September chords

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September chords are essentially tempered dyadic chords tempered by the september comma, 715/714.

There are 20 triads, 76 tetrads, 60 pentads, and 10 hexads as 17-odd-limit essentially tempered chords.

For triads, there are 10 pairs of chords in inverse relationship:

Inversely related pairs of triads
1 – 14/11 – 5/3
(steps 14/11, 17/13, 6/5)		 1 – 6/5 – 11/7
(steps 6/5, 17/13, 14/11)
1 – 12/11 – 10/7
(steps 12/11, 17/13, 7/5)		 1 – 7/5 – 11/6
(steps 7/5, 17/13, 12/11)
1 – 14/13 – 5/3
(steps 14/13, 17/11, 6/5)		 1 – 6/5 – 13/7
(steps 6/5, 17/11, 14/13)
1 – 13/12 – 7/5
(steps 13/12, 22/17, 10/7)		 1 – 10/7 – 24/13
(steps 10/7, 22/17, 13/12)
1 – 13/12 – 14/11
(steps 13/12, 20/17, 11/7)		 1 – 11/7 – 24/13
(steps 11/7, 20/17, 13/12)
1 – 12/11 – 13/7
(steps 12/11, 17/10, 14/13)		 1 – 14/13 – 11/6
(steps 14/13, 17/10, 12/11)
1 – 11/9 – 17/13
(steps 11/9, 15/14, 26/17)		 1 – 15/14 – 17/13
(steps 15/14, 11/9, 26/17)
1 – 13/9 – 17/11
(steps 13/9, 15/14, 22/17)		 1 – 15/14 – 17/11
(steps 15/14, 13/9, 22/17)
1 – 14/11 – 13/9
(steps 14/11, 17/15, 18/13)		 1 – 18/13 – 11/7
(steps 18/13, 17/15, 14/11)
1 – 18/11 – 13/7
(steps 18/11, 17/15, 14/13)		 1 – 14/13 – 11/9
(steps 14/13, 17/15, 18/11)

For tetrads, there are 38 pairs of chords in inverse relationship:

