Island chords: Difference between revisions

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'''Island chords''' are [[Dyadic chord|essentially tempered chords]] tempered by the island comma, [[676/675]].
'''Island chords''' are [[dyadic chord|essentially tempered chords]] tempered by the island comma, [[676/675]].


There are 9 triads, 37 tetrads, 51 pentads, 29 hexads and 6 heptads as 2.3.5.13 subgroup [[15-odd-limit]] essentially tempered chords.
There are 9 triads, 37 tetrads, 51 pentads, 29 hexads and 6 heptads as 2.3.5.13 subgroup [[15-odd-limit]] essentially tempered chords.
Line 6: Line 6:


The palindromic triad consists of two [[semifourth]]s and one [[perfect fifth]], splitting a fourth in two:  
The palindromic triad consists of two [[semifourth]]s and one [[perfect fifth]], splitting a fourth in two:  
* 1-15/13-4/3 with steps of 15/13-15/13-3/2.
* 1–15/13–4/3 with steps of 15/13, 15/13, 3/2.


The inversely related pairs of chords are  
The inversely related pairs of chords are  
* 1-5/4-13/9 with steps of 5/4-15/13-18/13, and its inverse
* 1–5/4–13/9 with steps of 5/4, 15/13, 18/13, and its inverse
* 1-15/13-13/9 with steps of 15/13-5/4-18/13;
* 1–15/13–13/9 with steps of 15/13, 5/4, 18/13;
* 1-13/10-18/13 with steps of 13/10-16/15-13/9, and its inverse
* 1–13/10–18/13 with steps of 13/10, 16/15, 13/9, and its inverse
* 1-16/15-18/13 with steps of 16/15-13/10-13/9;
* 1–16/15–18/13 with steps of 16/15, 13/10, 13/9;
* 1-15/13-13/10 with steps of 15/13-9/8-20/13, and its inverse
* 1–15/13–13/10 with steps of 15/13, 9/8, 20/13, and its inverse
* 1-9/8-13/10 with steps of 9/8-15/13-20/13;
* 1–9/8–13/10 with steps of 9/8, 15/13, 20/13;
* 1-13/12-15/13 with steps of 13/12-16/15-26/15, and its inverse
* 1–13/12–15/13 with steps of 13/12, 16/15, 26/15, and its inverse
* 1-16/15-15/13 with steps of 13/12-16/15-26/15.
* 1–16/15–15/13 with steps of 13/12, 16/15, 26/15.


For tetrads, there are seven palindromic chords and fifteen pairs of chords in inverse relationship.
For tetrads, there are seven palindromic chords and fifteen pairs of chords in inverse relationship.


One of the palindromic tetrads consists of three semifourths and one [[semisixth]],  
One of the palindromic tetrads consists of three semifourths and one [[semisixth]],  
* 1-13/10-3/2-26/15 with steps of 13/10-15/13-15/13-15/13.
* 1–13/10–3/2–26/15 with steps of 13/10, 15/13, 15/13, 15/13.


Aside from above, the following palindromic tetrad also contains a [[List of root-3rd-P5 triads in JI|barbados triad]] (otonal 1-13/10-3/2 chord) and its inversion (utonal 1-15/13-3/2 chord),  
Aside from above, the following palindromic tetrad also contains a barbados triad (otonal [[10:13:15|1–13/10–3/2]] chord) and its inversion (utonal [[26:30:39|1–15/13–3/2]] chord),  
* 1-15/13-13/10-3/2 with steps of 15/13-9/8-15/13-4/3.
* 1–15/13–13/10–3/2 with steps of 15/13, 9/8, 15/13, 4/3.


The rest five palindromic tetrads are  
The rest five palindromic tetrads are  
* 1-15/13-13/9-5/3 with steps of 15/13-5/4-15/13-6/5;
* 1–15/13–13/9–5/3 with steps of 15/13, 5/4, 15/13, 6/5;
* 1-10/9-13/9-20/13 with steps of 10/9-13/10-16/15-13/10;
* 1–10/9–13/9–20/13 with steps of 10/9, 13/10, 16/15, 13/10;
* 1-15/13-5/4-13/9 with steps of 15/13-13/12-15/13-18/13;
* 1–15/13–5/4–13/9 with steps of 15/13, 13/12, 15/13, 18/13;
* 1-13/12-15/13-5/4 with steps of 13/12-16/15-13/12-8/5;
* 1–13/12–15/13–5/4 with steps of 13/12, 16/15, 13/12, 8/5;
* 1-16/15-15/13-16/13 with steps of 16/15-13/12-16/15-13/8.
* 1–16/15–15/13–16/13 with steps of 16/15, 13/12, 16/15, 13/8.


