The Further Multitonic Change Probe
“John Coltrane’s “Giant Steps” is a notable example for the use of multitonic changes based on the notes of descending B (B-G-E♭/G-E♭-B) and ascending E♭ (E♭-G-B-E♭) augmented triads. One may notice that all the permutations of an augmented triad can be basically divided into ascending (1-3-♯5, 3-♯5-1 and ♯5-1-3) or descending (♯5-3-1, 3-1-♯5 and 1-♯5-3).
Taking this one step further, consider the other “four-note” symmetrical groups of limited transposition. There are only three types of four-note groupings that are symmetrical groups of limited transposition: (1) 1-♭3-♭5-6 <diminished>, (2) 1-4-♭5-7 <1-♭2-♭5-5 in inversion>, (3) 1-3-♭5-♭7 <1-2-♭5-♭6 in inversion>. The permutations of 1-4-♭5-7 and 1-3-♭5-♭7 can be schematized in Ex-1 and 2: (NOTE: 1-♭3-♭5-6 is omitted for lack of space).”