Inversely related pairs of tetrads
1 – 13/12 – 7/5 – 3/2
(steps 13/12, 22/17, 15/14, 4/3)		 1 – 15/14 – 18/13 – 3/2
(steps 15/14, 22/17, 13/12, 4/3)
1 – 7/5 – 3/2 – 11/6
(steps 7/5, 15/14, 11/9, 12/11)		 1 – 15/14 – 3/2 – 18/11
(steps 15/14, 7/5, 12/11, 11/9)
1 – 18/13 – 3/2 – 30/17
(steps 18/13, 13/12, 20/17, 17/15)		 1 – 13/12 – 3/2 – 17/10
(steps 13/12, 18/13, 17/15, 20/17)
1 – 17/15 – 11/9 – 4/3
(steps 17/15, 14/13, 12/11, 3/2)		 1 – 12/11 – 20/17 – 4/3
(steps 12/11, 14/13, 17/15, 3/2)
1 – 14/11 – 14/9 – 5/3
(steps 14/11, 11/9, 15/14, 6/5)		 1 – 6/5 – 9/7 – 11/7
(steps 6/5, 15/14, 11/9, 14/11)
1 – 14/11 – 17/11 – 5/3
(steps 14/11, 17/14, 14/13, 6/5)		 1 – 6/5 – 22/17 – 11/7
(steps 6/5, 14/13, 17/14, 14/11)
1 – 14/11 – 13/9 – 5/3
(steps 14/11, 17/15, 15/13, 6/5)		 1 – 6/5 – 18/13 – 11/7
(steps 6/5, 15/13, 17/15, 14/11)
1 – 14/11 – 15/11 – 5/3
(steps 14/11, 15/14, 11/9, 6/5)		 1 – 6/5 – 22/15 – 11/7
(steps 6/5, 11/9, 15/14, 14/11)
1 – 9/7 – 18/13 – 11/7
(steps 9/7, 14/13, 17/15, 14/11)		 1 – 14/11 – 13/9 – 14/9
(steps 14/11, 17/15, 14/13, 9/7)
1 – 14/11 – 13/9 – 17/11
(steps 14/11, 17/15, 15/14, 22/17)		 1 – 22/17 – 18/13 – 11/7
(steps 22/17, 15/14, 17/15, 14/11)
1 – 17/14 – 10/7 – 17/11
(steps 17/14, 20/17, 13/12, 22/17)		 1 – 17/14 – 11/7 – 17/10
(steps 17/14, 22/17, 13/12, 20/17)
1 – 13/11 – 13/9 – 17/11
(steps 13/11, 11/9, 15/14, 22/17)		 1 – 15/14 – 17/13 – 17/11
(steps 15/14, 11/9, 13/11, 22/17)
1 – 17/15 – 13/9 – 17/11
(steps 17/15, 14/11, 15/14, 22/17)		 1 – 17/15 – 22/15 – 11/7
(steps 17/15, 22/17, 15/14, 14/11)
1 – 17/13 – 17/12 – 11/6
(steps 17/13, 13/12, 22/17, 12/11)		 1 – 13/12 – 17/12 – 17/11
(steps 13/12, 17/13, 12/11, 22/17)
1 – 17/13 – 17/11 – 5/3
(steps 17/13, 13/11, 14/13, 6/5)		 1 – 6/5 – 22/17 – 26/17
(steps 6/5, 14/13, 13/11, 17/13)
1 – 17/13 – 20/13 – 5/3
(steps 17/13, 20/17, 13/12, 6/5)		 1 – 6/5 – 13/10 – 26/17
(steps 6/5, 13/12, 20/17, 17/13)
1 – 17/13 – 10/7 – 5/3
(steps 17/13, 12/11, 7/6, 6/5)		 1 – 6/5 – 7/5 – 26/17
(steps 6/5, 7/6, 12/11, 17/13)
1 – 17/13 – 17/12 – 5/3
(steps 17/13, 13/12, 20/17, 6/5)		 1 – 6/5 – 24/17 – 26/17
(steps 6/5, 20/17, 13/12, 17/13)
1 – 17/13 – 5/3 – 17/9
(steps 17/13, 14/11, 17/15, 18/17)		 1 – 17/13 – 18/13 – 11/7
(steps 17/13, 18/17, 17/15, 14/11)