The inversely related pairs of chords are  
The inversely related pairs of chords are  
* 1-9/8-13/10-13/8 with steps of 9/8-15/13-5/4-16/13, and its inverse
* 1–9/8–13/10–13/8 with steps of 9/8, 15/13, 5/4, 16/13, and its inverse
* 1-5/4-13/9-13/8 with steps of 5/4-15/13-9/8-16/13;
* 1–5/4–13/9–13/8 with steps of 5/4, 15/13, 9/8, 16/13;
* 1-5/4-3/2-26/15 with steps of 5/4-6/5-15/13-15/13, and its inverse
* 1–5/4–3/2–26/15 with steps of 5/4, 6/5, 15/13, 15/13, and its inverse
* 1-6/5-3/2-26/15 with steps of 6/5-5/4-15/13-15/13;
* 1–6/5–3/2–26/15 with steps of 6/5, 5/4, 15/13, 15/13;
* 1-5/4-13/9-20/13 with steps of 5/4-15/13-16/15-13/10, and its inverse
* 1–5/4–13/9–20/13 with steps of 5/4, 15/13, 16/15, 13/10, and its inverse
* 1-5/4-13/8-26/15 with steps of 5/4-13/10-16/15-15/13;
* 1–5/4–13/8–26/15 with steps of 5/4, 13/10, 16/15, 15/13;
* 1-13/10-3/2-15/8 with steps of 13/10-15/13-5/4-16/15, and its inverse
* 1–13/10–3/2–15/8 with steps of 13/10, 15/13, 5/4, 16/15, and its inverse
* 1-15/13-3/2-8/5 with steps of 15/13-13/10-16/15-5/4;
* 1–15/13–3/2–8/5 with steps of 15/13, 13/10, 16/15, 5/4;
* 1-13/10-18/13-3/2 with steps of 13/10-16/15-13/12-4/3, and its inverse
* 1–13/10–18/13–3/2 with steps of 13/10, 16/15, 13/12, 4/3, and its inverse
* 1-13/12-15/13-3/2 with steps of 13/12-16/15-13/10-4/3;
* 1–13/12–15/13–3/2 with steps of 13/12, 16/15, 13/10, 4/3;
* 1-15/13-4/3-3/2 with steps of 15/13-15/13-9/8-4/3, and its inverse
* 1–15/13–4/3–3/2 with steps of 15/13, 15/13, 9/8, 4/3, and its inverse
* 1-9/8-13/10-3/2 with steps of 9/8-15/13-15/13-4/3;
* 1–9/8–13/10–3/2 with steps of 9/8, 15/13, 15/13, 4/3;
* 1-18/13-3/2-8/5 with steps of 18/13-13/12-16/15-5/4, and its inverse
* 1–18/13–3/2–8/5 with steps of 18/13, 13/12, 16/15, 5/4, and its inverse
* 1-13/12-3/2-15/8 with steps of 13/12-18/13-5/4-16/15;
* 1–13/12–3/2–15/8 with steps of 13/12, 18/13, 5/4, 16/15;
* 1-15/13-13/10-13/9 with steps of 15/13-9/8-10/9-18/13, and its inverse
* 1–15/13–13/10–13/9 with steps of 15/13, 9/8, 10/9, 18/13, and its inverse
* 1-10/9-5/4-13/9 with steps of 10/9-9/8-15/13-18/13;
* 1–10/9–5/4–13/9 with steps of 10/9, 9/8, 15/13, 18/13;
* 1-18/13-3/2-26/15 with steps of 18/13-13/12-15/13-15/13, and its inverse
* 1–18/13–3/2–26/15 with steps of 18/13, 13/12, 15/13, 15/13, and its inverse
* 1-13/12-3/2-26/15 with steps of 13/12-18/13-15/13-15/13;
* 1–13/12–3/2–26/15 with steps of 13/12, 18/13, 15/13, 15/13;
* 1-6/5-13/10-18/13 with steps of 6/5-13/12-16/15-13/9, and its inverse
* 1–6/5–13/10–18/13 with steps of 6/5, 13/12, 16/15, 13/9, and its inverse
* 1-16/15-15/13-18/13 with steps of 16/15-13/12-6/5-13/9;
* 1–16/15–15/13–18/13 with steps of 16/15, 13/12, 6/5, 13/9;
* 1-15/13-13/10-18/13 with steps of 15/13-9/8-16/15-13/9, and its inverse
* 1–15/13–13/10–18/13 with steps of 15/13, 9/8, 16/15, 13/9, and its inverse
* 1-16/15-6/5-18/13 with steps of 16/15-9/8-15/13-13/9;
* 1–16/15–6/5–18/13 with steps of 16/15, 9/8, 15/13, 13/9;
* 1-9/8-13/10-18/13 with steps of 9/8-15/13-16/15-13/9, and its inverse
* 1–9/8–13/10–18/13 with steps of 9/8, 15/13, 16/15, 13/9, and its inverse
* 1-16/15-16/13-18/13 with steps of 16/15-15/13-9/8-13/9;
* 1–16/15–16/13–18/13 with steps of 16/15, 15/13, 9/8, 13/9;
* 1-15/13-16/13-4/3 with steps of 15/13-16/15-13/12-3/2, and its inverse
* 1–15/13–16/13–4/3 with steps of 15/13, 16/15, 13/12, 3/2, and its inverse
* 1-13/12-15/13-4/3 with steps of 13/12-16/15-15/13-3/2;
* 1–13/12–15/13–4/3 with steps of 13/12, 16/15, 15/13, 3/2;
* 1-15/13-5/4-4/3 with steps of 15/13-13/12-16/15-3/2, and its inverse
* 1–15/13–5/4–4/3 with steps of 15/13, 13/12, 16/15, 3/2, and its inverse
* 1-16/15-15/13-4/3 with steps of 16/15-13/12-15/13-3/2;
* 1–16/15–15/13–4/3 with steps of 16/15, 13/12, 15/13, 3/2;
* 1-9/8-6/5-13/10 with steps of 9/8-16/15-13/12-20/13, and its inverse
* 1–9/8–6/5–13/10 with steps of 9/8, 16/15, 13/12, 20/13, and its inverse
* 1-13/12-15/13-13/10 with steps of 13/12-16/15-9/8-20/13.
* 1–13/12–15/13–13/10 with steps of 13/12, 16/15, 9/8, 20/13.


