Xenharmonic Wiki

Worcester chords

Worcester chords are essentially tempered chords tempered by 576/575, the worcester comma.

There are 10 triads, 39 tetrads, 65 pentads, 45 hexads, 15 heptads and 2 octads as 2.3.5.23 subgroup 23-odd-limit essentially tempered chords.

For triads, there are two palindromic chords and four pairs of chords in inverse relationship:

Palindromic triads
1 – 5/4 – 36/23 (steps 5/4, 5/4, 23/18)
1 – 6/5 – 23/16 (steps 6/5, 6/5, 32/23)
Inversely related pairs of triads
1 – 5/4 – 32/23 (steps 5/4, 10/9, 23/16) 1 – 10/9 – 32/23 (steps 10/9, 5/4, 23/16)
1 – 6/5 – 23/18 (steps 6/5, 16/15, 36/23) 1 – 16/15 – 23/18 (steps 16/15, 6/5, 36/23)
1 – 6/5 – 5/4 (steps 6/5, 24/23, 8/5) 1 – 24/23 – 5/4 (steps 24/23, 6/5, 8/5)
1 – 16/15 – 10/9 (steps 16/15, 24/23, 9/5) 1 – 24/23 – 10/9 (steps 24/23, 16/15, 9/5)

For tetrads, there are seven palindromic chords and sixteen pairs of chords in inverse relationship:

Palindromic tetrads
1 – 23/20 – 23/16 – 8/5 (steps 23/20, 5/4, 10/9, 5/4)
1 – 6/5 – 23/18 – 23/15 (steps 6/5, 16/15, 6/5, 30/23)
1 – 6/5 – 5/4 – 3/2 (steps 6/5, 24/23, 6/5, 4/3)
1 – 10/9 – 5/4 – 32/23 (steps 10/9, 9/8, 10/9, 23/16)
1 – 24/23 – 5/4 – 30/23 (steps 24/23, 6/5, 24/23, 23/15)
1 – 16/15 – 6/5 – 23/18 (steps 16/15, 9/8, 16/15, 36/23)
1 – 24/23 – 6/5 – 5/4 (steps 24/23, 23/20, 24/23, 8/5)
Inversely related pairs of tetrads
1 – 5/4 – 3/2 – 9/5
(steps 5/4, 6/5, 6/5, 10/9)		 1 – 6/5 – 3/2 – 5/3
(steps 6/5, 5/4, 10/9, 6/5)
1 – 5/4 – 3/2 – 8/5
(steps 5/4, 6/5, 16/15, 5/4)		 1 – 6/5 – 3/2 – 15/8
(steps 6/5, 5/4, 5/4, 16/15)
1 – 5/4 – 3/2 – 36/23
(steps 5/4, 6/5, 24/23, 23/18)		 1 – 6/5 – 3/2 – 23/12
(steps 6/5, 5/4, 23/18, 24/23)
1 – 23/16 – 3/2 – 8/5
(steps 23/16, 24/23, 16/15, 5/4)		 1 – 24/23 – 3/2 – 15/8
(steps 24/23, 23/16, 5/4, 16/15)
1 – 6/5 – 23/16 – 3/2
(steps 6/5, 6/5, 24/23, 4/3)		 1 – 24/23 – 5/4 – 3/2
(steps 24/23, 6/5, 6/5, 4/3)
1 – 5/4 – 23/16 – 8/5
(steps 5/4, 23/20, 10/9, 5/4)		 1 – 5/4 – 32/23 – 8/5
(steps 5/4, 10/9, 23/20, 5/4)
1 – 5/4 – 32/23 – 36/23
(steps 5/4, 10/9, 9/8, 23/18)		 1 – 9/8 – 5/4 – 36/23
(steps 9/8, 10/9, 5/4, 23/18)
1 – 5/4 – 30/23 – 36/23
(steps 5/4, 24/23, 6/5, 23/18)		 1 – 6/5 – 5/4 – 36/23
(steps 6/5, 24/23, 5/4, 23/18)
1 – 6/5 – 23/16 – 23/15
(steps 6/5, 6/5, 16/15, 30/23)		 1 – 16/15 – 23/18 – 23/15
(steps 16/15, 6/5, 6/5, 30/23)
1 – 6/5 – 23/18 – 23/16
(steps 6/5, 16/15, 9/8, 32/23)		 1 – 9/8 – 6/5 – 23/16
(steps 9/8, 16/15, 6/5, 32/23)
1 – 6/5 – 5/4 – 23/16
(steps 6/5, 24/23, 23/20, 32/23)		 1 – 23/20 – 6/5 – 23/16
(steps 23/20, 24/23, 6/5, 32/23)
1 – 5/4 – 30/23 – 32/23
(steps 5/4, 24/23, 16/15, 23/16)		 1 – 16/15 – 10/9 – 32/23
(steps 16/15, 24/23, 5/4, 23/16)
1 – 6/5 – 23/18 – 4/3
(steps 6/5, 16/15, 24/23, 3/2)		 1 – 24/23 – 10/9 – 4/3
(steps 24/23, 16/15, 6/5, 3/2)
1 – 6/5 – 5/4 – 4/3
(steps 6/5, 24/23, 16/15, 3/2)		 1 – 16/15 – 10/9 – 4/3
(steps 16/15, 24/23, 6/5, 3/2)
1 – 23/20 – 6/5 – 23/18
(steps 23/20, 24/23, 16/15, 36/23)		 1 – 16/15 – 10/9 – 23/18
(steps 16/15, 24/23, 23/20, 36/23)
1 – 9/8 – 6/5 – 5/4
(steps 9/8, 16/15, 24/23, 8/5)		 1 – 24/23 – 10/9 – 5/4
(steps 24/23, 16/15, 9/8, 8/5)

