To be sure, this is just thought-in-progress and by no means a fully developed approach, so I’ll be glad to hear your opinions and suggestions. In case I might have unconsciously reproduced some recent findings from Neo-Riemannian Theory which I was unaware of, I’d appreciate if somebody pointed out a source to me. Also, if you can think of another more convincing 5×5 layout, please let me know.

To this end I will make use of tetrachordal theory. A tetrachord consists of four adjacent diatonic pitches in the range of a perfect or augmented fourth, coming in four possible variants which differ by the existence and position of half-tone steps: Ionian (2 2 1), Dorian (2 1 2), Phrygian (1 2 2), and Lydian (2 2 2: no half-tone step). If we conceive every diatonic mode as a combination of two disjunct tetrachords transposed by a perfect fifth, there are seven combinations making up the seven diatonic modes—for instance, the Aeolian mode consists of a lower Dorian and an upper Phrygian tetrachord, while the Mixolydian mode consists of a lower Ionian and an upper Dorian tetrachord. The eighth combination, though, joins two Lydian tetrachords to a non-diatonic whole-tone scale, provided we allow the upper tetrachord to transpose by a diminished (instead of a perfect) fifth. The resulting scale (2 2 2 0 2 2 2) has six pitches, but makes use of all seven scale degrees, with a diminished second F-sharp / G-flat at the center. In the plagal variant, the scale covers the range of an augmented seventh G-flat / F-sharp.