For pentads, there are one palindromic chord and twenty-five pairs of chords in inverse relationship.
For pentads, there are one palindromic chord and twenty-five pairs of chords in inverse relationship.


The palindromic pentad consists of four semifourths and one [[whole tone]],  
The palindromic pentad consists of four semifourths and one [[whole tone]],  
* 1-9/8-13/10-3/2-26/15 with steps of 9/8-15/13-15/13-15/13-15/13.
* 1–9/8–13/10–3/2–26/15 with steps of 9/8, 15/13, 15/13, 15/13, 15/13.


The inversely related pairs of chords are  
The inversely related pairs of chords are  
* 1-6/5-18/13-3/2-26/15 with steps of 6/5-15/13-13/12-15/13-15/13, and its inverse
* 1–6/5–18/13–3/2–26/15 with steps of 6/5, 15/13, 13/12, 15/13, 15/13, and its inverse
* 1-13/12-5/4-3/2-26/15 with steps of 13/12-15/13-6/5-15/13-15/13;
* 1–13/12–5/4–3/2–26/15 with steps of 13/12, 15/13, 6/5, 15/13, 15/13;
* 1-9/8-5/4-3/2-26/15 with steps of 9/8-10/9-6/5-15/13-15/13, and its inverse
* 1–9/8–5/4–3/2–26/15 with steps of 9/8, 10/9, 6/5, 15/13, 15/13, and its inverse
* 1-6/5-4/3-3/2-26/15 with steps of 6/5-10/9-9/8-15/13-15/13;
* 1–6/5–4/3–3/2–26/15 with steps of 6/5, 10/9, 9/8, 15/13, 15/13;
* 1-6/5-18/13-3/2-26/15 with steps of 6/5-13/12-15/13-15/13-15/13, and its inverse
* 1–6/5–18/13–3/2–26/15 with steps of 6/5, 13/12, 15/13, 15/13, 15/13, and its inverse
* 1-15/13-5/4-3/2-26/15 with steps of 15/13-13/12-6/5-15/13-15/13;
* 1–15/13–5/4–3/2–26/15 with steps of 15/13, 13/12, 6/5, 15/13, 15/13;
* 1-9/8-18/13-3/2-26/15 with steps of 9/8-16/13-13/12-15/13-15/13, and its inverse
* 1–9/8–18/13–3/2–26/15 with steps of 9/8, 16/13, 13/12, 15/13, 15/13, and its inverse
* 1-9/8-13/10-3/2-13/8 with steps of 9/8-15/13-15/13-13/12-16/13;
* 1–9/8–13/10–3/2–13/8 with steps of 9/8, 15/13, 15/13, 13/12, 16/13;
* 1-5/4-3/2-8/5-26/15 with steps of 5/4-6/5-16/15-13/12-15/13, and its inverse
* 1–5/4–3/2–8/5–26/15 with steps of 5/4, 6/5, 16/15, 13/12, 15/13, and its inverse
* 1-6/5-3/2-26/15-15/8 with steps of 6/5-5/4-15/13-13/12-16/15;
* 1–6/5–3/2–26/15–15/8 with steps of 6/5, 5/4, 15/13, 13/12, 16/15;
* 1-5/4-3/2-13/8-26/15 with steps of 5/4-6/5-13/12-16/15-15/13, and its inverse
* 1–5/4–3/2–13/8–26/15 with steps of 5/4, 6/5, 13/12, 16/15, 15/13, and its inverse
* 1-6/5-3/2-26/15-24/13 with steps of 6/5-5/4-15/13-16/15-13/12;
* 1–6/5–3/2–26/15–24/13 with steps of 6/5, 5/4, 15/13, 16/15, 13/12;
* 1-15/13-13/9-5/3-15/8 with steps of 15/13-5/4-15/13-9/8-16/15, and its inverse
* 1–15/13–13/9–5/3–15/8 with steps of 15/13, 5/4, 15/13, 9/8, 16/15, and its inverse
* 1-9/8-13/10-13/8-15/8 with steps of 9/8-15/13-5/4-15/13-16/15;
* 1–9/8–13/10–13/8–15/8 with steps of 9/8, 15/13, 5/4, 15/13, 16/15;
* 1-15/13-18/13-3/2-8/5 with steps of 15/13-6/5-13/12-16/15-5/4, and its inverse