For pentads, there are three palindromic chords and 31 pairs of chords in inverse relationship:

Palindromic pentads
1 – 6/5 – 4/3 – 3/2 – 5/3 (steps 6/5, 10/9, 9/8, 10/9, 6/5)
1 – 5/4 – 4/3 – 3/2 – 8/5 (steps 5/4, 16/15, 9/8, 16/15, 5/4)
1 – 24/23 – 5/4 – 3/2 – 36/23 (steps 24/23, 6/5, 6/5, 24/23, 23/18)
Inversely related pairs of pentads
1 – 23/16 – 3/2 – 9/5 – 15/8
(steps 23/16, 24/23, 6/5, 24/23, 16/15)		 1 – 24/23 – 3/2 – 8/5 – 5/3
(steps 24/23, 23/16, 16/15, 24/23, 6/5)
1 – 23/16 – 3/2 – 8/5 – 23/12
(steps 23/16, 24/23, 16/15, 6/5, 24/23)		 1 – 24/23 – 3/2 – 36/23 – 15/8
(steps 24/23, 23/16, 24/23, 6/5, 16/15)
1 – 23/16 – 3/2 – 8/5 – 9/5
(steps 23/16, 24/23, 16/15, 9/8, 10/9)		 1 – 24/23 – 3/2 – 5/3 – 15/8
(steps 24/23, 23/16, 10/9, 9/8, 16/15)
1 – 30/23 – 3/2 – 36/23 – 15/8
(steps 30/23, 23/20, 24/23, 6/5, 16/15)		 1 – 23/20 – 3/2 – 8/5 – 23/12
(steps 23/20, 30/23, 16/15, 6/5, 24/23)
1 – 30/23 – 3/2 – 36/23 – 5/3
(steps 30/23, 23/20, 24/23, 16/15, 6/5)		 1 – 23/20 – 3/2 – 9/5 – 23/12
(steps 23/20, 30/23, 6/5, 16/15, 24/23)
1 – 5/4 – 3/2 – 9/5 – 23/12
(steps 5/4, 6/5, 6/5, 16/15, 24/23)		 1 – 6/5 – 3/2 – 36/23 – 5/3
(steps 6/5, 5/4, 24/23, 16/15, 6/5)
1 – 5/4 – 3/2 – 9/5 – 15/8
(steps 5/4, 6/5, 6/5, 24/23, 16/15)		 1 – 6/5 – 3/2 – 8/5 – 5/3
(steps 6/5, 5/4, 16/15, 24/23, 6/5)
1 – 5/4 – 3/2 – 8/5 – 23/12
(steps 5/4, 6/5, 16/15, 6/5, 24/23)		 1 – 6/5 – 3/2 – 36/23 – 15/8
(steps 6/5, 5/4, 24/23, 6/5, 16/15)
1 – 5/4 – 3/2 – 8/5 – 9/5
(steps 5/4, 6/5, 16/15, 9/8, 10/9)		 1 – 6/5 – 3/2 – 5/3 – 15/8
(steps 6/5, 5/4, 10/9, 9/8, 16/15)
1 – 5/4 – 3/2 – 8/5 – 5/3
(steps 5/4, 6/5, 16/15, 24/23, 6/5)		 1 – 5/4 – 3/2 – 36/23 – 5/3
(steps 5/4, 6/5, 24/23, 16/15, 6/5)
1 – 5/4 – 3/2 – 36/23 – 15/8
(steps 5/4, 6/5, 24/23, 6/5, 16/15)		 1 – 6/5 – 3/2 – 8/5 – 23/12
(steps 6/5, 5/4, 16/15, 6/5, 24/23)
1 – 5/4 – 3/2 – 36/23 – 9/5
(steps 5/4, 6/5, 24/23, 23/20, 10/9)		 1 – 6/5 – 3/2 – 5/3 – 23/12
(steps 6/5, 5/4, 10/9, 23/20, 24/23)
1 – 5/4 – 23/16 – 3/2 – 9/5
(steps 5/4, 23/20, 24/23, 6/5, 10/9)		 1 – 24/23 – 6/5 – 3/2 – 5/3
(steps 24/23, 23/20, 5/4, 10/9, 6/5)
1 – 5/4 – 23/16 – 3/2 – 8/5
(steps 5/4, 23/20, 24/23, 16/15, 5/4)		 1 – 24/23 – 6/5 – 3/2 – 15/8
(steps 24/23, 23/20, 5/4, 5/4, 16/15)
1 – 5/4 – 23/16 – 3/2 – 36/23
