Six Cardinal Directions

Scored for flute & piano
Duration: 3′ 00″
PDF Score here.

Inspiration…
– The Compass School of Feng Shui “deals with numerology, calculation, orientation, and time… and utilises the Luo Shu, 3 × 3 magic square.
– Six cardinal directions = North, South, East, & West + Zenith & Nadir.
– Euler (magic) squares exist for all orders except 2 and 6.
– “In number theory, an integer… is called a quadratic residue modulo n if it is congruent to a perfect square modulo n” (wikipedia.org/wiki/Quadratic_residue).

Sequence of Number of non zero quadratic residues (mod n) –
first 36 (http://oeis.org/A105612)

0, 1, 1, 1, 2, 3,
3, 2, 3, 5, 5, 3,
6, 7, 5, 3, 8, 7,
9, 5, 7, 11, 11, 5,
10, 13, 10, 7, 14, 11,
15, 6, 11, 17, 11, 7,

18, 19, 13, 8, 20, 15,
21, 11, 11, 23, 23, 7,
21, 21, 17, 13, 26, 21,
17, 11, 19, 29, 29, 11,
30, 31, 15, 11, 20, 23,
33, 17, 23, 23, 35, 11, 36,

If C = 0, 12, 24…, B =11, 23, 35…

C4 -> C7 (flute range): Central point of reference = G#5
Potential isomorphic LIE scale (containign IC 2 & 6) = +1+5, “+2+6”, +3+7, +4+8…
Pitches ascending from G#4: (n=+1+2…)
G#4 (+1) A4 (+2 ) B4 (+3) D5 (+4) F#5 (+5) B5 (+6) F6 (+7) C7 (+8) G#7…
Pitches ascending from G#4: (n=+5+6…)
G#4 (+5) C#5 (+6 ) G5 (+7) D6 (+8) Bb6 (+9) G7…
Combinatory ascending scale:
G#4 (+1) A4 (+2 ) B4 (+2) C#5 (+1) D5 (+4) F#5 (+1) G5 (+4) B5 (+3) D6 (+3) F6 (+5) Bb6 (+2) C7 (+7) G7 (+1 ) G#7…

Pitches descending from G#4: (n=+1+2…)
G#4 (+1) G4 (+2 ) F4 (+3) D4 (+4) Bb3 (+5) F3 (+6) B2 (+7) E2 (+8) G#1 (+9) B0…
Pitches descending from G#4: (n=+5+6…)
G#4 (+5) Eb4 (+6 ) A3 (+7) D3 (+8) F#2 (+9) A1 (+10) B0…
Combinatory descending scale:
B0 G#1 (+1 ) A1 (+7) E2 (+2) F#2 (+5) B2 (+3) D3 (+3) F3 (+4) A3 (+1) Bb3 (+4) D4 (+1) Eb4 (+2) F4 (+2) G4 (+1) G#4

IC 1,7,2,5,3,3,4,1 = Decimal expansion of 20th root of 2.
Corresponds to unique integer sequence: https://oeis.org/A010782
IC 1,2,2,1,4,1,4,3,3
Corresponds to unique integer sequence: https://oeis.org/A286582