* 1–15/13–18/13–3/2–8/5 with steps of 15/13, 6/5, 13/12, 16/15, 5/4, and its inverse
* 1-13/12-13/10-3/2-15/8 with steps of 13/12-6/5-15/13-5/4-16/15;
* 1–13/12–13/10–3/2–15/8 with steps of 13/12, 6/5, 15/13, 5/4, 16/15;
* 1-5/4-4/3-3/2-26/15 with steps of 5/4-16/15-9/8-15/13-15/13, and its inverse
* 1–5/4–4/3–3/2–26/15 with steps of 5/4, 16/15, 9/8, 15/13, 15/13, and its inverse
* 1-9/8-6/5-3/2-26/15 with steps of 9/8-16/15-5/4-15/13-15/13;
* 1–9/8–6/5–3/2–26/15 with steps of 9/8, 16/15, 5/4, 15/13, 15/13;
* 1-15/13-13/10-3/2-15/8 with steps of 15/13-9/8-15/13-5/4-16/15, and its inverse
* 1–15/13–13/10–3/2–15/8 with steps of 15/13, 9/8, 15/13, 5/4, 16/15, and its inverse
* 1-15/13-13/10-3/2-8/5 with steps of 15/13-9/8-15/13-16/15-5/4;
* 1–15/13–13/10–3/2–8/5 with steps of 15/13, 9/8, 15/13, 16/15, 5/4;
* 1-13/10-18/13-3/2-8/5 with steps of 13/10-16/15-13/12-16/15-5/4, and its inverse
* 1–13/10–18/13–3/2–8/5 with steps of 13/10, 16/15, 13/12, 16/15, 5/4, and its inverse
* 1-13/12-15/13-3/2-15/8 with steps of 13/12-16/15-13/10-5/4-16/15;
* 1–13/12–15/13–3/2–15/8 with steps of 13/12, 16/15, 13/10, 5/4, 16/15;
* 1-13/10-18/13-3/2-9/5 with steps of 13/10-16/15-13/12-6/5-10/9, and its inverse
* 1–13/10–18/13–3/2–9/5 with steps of 13/10, 16/15, 13/12, 6/5, 10/9, and its inverse
* 1-13/12-15/13-3/2-5/3 with steps of 13/12-16/15-13/10-10/9-6/5;
* 1–13/12–15/13–3/2–5/3 with steps of 13/12, 16/15, 13/10, 10/9, 6/5;
* 1-9/8-5/4-13/8-26/15 with steps of 9/8-10/9-13/10-16/15-15/13, and its inverse
* 1–9/8–5/4–13/8–26/15 with steps of 9/8, 10/9, 13/10, 16/15, 15/13, and its inverse
* 1-15/13-16/13-8/5-16/9 with steps of 15/13-16/15-13/10-10/9-9/8;
* 1–15/13–16/13–8/5–16/9 with steps of 15/13, 16/15, 13/10, 10/9, 9/8;
* 1-13/10-3/2-13/8-26/15 with steps of 13/10-15/13-13/12-16/15-15/13, and its inverse
* 1–13/10–3/2–13/8–26/15 with steps of 13/10, 15/13, 13/12, 16/15, 15/13, and its inverse
* 1-13/10-3/2-8/5-26/15 with steps of 13/10-15/13-16/15-13/12-15/13;
* 1–13/10–3/2–8/5–26/15 with steps of 13/10, 15/13, 16/15, 13/12, 15/13;
* 1-13/10-3/2-26/15-15/8 with steps of 13/10-15/13-15/13-13/12-16/15, and its inverse
* 1–13/10–3/2–26/15–15/8 with steps of 13/10, 15/13, 15/13, 13/12, 16/15, and its inverse
* 1-13/10-18/13-3/2-26/15 with steps of 13/10-16/15-13/12-15/13-15/13;
* 1–13/10–18/13–3/2–26/15 with steps of 13/10, 16/15, 13/12, 15/13, 15/13;
* 1-13/10-3/2-13/8-15/8 with steps of 13/10-15/13-13/12-15/13-16/15, and its inverse
* 1–13/10–3/2–13/8–15/8 with steps of 13/10, 15/13, 13/12, 15/13, 16/15, and its inverse
* 1-15/13-3/2-8/5-24/13 with steps of 15/13-13/10-16/15-15/13-13/12;
* 1–15/13–3/2–8/5–24/13 with steps of 15/13, 13/10, 16/15, 15/13, 13/12;