1 – 17/13 – 5/3 – 11/6
(steps 17/13, 14/11, 11/10, 12/11)		 1 – 17/13 – 10/7 – 11/7
(steps 17/13, 12/11, 11/10, 14/11)
1 – 17/13 – 17/10 – 11/6
(steps 17/13, 13/10, 14/13, 12/11)		 1 – 14/13 – 7/5 – 11/6
(steps 14/13, 13/10, 17/13, 12/11)
1 – 17/13 – 22/13 – 11/6
(steps 17/13, 22/17, 13/12, 12/11)		 1 – 13/12 – 7/5 – 11/6
(steps 13/12, 22/17, 17/13, 12/11)
1 – 15/11 – 17/11 – 5/3
(steps 15/11, 17/15, 14/13, 6/5)		 1 – 6/5 – 22/17 – 22/15
(steps 6/5, 14/13, 17/15, 15/11)
1 – 13/11 – 14/11 – 13/9
(steps 13/11, 14/13, 17/15, 18/13)		 1 – 17/15 – 11/9 – 13/9
(steps 17/15, 14/13, 13/11, 18/13)
1 – 7/5 – 17/10 – 11/6
(steps 7/5, 17/14, 14/13, 12/11)		 1 – 17/14 – 17/10 – 13/7
(steps 17/14, 7/5, 12/11, 14/13)
1 – 17/12 – 17/10 – 11/6
(steps 17/12, 6/5, 14/13, 12/11)		 1 – 6/5 – 17/10 – 13/7
(steps 6/5, 17/12, 12/11, 14/13)
1 – 6/5 – 22/17 – 7/5
(steps 6/5, 14/13, 13/12, 10/7)		 1 – 13/12 – 7/6 – 7/5
(steps 13/12, 14/13, 6/5, 10/7)
1 – 11/10 – 22/17 – 7/5
(steps 11/10, 20/17, 13/12, 10/7)		 1 – 13/12 – 14/11 – 7/5
(steps 13/12, 20/17, 11/10, 10/7)
1 – 13/12 – 13/10 – 7/5
(steps 13/12, 6/5, 14/13, 10/7)		 1 – 14/13 – 22/17 – 7/5
(steps 14/13, 6/5, 13/12, 10/7)
1 – 13/9 – 17/11 – 17/9
(steps 13/9, 15/14, 11/9, 18/17)		 1 – 11/9 – 17/13 – 17/9
(steps 11/9, 15/14, 13/9, 18/17)
1 – 13/9 – 17/11 – 13/7
(steps 13/9, 15/14, 6/5, 14/13)		 1 – 15/14 – 17/11 – 5/3
(steps 15/14, 13/9, 14/13, 6/5)
1 – 13/9 – 17/11 – 5/3
(steps 13/9, 15/14, 14/13, 6/5)		 1 – 14/13 – 15/13 – 5/3
(steps 14/13, 15/14, 13/9, 6/5)
1 – 17/15 – 11/9 – 17/13
(steps 17/15, 14/13, 15/14, 26/17)		 1 – 15/14 – 15/13 – 17/13
(steps 15/14, 14/13, 17/15, 26/17)
1 – 14/13 – 11/9 – 17/13
(steps 14/13, 17/15, 15/,14 26/17)		 1 – 15/14 – 17/14 – 17/13
(steps 15/14, 17/15, 14/13, 26/17)
1 – 17/11 – 5/3 – 17/9
(steps 17/11, 14/13, 17/15, 18/17)		 1 – 17/15 – 11/9 – 17/9
(steps 17/15, 14/13, 17/11, 18/17)
1 – 17/11 – 17/10 – 13/7
(steps 17/11, 11/10, 12/11, 14/13)		 1 – 12/11 – 6/5 – 13/7
(steps 12/11, 11/10, 17/11, 14/13)
1 – 11/7 – 17/10 – 13/7
(steps 11/7, 13/12, 12/11, 14/13)		 1 – 13/12 – 17/10 – 11/6
(steps 13/12, 11/7, 14/13, 12/11)
1 – 11/7 – 17/10 – 11/6
(steps 11/7, 13/12, 14/13, 12/11)		 1 – 13/12 – 17/10 – 13/7
(steps 13/12, 11/7, 12/11, 14/13)

