Considering the three different Octatonic Scales, each has a sort of tell-tale characteristic that is easy to spot: the appearance of a particular note as both flat and sharp but not natural. For example, one Octatonic scale has both Db and D# as
[ A Bb C Db D# E F# G ]
So a shorthand “key-signature” for each octatonic can be easily seen as:
[Eb Gb G#] [Bb Db D# F#] [Ab A# C#]
The relationships between the octatonics and the 12 transpositions of the Acoustic Scale form a fascinating lattice of symmetries, which underlie this triangular axial system.
An Acoustic Scale may be derived from an octatonic by collapsing any whole step of an octatonic into the in-between pitch. For example, starting from the octatonic that has a characteristic key signature of [ Bb Db D# F# ] … but altering it by collapsing the D# and Db into the single pitch D between, yields the Acoustic Scale centered on D
Octatonic [ A Bb C Db D# E F# G ]
becomes Acoustic [ A Bb C D E F# G ]
Starting from another octatonic with key-signature [ Eb Gb G# ] but this time collapsing the whole step of [ A B ] into the pitch Bb between (and notating the G# as enharmonic equivalent Ab ), arrives at the Acoustic Scale centered on Bb
Octatonic [ F Gb G# A B C D Eb ]
becomes Acoustic [ F Gb Ab Bb C D Eb F ]
Starting from the remaining octatonic [Ab A# C#] and this time collapsing the whole step [ F G ] to the F# between (and notating Ab with enharmonic G#), gives the Acoustic Scale centered on F#
Octatonic [ C# D E F G Ab A# B ]
becomes Acoustic [ C# D E F# G# A# B]
Taken together the above three Acoustic Scales form one triangular axis with corresponding relationships to each of the three differing octatonics:
Bb-acoustic D-acoustic F#-acoustic
This same pattern may be worked out in four different permutations to yield four triangular axis of Acoustic Scales, so each triangular axis has corresponding relations to the three octatonics. This may seem complex in descriptive terms, but looking at the diagram makes it fairly simple really.
Coming at it from such a theoretical perspective like this, it all seems like just a bunch of geometric patterns, but keep in mind that is not at all how this system was developed. First there was an intuitive exploration, wandering the piano without thinking about it at all, nothing but ears to guide me. This system emerged gradually through several years of composing music that was created entirely only by listening and working with harmonic intervals according to how they sounded. The analytical patterns were later found only afterwards in trying to figure out what had made the music sound the way it did, a distinctive musical character and quality which first and foremost came purely from subconscious. I just listened in my head and wrote what I heard in my imagination without any clue what I were doing. Years later exploring why and how that music “worked” is the real point of this whole system’s emergence.
These sorts of implicit octatonic relationships having been found within all manner of shifts, pivots, modulations, and substitutions between the Acoustic Scales … opened up a vast realm of fresh musical sounds with a whole spectrum of subtle hues.
There is a reason why the sound is distinctively fresh and so extremely flexible. The traditional dominant/subdominant chord relations are not present within this system, so you never get locked into the expectations of traditional functional harmony. In other words, any progression of tonalities can be easily made from this system with results that will sound good, so you can move very freely in any harmonic direction with very fine gradations across a full spectrum of hues and textures. The flip side of that freedom is that in using this system requires careful attention to create a sense of motion that sounds more or less dramatic or smooth according to your compositional intentions, and without any of the rules of traditional functional harmonic practice you have only your intuition and your ears to guide you … but that’s the best part of it.