* 1-6/5-13/10-18/13-3/2 with steps of 6/5-13/12-16/15-13/12-4/3, and its inverse
* 1–6/5–13/10–18/13–3/2 with steps of 6/5, 13/12, 16/15, 13/12, 4/3, and its inverse
* 1-13/12-15/13-5/4-3/2 with steps of 13/12-16/15-13/12-6/5-4/3;
* 1–13/12–15/13–5/4–3/2 with steps of 13/12, 16/15, 13/12, 6/5, 4/3;
* 1-15/13-5/4-4/3-3/2 with steps of 15/13-13/12-16/15-9/8-4/3, and its inverse
* 1–15/13–5/4–4/3–3/2 with steps of 15/13, 13/12, 16/15, 9/8, 4/3, and its inverse
* 1-9/8-6/5-13/10-3/2 with steps of 9/8-16/15-13/12-15/13-4/3;
* 1–9/8–6/5–13/10–3/2 with steps of 9/8, 16/15, 13/12, 15/13, 4/3;
* 1-9/8-13/10-18/13-3/2 with steps of 9/8-15/13-16/15-13/12-4/3, and its inverse
* 1–9/8–13/10–18/13–3/2 with steps of 9/8, 15/13, 16/15, 13/12, 4/3, and its inverse
* 1-13/12-15/13-4/3-3/2 with steps of 13/12-16/15-15/13-9/8-4/3;
* 1–13/12–15/13–4/3–3/2 with steps of 13/12, 16/15, 15/13, 9/8, 4/3;
* 1-15/13-13/10-18/13-3/2 with steps of 15/13-9/8-16/15-13/12-4/3, and its inverse
* 1–15/13–13/10–18/13–3/2 with steps of 15/13, 9/8, 16/15, 13/12, 4/3, and its inverse
* 1-13/12-15/13-13/10-3/2 with steps of 13/12-16/15-9/8-15/13-4/3;
* 1–13/12–15/13–13/10–3/2 with steps of 13/12, 16/15, 9/8, 15/13, 4/3;
* 1-18/13-3/2-8/5-26/15 with steps of 18/13-13/12-16/15-13/12-15/13, and its inverse
* 1–18/13–3/2–8/5–26/15 with steps of 18/13, 13/12, 16/15, 13/12, 15/13, and its inverse
* 1-13/12-3/2-26/15-15/8 with steps of 13/12-18/13-15/13-13/12-16/15;
* 1–13/12–3/2–26/15–15/8 with steps of 13/12, 18/13, 15/13, 13/12, 16/15;
* 1-18/13-3/2-8/5-9/5 with steps of 18/13-13/12-16/15-9/8-10/9, and its inverse
* 1–18/13–3/2–8/5–9/5 with steps of 18/13, 13/12, 16/15, 9/8, 10/9, and its inverse
* 1-13/12-3/2-5/3-15/8 with steps of 13/12-18/13-10/9-9/8-16/15;
* 1–13/12–3/2–5/3–15/8 with steps of 13/12, 18/13, 10/9, 9/8, 16/15;
* 1-13/12-3/2-13/8-15/8 with steps of 13/12-18/13-13/12-15/13-16/15, and its inverse
* 1–13/12–3/2–13/8–15/8 with steps of 13/12, 18/13, 13/12, 15/13, 16/15, and its inverse
* 1-13/12-3/2-13/8-26/15 with steps of 13/12-18/13-13/12-16/15-15/13;
* 1–13/12–3/2–13/8–26/15 with steps of 13/12, 18/13, 13/12, 16/15, 15/13;
* 1-9/8-6/5-13/10-18/13 with steps of 9/8-16/15-13/12-16/15-13/9, and its inverse
* 1–9/8–6/5–13/10–18/13 with steps of 9/8, 16/15, 13/12, 16/15, 13/9, and its inverse
* 1-16/15-15/13-16/13-18/13 with steps of 16/15-13/12-16/15-9/8-13/9;
* 1–16/15–15/13–16/13–18/13 with steps of 16/15, 13/12, 16/15, 9/8, 13/9;
* 1-13/12-15/13-5/4-4/3 with steps of 13/12-16/15-13/12-16/15-3/2, and its inverse
* 1–13/12–15/13–5/4–4/3 with steps of 13/12, 16/15, 13/12, 16/15, 3/2, and its inverse
* 1-16/15-15/13-16/13-4/3 with steps of 16/15-13/12-16/15-13/12-3/2.
* 1–16/15–15/13–16/13–4/3 with steps of 16/15, 13/12, 16/15, 13/12, 3/2.