(steps 5/4, 23/20, 24/23, 24/23, 23/18)		 1 – 24/23 – 6/5 – 3/2 – 23/12
(steps 24/23, 23/20, 5/4, 23/18, 24/23)
1 – 6/5 – 23/16 – 3/2 – 15/8
(steps 6/5, 6/5, 24/23, 5/4, 16/15)		 1 – 24/23 – 5/4 – 3/2 – 8/5
(steps 24/23, 6/5, 6/5, 16/15, 5/4)
1 – 6/5 – 4/3 – 3/2 – 23/12
(steps 6/5, 10/9, 9/8, 23/18, 24/23)		 1 – 9/8 – 5/4 – 3/2 – 36/23
(steps 9/8, 10/9, 6/5, 24/23, 23/18)
1 – 6/5 – 5/4 – 3/2 – 23/12
(steps 6/5, 24/23, 6/5, 23/18, 24/23)		 1 – 6/5 – 5/4 – 3/2 – 36/23
(steps 6/5, 24/23, 6/5, 24/23, 23/18)
1 – 6/5 – 5/4 – 3/2 – 9/5
(steps 6/5, 24/23, 6/5, 6/5, 10/9)		 1 – 6/5 – 5/4 – 3/2 – 5/3
(steps 6/5, 24/23, 6/5, 10/9, 6/5)
1 – 23/20 – 23/16 – 3/2 – 9/5
(steps 23/20, 5/4, 24/23, 6/5, 10/9)		 1 – 24/23 – 30/23 – 3/2 – 5/3
(steps 24/23, 5/4, 23/20, 10/9, 6/5)
1 – 23/20 – 23/16 – 3/2 – 8/5
(steps 23/20, 5/4, 24/23, 16/15, 5/4)		 1 – 24/23 – 30/23 – 3/2 – 15/8
(steps 24/23, 5/4, 23/20, 5/4, 16/15)
1 – 6/5 – 5/4 – 23/16 – 3/2
(steps 6/5, 24/23, 23/20, 24/23, 4/3)		 1 – 24/23 – 6/5 – 5/4 – 3/2
(steps 24/23, 23/20, 24/23, 6/5, 4/3)
1 – 6/5 – 5/4 – 4/3 – 3/2
(steps 6/5, 24/23, 16/15, 9/8, 4/3)		 1 – 9/8 – 6/5 – 5/4 – 3/2
(steps 9/8, 16/15, 24/23, 6/5, 4/3)
1 – 23/20 – 6/5 – 23/16 – 3/2
(steps 23/20, 24/23, 6/5, 24/23, 4/3)		 1 – 24/23 – 5/4 – 30/23 – 3/2
(steps 24/23, 6/5, 24/23, 23/20, 4/3)
1 – 9/8 – 6/5 – 23/16 – 3/2
(steps 9/8, 16/15, 6/5, 24/23, 4/3)		 1 – 24/23 – 5/4 – 4/3 – 3/2
(steps 24/23, 6/5, 16/15, 9/8, 4/3)
1 – 23/20 – 23/18 – 23/16 – 8/5
(steps 23/20, 10/9, 9/8, 10/9, 5/4)		 1 – 10/9 – 5/4 – 32/23 – 8/5
(steps 10/9, 9/8, 10/9, 23/20, 5/4)
1 – 5/4 – 30/23 – 32/23 – 36/23
(steps 5/4, 24/23, 16/15, 9/8, 23/18)		 1 – 9/8 – 6/5 – 5/4 – 36/23
(steps 9/8, 16/15, 24/23, 5/4, 23/18)
1 – 6/5 – 23/18 – 23/16 – 23/15
(steps 6/5, 16/15, 9/8, 16/15, 30/23)		 1 – 16/15 – 6/5 – 23/18 – 23/15
(steps 16/15, 9/8, 16/15, 6/5, 30/23)
1 – 23/20 – 6/5 – 23/18 – 23/16
(steps 23/20, 24/23, 16/15, 9/8, 32/23)		 1 – 9/8 – 6/5 – 5/4 – 23/16
(steps 9/8, 16/15, 24/23, 23/20, 32/23)
1 – 16/15 – 6/5 – 23/18 – 4/3
(steps 16/15, 9/8, 16/15, 24/23, 3/2)		 1 – 24/23 – 10/9 – 5/4 – 4/3
(steps 24/23, 16/15, 9/8, 16/15, 3/2)
1 – 16/15 – 10/9 – 23/18 – 4/3
(steps 16/15, 24/23, 23/20, 24/23, 3/2)		 1 – 24/23 – 6/5 – 5/4 – 4/3
(steps 24/23, 23/20, 24/23, 16/15, 3/2)

