A0-(A3)-A6

EXPANDING intervallic pairs (A0 – A6), 6 x Octave Symmetry:
^{+1 (x2), +2 (x2), +3 (x2), +4 (x2), +5 (x2), +6 (x2), +7 (x2), +8 (x2)}
Ascending:
^{A0 (+1), Bb0 (+1), B0 (+2), C#1 (+2), Eb1 (+3), F#1 (+3), A1 (+4), C#2 (+4),}
^{F2 (+5), Bb2 (+5),Eb3 (+6), A3 (+6), Eb4 (+7), Bb4 (+7), F5 (+8), C#6 (+8), A6}
Descending:
^{A0 (+8), F1 (+8), C#2 (+7), G#2 (+7), Eb3 (+6), A3 (+6), Eb4 (+5), G#4 (+5),}
^{C#5 (+4), F5 (+4), A5 (+3), C6 (+3), Eb6 (+2), F6 (+2), G6 (+1), G#6 (+1), A6}

Complete EXTERNAL Scale:
^{A0, Bb0, B0, C#1, Eb1, F1, F#1, A1, C#2, F2, G#2, Bb2,

Eb3, A3, Eb4,

G#4, Bb4, C#5, F5, A5, C6, C#6, Eb6, F6, G6, G#6, A6}

26 x symmetrical IC’s:
^{1, 1, 2, 2, 2, 1, 3, 4, 4, 3, 2, 5, 6|6, 5, 2, 3, 4, 4, 3, 1, 2, 2, 2, 1, 1}
OEIS A105777 = 4, 4, 3, 1, 2, 2, 2, 1, 1
OEIS A200631 = 2, 5, 6, 6, 5, 2, 3

Outagraphic Scale:
^{C1, D1, E1, G1, G#1, Bb1, B1, C2, D2, Eb2, E2, F#2, G2, A2, B2, C3, C#3, D3, E3, F3, F#3, G3, G#3, Bb3, B3, C4, C#4, D4, E4, F4, F#4, G4, A4, B4, C5, D5, Eb5, E5, F#5, G5, G#5, Bb5, B5, D6, E6, F#6}

45 x symmetrical IC’s:
^{2, 2, 3, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 3, 2, 2}
OEIS A030348 = 2, 2, 3, 1, 2, 1, 1, 2, 1, 1,
OEIS A250007 = 1, 1, 2, 1, 1, 2, 1, 3, 2, 2
OEIS A205600 = 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2

Internal/central expansion from A3:
^{+1 (x2), +2 (x2), +3 (x2), +4 (x2), +5 (x2), +6 (x2), +7 (x2), +8 (x2)}
Ascending:
^{A3 (+1), Bb3 (+1), B3 (+2), C#4 (+2), Eb4 (+3), F#4 (+3), A4 (+4), C#5 (+4), F5 (+5), Bb5 (+5), Eb6 (+6), A6}
Descending:
^{A0 (+6), Eb1 (+5), G#1 (+5), C#2 (+4), F2 (+4), A2 (+3), C3 (+3), Eb3 (+2), F3 (+2), G3 (+1), G#3 (+1), A3}

Complete INTERNAL Scale:
^{A0, Eb1, G#1, C#2, F2, A2, C3, Eb3, F3, G3, G#3, A3, Bb3, B3, C#4, Eb4, F#4, A4, C#5, F5, Bb5, Eb6, A6}
22 x symmetrical IC’s:
^{6, 5, 5, 4, 4, 3, 3, 2, 2, 1, 1 | 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6}

Complete (39 pitch) INTERNAL+ EXTERNAL Scale:
^{A0, Bb0, B0, C#1, Eb1, F1, F#1, G#1, A1, C#2, F2, G#2, A2, Bb2, C3, Eb3, F3, G3, G#3,

A3, Bb3, B3, C#4, Eb4, F#4, G#4, A4, Bb4, C#5, F5, A5, Bb5, C6, C#6, Eb6, F6, G6, G#6, A6}

Musical example that utilises this scale:
Piano realisation (score here )

38 x symmetrical IC’s:
^{1, 1, 2, 2, 2, 1, 2, 1, 4, 4, 3, 1, 1, 2, 3, 2, 2, 1, 1 | 1, 1, 2, 2, 3, 2, 1, 1, 3, 4, 4, 1, 2, 1, 2, 2, 2, 1, 1}
OEIS A265671 = 3, 1, 1, 2, 3, 2, 2, 1, 1,
OEIS A072699 = 1, 1, 2, 2, 3, 2, 1, 1, 3, 4