For hexads, there are three palindromic chords and thirteen pairs of chords in inverse relationship. The palindromic chords are  
For hexads, there are three palindromic chords and thirteen pairs of chords in inverse relationship. The palindromic chords are  
* 1-13/12-5/4-3/2-26/15-15/8 with steps of 13/12-15/13-6/5-15/13-13/12-16/15;
* 1–13/12–5/4–3/2–26/15–15/8 with steps of 13/12, 15/13, 6/5, 15/13, 13/12, 16/15;
* 1-13/12-15/13-5/4-3/2-5/3 with steps of 13/12-16/15-13/12-6/5-10/9-6/5;
* 1–13/12–15/13–5/4–3/2–5/3 with steps of 13/12, 16/15, 13/12, 6/5, 10/9, 6/5;
* 1-13/12-3/2-13/8-26/15-15/8 with steps of 13/12-18/13-13/12-16/15-13/12-16/15.
* 1–13/12–3/2–13/8–26/15–15/8 with steps of 13/12, 18/13, 13/12, 16/15, 13/12, 16/15.


The inversely related pairs of chords are  
The inversely related pairs of chords are  
* 1-15/13-5/4-4/3-3/2-26/15 with steps of 15/13-13/12-16/15-9/8-15/13-15/13, and its inverse
* 1–15/13–5/4–4/3–3/2–26/15 with steps of 15/13, 13/12, 16/15, 9/8, 15/13, 15/13, and its inverse
* 1-9/8-6/5-13/10-3/2-26/15 with steps of 9/8-16/15-13/12-15/13-15/13-15/13;
* 1–9/8–6/5–13/10–3/2–26/15 with steps of 9/8, 16/15, 13/12, 15/13, 15/13, 15/13;
* 1-9/8-13/10-3/2-13/8-26/15 with steps of 9/8-15/13-15/13-13/12-16/15-15/13, and its inverse
* 1–9/8–13/10–3/2–13/8–26/15 with steps of 9/8, 15/13, 15/13, 13/12, 16/15, 15/13, and its inverse
* 1-9/8-13/10-18/13-3/2-26/15 with steps of 9/8-15/13-16/15-13/12-15/13-15/13;
* 1–9/8–13/10–18/13–3/2–26/15 with steps of 9/8, 15/13, 16/15, 13/12, 15/13, 15/13;
* 1-9/8-6/5-18/13-3/2-26/15 with steps of 9/8-16/15-15/13-13/12-15/13-15/13, and its inverse
* 1–9/8–6/5–18/13–3/2–26/15 with steps of 9/8, 16/15, 15/13, 13/12, 15/13, 15/13, and its inverse
* 1-13/12-5/4-4/3-3/2-26/15 with steps of 13/12-15/13-16/15-9/8-15/13-15/13;
* 1–13/12–5/4–4/3–3/2–26/15 with steps of 13/12, 15/13, 16/15, 9/8, 15/13, 15/13;
* 1-6/5-4/3-3/2-8/5-26/15 with steps of 6/5-10/9-9/8-16/15-13/12-15/13, and its inverse
* 1–6/5–4/3–3/2–8/5–26/15 with steps of 6/5, 10/9, 9/8, 16/15, 13/12, 15/13, and its inverse
* 1-9/8-5/4-3/2-26/15-15/8 with steps of 9/8-10/9-6/5-15/13-13/12-16/15;
* 1–9/8–5/4–3/2–26/15–15/8 with steps of 9/8, 10/9, 6/5, 15/13, 13/12, 16/15;
* 1-6/5-13/10-18/13-3/2-26/15 with steps of 6/5-13/12-16/15-13/12-15/13-15/13, and its inverse
* 1–6/5–13/10–18/13–3/2–26/15 with steps of 6/5, 13/12, 16/15, 13/12, 15/13, 15/13, and its inverse
* 1-13/12-15/13-5/4-3/2-26/15 with steps of 13/12-16/15-13/12-6/5-15/13-15/13;
* 1–13/12–15/13–5/4–3/2–26/15 with steps of 13/12, 16/15, 13/12, 6/5, 15/13, 15/13;
* 1-6/5-18/13-3/2-26/15-24/13 with steps of 6/5-15/13-13/12-15/13-16/15-13/12, and its inverse
