The aforementioned axial system covers many kinds of tonality, with more or less “complete closure” of a full system. It also provides hues in gradient shades of modality neither major nor minor but an infinite spectrum between. The addition of the Double Harmonic scales provides all sorts of angular twists to further spice things up. All of this symmetry can leave the listener with a sense of floating in space without any ground under their feet. There is often (though not always) an aesthetic need to provide least a hint of contextual “grounding”, else the listener may become so disoriented that they are unable to reconcile their perceptual bearings sufficiently to “get it” musically.
So … occasional reference to the underlying fundamentals becomes necessary: the three Octatonic scales and/or the two Whole-Tone scales. The trick is to make it so subtle as to be unnoticed. There are distinctly audible fragments and implicit sort of “parentage” of both Octatonic and Whole-Tone subsets easily found within the Acoustic and Double-Harmonic scales.
For sake of explanation, it is convenient to give a more or less arbitrary name to each of the three distinct Octatonics. In most instances that occur in real music, the give-away clue is the presence of a particular note with both sharps and flats appearing frequently and alternately on the same pitch, but never that natural pitch. There is one Octatonic with D-flat and D-sharp, another with A-flat and A-sharp, and one more with G-flat and G-sharp. So it is intuitive enough to use those thumb prints as labels
Ab#8 Db#8 Gb#8
This does not in any way imply a perceptible root tone. The complete ambiguity of the Octatonic scales yields no clear center of symmetry. Paradoxically the reduced dimensionality of only three Octatonics is precisely what can provide the most easily grasped contextual “ground” for listeners to get a frame of reference. It also can provide a strong “jumping off platform” from which to launch into flights through the space of the axial system. The Octatonic parentage from which the Acoustic scales are born may not be obvious, but once you see it then everything begins to fall into place, and once you hear it the musical implications are quite striking.
Consider the example of just one triangular axis of three Acoustic scales: D, Bb, and F# and how they can be derived by collapsing the center of each of the three Octatonics as follows…
Octatonic Db#8 [ A Bb C Db D# E F# G ]
is parent of Acoustic centered on D [ A Bb C D E F# G ]
Octatonic Gb#8 [ F Gb G# A B C D Eb ]
is parent of Acoustic centered on Bb [ F Gb Ab Bb C D Eb ]
Octatonic Ab#8 [ C# D E F G Ab A# B ]
is parent of Acoustic centered on F# [ C# D E F# G# A# B ]
The above forms one triangular axis of this system. Each triangular axis has the three Octatonics as parents of its three Acoustic scales which form one triangle. In other words, each triangular axis is derived from the three octatonics. There are four different locations within one Octatonic that can be collapsed to form an Acoustic scale. Thus we have the four rotations of a triangular axis in this system.
So how can any sense of “grounding” be had amidst all this ambiguous symmetry? Pick any one of the Octatonics. You might not necessarily need to state it explicitly (unless you like that). It can be more effective to subtly establish a “ground” implicitly by creating a musical motif or theme which encompasses all four Acoustics that are derived from the same Octatonic. If done in a balanced way it can initiate a very clear sense of grounding context. Then the triangular axis can easily be rotated in any permutations you like to create a sense of motion, and occasional reference to any one of the related Octatonics can provide a sense of ground if/when needed or desired.
Likewise, there are symmetric relationships between the two Whole-Tone scales and the Acoustic scales. The Whole-Tone scales are so devoid of any harsh dissonance, and so ambiguous without any tonal center, that they may be exploited to sort of “explode” any implicit tonality into an extremely bright wide open space. Used judiciously, the sense of release and brilliance of the sound can be quite extraordinary.