For hexads, there are three palindromic chords and 21 pairs of chords in inverse relationship:

Palindromic hexads
1 – 23/16 – 3/2 – 8/5 – 9/5 – 23/12 (steps 23/16, 24/23, 16/15, 9/8, 16/15, 24/23)
1 – 6/5 – 5/4 – 3/2 – 36/23 – 15/8 (steps 6/5, 24/23, 6/5, 24/23, 6/5, 16/15)
1 – 6/5 – 5/4 – 23/16 – 3/2 – 9/5 (steps 6/5, 24/23, 23/20, 24/23, 6/5, 10/9)
Inversely related pairs of hexads
1 – 5/4 – 3/2 – 8/5 – 9/5 – 23/12
(steps 5/4, 6/5, 16/15, 9/8, 16/15, 24/23)		 1 – 6/5 – 3/2 – 36/23 – 5/3 – 15/8
(steps 6/5, 5/4, 24/23, 16/15, 9/8, 16/15)
1 – 30/23 – 3/2 – 36/23 – 5/3 – 15/8
(steps 30/23, 23/20, 24/23, 16/15, 9/8, 16/15)		 1 – 23/20 – 3/2 – 8/5 – 9/5 – 23/12
(steps 23/20, 30/23, 16/15, 9/8, 16/15, 24/23)
1 – 5/4 – 23/16 – 3/2 – 9/5 – 23/12
(steps 5/4, 23/20, 24/23, 6/5, 16/15, 24/23)		 1 – 23/20 – 23/16 – 3/2 – 8/5 – 23/12
(steps 23/20, 5/4, 24/23, 16/15, 6/5, 24/23)
1 – 5/4 – 23/16 – 3/2 – 8/5 – 23/12
(steps 5/4, 23/20, 24/23, 16/15, 6/5, 24/23)		 1 – 23/20 – 23/16 – 3/2 – 9/5 – 23/12
(steps 23/20, 5/4, 24/23, 6/5, 16/15, 24/23)
1 – 5/4 – 23/16 – 3/2 – 8/5 – 9/5
(steps 5/4, 23/20, 24/23, 16/15, 9/8, 10/9)		 1 – 24/23 – 6/5 – 3/2 – 5/3 – 15/8
(steps 24/23, 23/20, 5/4, 10/9, 9/8, 16/15)
1 – 5/4 – 30/23 – 3/2 – 36/23 – 5/3
(steps 5/4, 24/23, 23/20, 24/23, 16/15, 6/5)		 1 – 23/20 – 6/5 – 3/2 – 9/5 – 23/12
(steps 23/20, 24/23, 5/4, 6/5, 16/15, 24/23)
1 – 6/5 – 23/16 – 3/2 – 9/5 – 23/12
(steps 6/5, 6/5, 24/23, 6/5, 16/15, 24/23)		 1 – 6/5 – 23/16 – 3/2 – 8/5 – 23/12
(steps 6/5, 6/5, 24/23, 16/15, 6/5, 24/23)
1 – 6/5 – 23/16 – 3/2 – 9/5 – 15/8
(steps 6/5, 6/5, 24/23, 6/5, 24/23, 16/15)		 1 – 6/5 – 5/4 – 3/2 – 9/5 – 23/12
(steps 6/5, 24/23, 6/5, 6/5, 16/15, 24/23)
1 – 23/20 – 23/16 – 3/2 – 8/5 – 9/5
(steps 23/20, 5/4, 24/23, 16/15, 9/8, 10/9)		 1 – 24/23 – 30/23 – 3/2 – 5/3 – 15/8
(steps 24/23, 5/4, 23/20, 10/9, 9/8, 16/15)
1 – 6/5 – 5/4 – 23/16 – 3/2 – 8/5