* 1–6/5–18/13–3/2–26/15–24/13 with steps of 6/5, 15/13, 13/12, 15/13, 16/15, 13/12, and its inverse
* 1-13/12-5/4-3/2-13/8-26/15 with steps of 13/12-15/13-6/5-13/12-16/15-15/13;
* 1–13/12–5/4–3/2–13/8–26/15 with steps of 13/12, 15/13, 6/5, 13/12, 16/15, 15/13;
* 1-6/5-13/10-3/2-8/5-26/15 with steps of 6/5-13/12-15/13-16/15-13/12-15/13, and its inverse
* 1–6/5–13/10–3/2–8/5–26/15 with steps of 6/5, 13/12, 15/13, 16/15, 13/12, 15/13, and its inverse
* 1-15/13-5/4-3/2-26/15-15/8 with steps of 15/13-13/12-6/5-15/13-13/12-16/15;
* 1–15/13–5/4–3/2–26/15–15/8 with steps of 15/13, 13/12, 6/5, 15/13, 13/12, 16/15;
* 1-6/5-4/3-3/2-26/15-24/13 with steps of 6/5-10/9-9/8-15/13-16/15-13/12, and its inverse
* 1–6/5–4/3–3/2–26/15–24/13 with steps of 6/5, 10/9, 9/8, 15/13, 16/15, 13/12, and its inverse
* 1-9/8-5/4-3/2-8/5-26/15 with steps of 9/8-10/9-6/5-13/12-16/15-15/13;
* 1–9/8–5/4–3/2–8/5–26/15 with steps of 9/8, 10/9, 6/5, 13/12, 16/15, 15/13;
* 1-5/4-3/2-13/8-26/15-15/8 with steps of 5/4-6/5-13/12-16/15-13/12-16/15, and its inverse
* 1–5/4–3/2–13/8–26/15–15/8 with steps of 5/4, 6/5, 13/12, 16/15, 13/12, 16/15, and its inverse
* 1-6/5-3/2-8/5-26/15-24/13 with steps of 6/5-5/4-16/15-13/12-16/15-13/12;
* 1–6/5–3/2–8/5–26/15–24/13 with steps of 6/5, 5/4, 16/15, 13/12, 16/15, 13/12;
* 1-15/13-13/10-18/13-3/2-8/5 with steps of 15/13-9/8-16/15-13/12-16/15-5/4, and its inverse
* 1–15/13–13/10–18/13–3/2–8/5 with steps of 15/13, 9/8, 16/15, 13/12, 16/15, 5/4, and its inverse
* 1-13/12-15/13-13/10-3/2-15/8 with steps of 13/12-16/15-9/8-15/13-5/4-16/15;
* 1–13/12–15/13–13/10–3/2–15/8 with steps of 13/12, 16/15, 9/8, 15/13, 5/4, 16/15;
* 1-13/10-18/13-3/2-8/5-26/15 with steps of 13/10-16/15-13/12-16/15-13/12-15/13, and its inverse
* 1–13/10–18/13–3/2–8/5–26/15 with steps of 13/10, 16/15, 13/12, 16/15, 13/12, 15/13, and its inverse
* 1-13/12-15/13-3/2-26/15-15/8 with steps of 13/12-16/15-13/10-15/13-13/12-16/15;
* 1–13/12–15/13–3/2–26/15–15/8 with steps of 13/12, 16/15, 13/10, 15/13, 13/12, 16/15;
* 1-13/10-18/13-3/2-8/5-9/5 with steps of 13/10-16/15-13/12-16/15-9/8-10/9, and its inverse
* 1–13/10–18/13–3/2–8/5–9/5 with steps of 13/10, 16/15, 13/12, 16/15, 9/8, 10/9, and its inverse
* 1-13/12-15/13-3/2-5/3-15/8 with steps of 13/12-16/15-13/10-10/9-9/8-16/15;
* 1–13/12–15/13–3/2–5/3–15/8 with steps of 13/12, 16/15, 13/10, 10/9, 9/8, 16/15;
* 1-9/8-6/5-13/10-18/13-3/2 with steps of 9/8-16/15-13/12-16/15-13/12-4/3, and its inverse
* 1–9/8–6/5–13/10–18/13–3/2 with steps of 9/8, 16/15, 13/12, 16/15, 13/12, 4/3, and its inverse
* 1-13/12-15/13-5/4-4/3-3/2 with steps of 13/12-16/15-13/12-16/15-9/8-4/3.
* 1–13/12–15/13–5/4–4/3–3/2 with steps of 13/12, 16/15, 13/12, 16/15, 9/8, 4/3.


