I didn't know whether to get into a larger kettle of fish. I am glad to have been shown alternative views. Here goes:
In traditional theory, the dim7 is shown as stacked thirds, as "root position". BDFAb is the easiest since BDF (the viio of C major) is all white keys - the Ab is no longer part of the C major scale or key, but easy to hold on to. As someone said, you have a series of m3's. On the piano, if you play an m3, there are always two unplayed piano keys in between - a visual way of seeing m3 - and it always involves 4 semitones (B, C, C#, D). ...... If you play an "inversion" .... DFAbB (Bdim7/D) ... your notation will show one aug2 - a 2nd, line then space, or space then line .... but your piano will show a smooth continuation of these m3's (sometimes spelled aug2). Your 4 semitones are always there.
Let's take that same BDFAb, which in this case is sort of "built on B", with skipped letter names, which show up as notes on adjacent lines or spaces.
What if I build a dim7 on D? I'll have D F Ab Cb. If you want to see it as being built on the "leading tone" of a key as per bSharp(C)yclist - D is the leading tone of Eb major. Not that it matters here. What if I invert this Ddim7 with the Cb on the bottom? CbDFAb or Ddim7/Cb? Play it on the piano. You get the same notes as our Bdim7 -- only that the B is now spelled Cb.
You can do this with the other two notes of your Bdim7. How about G# B D F? (Your Ab is now G# - same piano key). Its "inversion" BDFG# is the same as our Bdim7.
So - an entirely different way of viewing a Dim7.
You have a series of "stacked m3's" which you can see on the piano - which in notation might show up as an aug2 (always only one) - If you stack one more m3 or "m3" (aug2) you're back at your starting note. It's a kind of "infinite stacking of m3's" and the interval (real) never changes in "inversions".
At the piano, build successive dim3's, observing the piano keys you use. Bdim7, Cdim7, C#dim7 ..... If you go any further than that, you'll be at D dim7, which uses the same piano keys as our Bdim7 does. So in essence, "There are only 3 dim7's - but you have numerous spellings for them." I gave the reason for the spellings in the first part, in the inversions.
This is another, more fluid or amoeba-like way of seeing the dim7. Which spelling to choose - depends on function and context. This moves into the conversation between Sidokar and bSharp(C)yclist (he of the hard-to-spell name
Some other for-nerds cool aspects of the dim7:
- It contains two tritones. (In B dim7, that's BF and DAb)
- Bdim could be seen as the top part of G7 (GBDF
) ..... Bdim7 could be seen as the top part of G7b9 (GBDF
If at the piano you lower any of the four notes of a Bdim7, you end up with a V7. Example: BDFAb - lower B to Bb - you get Bb7. For some you may have to spell the letters enharmonically - G# for Ab, etc. - but at the piano you'll hear a V7 in some inversion. And of course as soon as you have a V7, you can do V7-I, or V7-vi etc. if you have some theory or feel for progressions.
I sort of like adding the amoeba view of the dim7, as will as the traditional stacked third way.