For pentads, there are 30 pairs of chords in inverse relationship:

Inversely related pairs of pentads
1 – 13/12 – 13/10 – 7/5 – 3/2
(steps 13/12, 6/5, 14/13, 15/14, 4/3)		 1 – 15/14 – 15/13 – 18/13 – 3/2
(steps 15/14, 14/13, 6/5, 13/12, 4/3)
1 – 13/12 – 7/6 – 7/5 – 3/2
(steps 13/12, 14/13, 6/5, 15/14, 4/3)		 1 – 15/14 – 9/7 – 18/13 – 3/2
(steps 15/14, 6/5, 14/13, 13/12, 4/3)
1 – 7/6 – 7/5 – 3/2 – 11/6
(steps 7/6, 6/5, 15/14, 11/9, 12/11)		 1 – 15/14 – 9/7 – 3/2 – 18/11
(steps 15/14, 6/5, 7/6, 12/11, 11/9)
1 – 15/13 – 18/13 – 3/2 – 30/17
(steps 15/13, 6/5, 13/12, 20/17, 17/15)		 1 – 13/12 – 13/10 – 3/2 – 17/10
(steps 13/12, 6/5, 15/13, 17/15, 20/17)
1 – 11/10 – 7/5 – 3/2 – 11/6
(steps 11/10, 14/11, 15/14, 11/9, 12/11)		 1 – 15/14 – 15/11 – 3/2 – 18/11
(steps 15/14, 14/11, 11/10, 12/11, 11/9)
1 – 13/12 – 17/12 – 3/2 – 17/10
(steps 13/12, 17/13, 18/17, 17/15, 20/17)		 1 – 18/17 – 18/13 – 3/2 – 30/17
(steps 18/17, 17/13, 13/12, 20/17, 17/15)
1 – 13/12 – 7/5 – 3/2 – 11/6
(steps 13/12, 22/17, 15/14, 11/9, 12/11)		 1 – 15/14 – 18/13 – 3/2 – 18/11
(steps 15/14, 22/17, 13/12, 12/11, 11/9)
1 – 13/12 – 7/5 – 3/2 – 17/10
(steps 13/12, 22/17, 15/14, 17/15, 20/17)		 1 – 15/14 – 18/13 – 3/2 – 30/17
(steps 15/14, 22/17, 13/12, 20/17, 17/15)
1 – 17/12 – 3/2 – 17/10 – 11/6
(steps 17/12, 18/17, 17/15, 14/13, 12/11)		 1 – 18/17 – 3/2 – 18/11 – 30/17
(steps 18/17, 17/12, 12/11, 14/13, 17/15)
1 – 7/5 – 3/2 – 17/10 – 11/6
(steps 7/5, 15/14, 17/15, 14/13, 12/11)		 1 – 15/14 – 3/2 – 18/11 – 30/17
(steps 15/14, 7/5, 12/11, 14/13, 17/15)
1 – 18/13 – 3/2 – 18/11 – 30/17
(steps 18/13, 13/12, 12/11, 14/13, 17/15)		 1 – 13/12 – 3/2 – 17/10 – 11/6
(steps 13/12, 18/13, 17/15, 14/13, 12/11)
1 – 15/11 – 3/2 – 18/11 – 30/17
(steps 15/11, 11/10, 12/11, 14/13, 17/15)		 1 – 11/10 – 3/2 – 17/10 – 11/6
(steps 11/10, 15/11, 17/15, 14/13, 12/11)
1 – 13/12 – 14/11 – 17/11 – 5/3
(steps 13/12, 20/17, 17/14, 14/13, 6/5)		 1 – 14/13 – 17/13 – 20/13 – 5/3
(steps 14/13, 17/14, 20/17, 13/12, 6/5)
1 – 14/13 – 14/11 – 14/9 – 5/3
(steps 14/13, 13/11, 11/9, 15/14, 6/5)		 1 – 15/14 – 17/13 – 17/11 – 5/3
(steps 15/14, 11/9, 13/11, 14/13, 6/5)
1 – 17/14 – 17/13 – 10/7 – 11/7
(steps 17/14, 14/13, 12/11, 11/10, 14/11)		 1 – 11/10 – 6/5 – 22/17 – 11/7
(steps 11/10, 12/11, 14/13, 17/14, 14/11)
1 – 6/5 – 22/17 – 22/15 – 11/7
(steps 6/5, 14/13, 17/15, 15/14, 14/11)		 1 – 15/14 – 17/14 – 17/13 – 11/7
(steps 15/14, 17/15, 14/13, 6/5, 14/11)
1 – 6/5 – 22/17 – 18/13 – 11/7
(steps 6/5, 14/13, 15/14, 17/15, 14/11)		 1 – 17/15 – 17/14 – 17/13 – 11/7
(steps 17/15, 15/14, 14/13, 6/5, 14/11)
1 – 6/5 – 9/7 – 18/13 – 11/7
(steps 6/5, 15/14, 14/13, 17/15, 14/11)		 1 – 17/15 – 11/9 – 17/13 – 11/7
(steps 17/15, 14/13, 15/14, 6/5, 14/11)
1 – 17/15 – 11/9 – 22/15 – 11/7
(steps 17/15, 14/13, 6/5, 15/14, 14/11)		 1 – 15/14 – 9/7 – 18/13 – 11/7
(steps 15/14, 6/5, 14/13, 17/15, 14/11)
1 – 17/15 – 6/5 – 22/15 – 11/7
(steps 17/15, 18/17, 11/9, 15/14, 14/11)		 1 – 15/14 – 17/13 – 18/13 – 11/7
(steps 15/14, 11/9, 18/17, 17/15, 14/11)
1 – 17/14 – 17/13 – 10/7 – 17/11
(steps 17/14, 14/13, 12/11, 13/12, 22/17)		 1 – 13/12 – 13/11 – 14/11 – 17/11
(steps 13/12, 12/11, 14/13, 17/14, 22/17)
1 – 13/11 – 14/11 – 13/9 – 17/11
(steps 13/11, 14/13, 17/15, 15/14, 22/17)		 1 – 15/14 – 17/14 – 17/13 – 17/11
(steps 15/14, 17/15, 14/13, 13/11, 22/17)
1 – 17/13 – 17/11 – 5/3 – 17/9
(steps 17/13, 13/11, 14/13, 17/15, 18/17)		 1 – 13/11 – 17/11 – 18/11 – 13/7
(steps 13/11, 17/13, 18/17, 17/15, 14/13)
1 – 17/13 – 17/12 – 5/3 – 11/6
(steps 17/13, 13/12, 20/17, 11/10, 12/11)		 1 – 13/12 – 17/12 – 17/11 – 17/10
(steps 13/12, 17/13, 12/11, 11/10, 20/17)
1 – 17/13 – 10/7 – 17/11 – 5/3
(steps 17/13, 12/11, 13/12, 14/13, 6/5)		 1 – 14/13 – 7/6 – 14/11 – 5/3
(steps 14/13, 13/12, 12/11, 17/13, 6/5)
1 – 17/13 – 10/7 – 20/13 – 5/3
(steps 17/13, 12/11, 14/13, 13/12, 6/5)		 1 – 13/12 – 7/6 – 14/11 – 5/3
(steps 13/12, 14/13, 12/11, 17/13, 6/5)
1 – 17/13 – 17/12 – 17/11 – 5/3
(steps 17/13, 13/12, 12/11, 14/13, 6/5)		 1 – 14/13 – 20/17 – 14/11 – 5/3
(steps 14/13, 12/11, 13/12, 17/13, 6/5)
1 – 13/12 – 17/12 – 17/11 – 5/3
(steps 13/12, 17/13, 12/11, 14/13, 6/5)		 1 – 14/13 – 20/17 – 20/13 – 5/3
(steps 14/13, 12/11, 17/13, 13/12, 6/5)
1 – 13/9 – 17/11 – 5/3 – 17/9
(steps 13/9, 15/14, 14/13, 17/15, 18/17)		 1 – 17/15 – 11/9 – 17/13 – 17/9
(steps 17/15, 14/13, 15/14, 13/9, 18/17)
1 – 11/10 – 6/5 – 22/17 – 7/5
(steps 11/10, 12/11, 14/13, 13/12, 10/7)		 1 – 13/12 – 7/6 – 14/11 – 7/5
(steps 13/12, 14/13, 12/11, 11/10, 10/7)