(steps 6/5, 24/23, 23/20, 24/23, 16/15, 5/4)		 1 – 24/23 – 6/5 – 5/4 – 3/2 – 15/8
(steps 24/23, 23/20, 24/23, 6/5, 5/4, 16/15)
1 – 6/5 – 5/4 – 23/16 – 3/2 – 15/8
(steps 6/5, 24/23, 23/20, 24/23, 5/4, 16/15)		 1 – 24/23 – 6/5 – 5/4 – 3/2 – 8/5
(steps 24/23, 23/20, 24/23, 6/5, 16/15, 5/4)
1 – 6/5 – 5/4 – 23/16 – 3/2 – 23/12
(steps 6/5, 24/23, 23/20, 24/23, 23/18, 24/23)		 1 – 23/20 – 6/5 – 23/16 – 3/2 – 23/12
(steps 23/20, 24/23, 6/5, 24/23, 23/18, 24/23)
1 – 6/5 – 5/4 – 4/3 – 3/2 – 23/12
(steps 6/5, 24/23, 16/15, 9/8, 23/18, 24/23)		 1 – 9/8 – 6/5 – 5/4 – 3/2 – 36/23
(steps 9/8, 16/15, 24/23, 6/5, 24/23, 23/18)
1 – 6/5 – 5/4 – 4/3 – 3/2 – 5/3
(steps 6/5, 24/23, 16/15, 9/8, 10/9, 6/5)		 1 – 9/8 – 6/5 – 5/4 – 3/2 – 9/5
(steps 9/8, 16/15, 24/23, 6/5, 6/5, 10/9)
1 – 6/5 – 5/4 – 4/3 – 3/2 – 8/5
(steps 6/5, 24/23, 16/15, 9/8, 16/15, 5/4)		 1 – 9/8 – 6/5 – 5/4 – 3/2 – 15/8
(steps 9/8, 16/15, 24/23, 6/5, 5/4, 16/15)
1 – 23/20 – 6/5 – 23/16 – 3/2 – 9/5
(steps 23/20, 24/23, 6/5, 24/23, 6/5, 10/9)		 1 – 24/23 – 5/4 – 30/23 – 3/2 – 5/3
(steps 24/23, 6/5, 24/23, 23/20, 10/9, 6/5)
1 – 23/20 – 6/5 – 23/16 – 3/2 – 8/5
(steps 23/20, 24/23, 6/5, 24/23, 16/15, 5/4)		 1 – 24/23 – 5/4 – 30/23 – 3/2 – 15/8
(steps 24/23, 6/5, 24/23, 23/20, 5/4, 16/15)
1 – 9/8 – 5/4 – 23/16 – 3/2 – 9/5
(steps 9/8, 10/9, 23/20, 24/23, 6/5, 10/9)		 1 – 24/23 – 6/5 – 4/3 – 3/2 – 5/3
(steps 24/23, 23/20, 10/9, 9/8, 10/9, 6/5)
1 – 9/8 – 6/5 – 23/16 – 3/2 – 9/5
(steps 9/8, 16/15, 6/5, 24/23, 6/5, 10/9)		 1 – 24/23 – 5/4 – 4/3 – 3/2 – 5/3
(steps 24/23, 6/5, 16/15, 9/8, 10/9, 6/5)
1 – 9/8 – 6/5 – 23/16 – 3/2 – 15/8
(steps 9/8, 16/15, 6/5, 24/23, 5/4, 16/15)		 1 – 24/23 – 5/4 – 4/3 – 3/2 – 8/5
(steps 24/23, 6/5, 16/15, 9/8, 16/15, 5/4)
1 – 9/8 – 6/5 – 5/4 – 23/16 – 3/2
(steps 9/8, 16/15, 24/23, 23/20, 24/23, 4/3)		 1 – 24/23 – 6/5 – 5/4 – 4/3 – 3/2
(steps 24/23, 23/20, 24/23, 16/15, 9/8, 4/3)

