So I'm cruising these music sites trying to PR things and publish a few pertinent facts re: jazz piano theory. One of my first forays was into the area of diminished 7th chord function and I actually got a comment from a deeper music theory guy and I quote:
Even though the author so liberally uses the idea of symmetrical divisions of the octave, it's interesting to know that this kind of symmetry, while clearly audible, is in fact not possible to notate. For example, B-D-F-Ab is a set of pitch-classes for a diminished seventh chord, each separated by a minor 3rd interval. But if you consider the wrap-around interval Ab-B, you've got a augmented 2nd interval. If one were to continue stacking minor thirds to that collection, it would look like B-D-F-Ab-Cb-Ebb-Gbb-... etc. The same issue arises in the so-called symmetrical divisions of the octave by major 3rds. Richard Cohn talks about this in his Maximally Smooth Cycles article, if anyone's interested in some more thoughts on the matter.
Now that's cool actually and I hope I responded graciously especially since I ran down a couple of his leads and found some interesting and even inspiring graphic theory stuff that related to the world of improv piano (i.e. jazz). My response:
Thank you for your comment re: the notation of diminished "7th" pitch classes. Yes, a so-called B diminished 7th chord can't be written as successive minor 3rds, without producing enharmonic descriptions of the same thing i.e. Bdim7, Cbdim7, Dbbbdim7 ect. and this is a written musical fact, a thorny issue in an "equal" interval system to be sure. From an improviser/composer's point of view, especially a pianist's point of view, the beauty of this symmetrical/enharmonic/aural pattern phenomena whether described as stacked augmented 2nds or minor 3rds (or "double diminished" 4ths?) or combinations of the above will still allow, through the symmetrical diminished scale (2 minor tetrachords a tritone apart), all the potentialities for chord voicing, melody, and harmonic function within a given chord progression and voicing texture that the symmetrical diminished scale has to offer. The idea of the article was to discuss the flexibility of this set of pitch classes. It's a difficult enough task as a jazz pianist to obtain these useful diminished scale textures between the hands without being concerned about the infinite possibilities of writing them in a certain intervallic way. However it is worth exploring. There is, no doubt, some benefit that can be gained by studying this notation aspect. Thanks again for your comments which are educational and helpful. I hope that the point I was making can be recognized in the context intended. I will read the article by Richard Cohn with interest.
Here's the link I found relating to this: http://en.wikipedia.org/wiki/Neo-Riemannian_theory
Now in this first page scrolling, down low and behold: an active "torridial" graphic which links major and minor triads augment triad, the diminished 7ths, and the cycle of 5ths all in this spinning/revolving TORUS structure, not unlike the Torus device described in the controversial video: Thrive. In this case it's color coded to make it more accessible. There's another graphic just above which is kind of the prep course for the major—minor—augmented—diminished 7ths—5ths-cycle Torus demonstration. It's really quite brilliant. There's some interesting "transformational theory" (an academic sounding name that codifies little musical moves that we all make) that has piqued my interest too.
So given this Torus graphic I feel confident regarding the diminished 7th numbers game and the exploration of the Symmetrical Diminished Scale (i.e. 3 separate diminished 7ths each containing the 4 rotations that themselves are diminished). On to the augmented chords of 3 equal intervals (ma 3rd) with a total only 4 different augmented triads. So this is starting to get good !!