jumping roots from the dominant to tonic (which more usually the bass does), but the other voices build up some extra tension by forming part of a diminished seventh chord underneath:
X:1 L:1/8 M:2/2 K:C %%score T1 T2 A B V:T1 clef=treble V:T2 clef=treble V:A clef=alto V:B clef=bass % 1 [V:T1] cd cB A2 "First"G2 | "cadence"c4 zB cd | ef ed f2 "Once"g2 | "more"c'4 [V:T2] G2 A2 F2 F2 | E4 G2 AB | cd cB c2 B2 | c4 [V:A] EF ED C2 B,2| C2 CD E2 F2 | G2 A2 F2 "Extra"_A2 | "tension"G4 [V:B] C4 F,2 G,2| zG, A,B, C3 G,| C,4 A,,2 D,2 | E,2 C,2
I can't come up with an example of this by a notable composer, but I think I have encountered this in a few places.
It can be used in passing. But it's seldom notated as it's not held out long. It "resolves" in my example to a
Cdmin7 which which in turn would resolve to a
C. there may be other uses but this is one that I've heard in more reflective jazz pieces.
If you think of a Dim7 as being tension, then this is "tension on tension". Held too long would probably sound "soap-opera"-y.
In a jazz context, I've come across that chord as an upper structure of a more complex harmony. E.g., a G diminished major seventh chord (enharmonically equivalent) as the upper structure of an A13(b9) chord (the fifth is commonly left out):
[A] G Bb C# F# [R] b7 b9 3 13
Another example would be a G diminished major chord over as an upper structure of an Eb7(#9) chord:
[Eb] G Bb Db F# [R] 3 5 b7 #9
I have never come across this chord actually used as a diminished major seventh chord.