For hexads, there are 5 pairs of chords in inverse relationship:

Inversely related pairs of hexads
1 – 13/12 – 13/10 – 7/5 – 3/2 – 17/10
(steps 13/12, 6/5, 14/13, 15/14, 17/15, 20/17)		 1 – 15/14 – 15/13 – 18/13 – 3/2 – 30/17
(steps 15/14, 14/13, 6/5, 13/12, 20/17, 17/15)
1 – 13/12 – 7/6 – 7/5 – 3/2 – 11/6
(steps 13/12, 14/13, 6/5, 15/14, 11/9, 12/11)		 1 – 15/14 – 9/7 – 18/13 – 3/2 – 18/11
(steps 15/14, 6/5, 14/13, 13/12, 12/11, 11/9)
1 – 11/10 – 7/5 – 3/2 – 17/10 – 11/6
(steps 11/10, 14/11, 15/14, 17/15, 14/13, 12/11)		 1 – 15/14 – 15/11 – 3/2 – 18/11 – 30/17
(steps 15/14, 14/11, 11/10, 12/11, 14/13, 17/15)
1 – 13/12 – 17/12 – 3/2 – 17/10 – 11/6
(steps 13/12, 17/13, 18/17, 17/15, 14/13, 12/11)		 1 – 18/17 – 18/13 – 3/2 – 18/11 – 30/17
(steps 18/17, 17/13, 13/12, 12/11, 14/13, 17/15)
1 – 13/12 – 7/5 – 3/2 – 17/10 – 11/6
(steps 13/12, 22/17, 15/14, 17/15, 14/13, 12/11)		 1 – 15/14 – 18/13 – 3/2 – 18/11 – 30/17
(steps 15/14, 22/17, 13/12, 12/11, 14/13, 17/15)

Equal temperaments with september chords include 31, 41, 46, 72, 87, 94, 103, 111, 121, 140, 183, 224 and 270.

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