For heptads, there are one palindromic chord and seven pairs of chords in inverse relationship:

Palindromic heptad
1 – 6/5 – 5/4 – 4/3 – 3/2 – 8/5 – 5/3 (steps 6/5, 24/23, 16/15, 9/8, 16/15, 24/23, 6/5)
Inversely related pairs of heptads
1 – 5/4 – 23/16 – 3/2 – 8/5 – 9/5 – 23/12
(steps 5/4, 23/20, 24/23, 16/15, 9/8, 16/15, 24/23)		 1 – 23/20 – 23/16 – 3/2 – 8/5 – 9/5 – 23/12
(steps 23/20, 5/4, 24/23, 16/15, 9/8, 16/15, 24/23)
1 – 6/5 – 5/4 – 4/3 – 3/2 – 5/3 – 23/12
(steps 6/5, 24/23, 16/15, 9/8, 10/9, 23/20, 24/23)		 1 – 9/8 – 6/5 – 5/4 – 3/2 – 36/23 – 9/5
(steps 9/8, 16/15, 24/23, 6/5, 24/23, 23/20, 10/9)
1 – 6/5 – 5/4 – 4/3 – 3/2 – 8/5 – 23/12
(steps 6/5, 24/23, 16/15, 9/8, 16/15, 6/5, 24/23)		 1 – 9/8 – 6/5 – 5/4 – 3/2 – 36/23 – 15/8
(steps 9/8, 16/15, 24/23, 6/5, 24/23, 6/5, 16/15)
1 – 6/5 – 5/4 – 23/16 – 3/2 – 9/5 – 23/12
(steps 6/5, 24/23, 23/20, 24/23, 6/5, 16/15, 24/23)		 1 – 6/5 – 5/4 – 23/16 – 3/2 – 9/5 – 15/8
(steps 6/5, 24/23, 23/20, 24/23, 6/5, 24/23, 16/15)
1 – 6/5 – 5/4 – 23/16 – 3/2 – 8/5 – 23/12
(steps 6/5, 24/23, 23/20, 24/23, 16/15, 6/5, 24/23)		 1 – 23/20 – 6/5 – 23/16 – 3/2 – 9/5 – 23/12
(steps 23/20, 24/23, 6/5, 24/23, 6/5, 16/15, 24/23)
1 – 9/8 – 6/5 – 5/4 – 23/16 – 3/2 – 15/8
(steps 9/8, 16/15, 24/23, 23/20, 24/23, 5/4, 16/15)		 1 – 24/23 – 6/5 – 5/4 – 4/3 – 3/2 – 8/5
(steps 24/23, 23/20, 24/23, 16/15, 9/8, 16/15, 5/4)
1 – 9/8 – 6/5 – 5/4 – 23/16 – 3/2 – 9/5
(steps 9/8, 16/15, 24/23, 23/20, 24/23, 6/5, 10/9)		 1 – 24/23 – 6/5 – 5/4 – 4/3 – 3/2 – 5/3
(steps 24/23, 23/20, 24/23, 16/15, 9/8, 10/9, 6/5)

Finally, there is a pair of octads in inverse relationship:

Inversely related pair of octads
1 – 9/8 – 6/5 – 5/4 – 23/16 – 3/2 – 9/5 – 15/8
(steps 9/8, 16/15, 24/23, 23/20, 24/23, 6/5, 24/23, 16/15)		 1 – 24/23 – 6/5 – 5/4 – 4/3 – 3/2 – 8/5 – 5/3
(steps 24/23, 23/20, 24/23, 16/15, 9/8, 16/15, 24/23, 6/5)

Equal temperaments with worcester chords include 12, 15, 19, 22, 31, 34, 46, 53, 65, 84, 99, 118, 164 and 183.

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