Finally, there are three pairs of heptads in inverse relationship:  
Finally, there are three pairs of heptads in inverse relationship:  
* 1-9/8-6/5-13/10-18/13-3/2-26/15 with steps of 9/8-16/15-13/12-16/15-13/12-15/13-15/13, and its inverse
* 1–9/8–6/5–13/10–18/13–3/2–26/15 with steps of 9/8, 16/15, 13/12, 16/15, 13/12, 15/13, 15/13, and its inverse
* 1-13/12-15/13-5/4-4/3-3/2-26/15 with steps of 13/12-16/15-13/12-16/15-9/8-15/13-15/13;
* 1–13/12–15/13–5/4–4/3–3/2–26/15 with steps of 13/12, 16/15, 13/12, 16/15, 9/8, 15/13, 15/13;
* 1-13/12-5/4-3/2-13/8-26/15-15/8 with steps of 13/12-15/13-6/5-13/12-16/15-13/12-16/15, and its inverse
* 1–13/12–5/4–3/2–13/8–26/15–15/8 with steps of 13/12, 15/13, 6/5, 13/12, 16/15, 13/12, 16/15, and its inverse
* 1-13/12-15/13-5/4-3/2-26/15-15/8 with steps of 13/12-16/15-13/12-6/5-15/13-13/12-16/15;
* 1–13/12–15/13–5/4–3/2–26/15–15/8 with steps of 13/12, 16/15, 13/12, 6/5, 15/13, 13/12, 16/15;
* 1-13/12-15/13-5/4-4/3-3/2-5/3 with steps of 13/12-16/15-13/12-16/15-9/8-10/9-6/5, and its inverse
* 1–13/12–15/13–5/4–4/3–3/2–5/3 with steps of 13/12, 16/15, 13/12, 16/15, 9/8, 10/9, 6/5, and its inverse
* 1-13/12-15/13-5/4-3/2-5/3-15/8 with steps of 13/12-16/15-13/12-6/5-10/9-9/8-16/15.
* 1–13/12–15/13–5/4–3/2–5/3–15/8 with steps of 13/12, 16/15, 13/12, 6/5, 10/9, 9/8, 16/15.


Equal temperaments with island chords include {{Optimal ET sequence|10, 15, 19, 24, 29, 34, 43, 53, 58, 72, 77, 87, 111, 130, 140, 164, 183 and 217}}.
Equal temperaments with island chords include {{Optimal ET sequence| 10, 15, 19, 24, 29, 34, 43, 53, 58, 72, 77, 87, 111, 130, 140, 164, 183 and 217 }}.


== See also ==
== See also ==
Line 169: Line 169:
* [[The Archipelago]]
* [[The Archipelago]]


[[Category:15-odd-limit]]
[[Category:15-odd-limit chords]]
[[Category:Essentially tempered chords]]
[[Category:Essentially tempered chords]]
[[Category:Triads]]
[[Category:Triads]]

Revision as of 14:19, 20 March 2025

Island chords are essentially tempered chords tempered by the island comma, 676/675.

There are 9 triads, 37 tetrads, 51 pentads, 29 hexads and 6 heptads as 2.3.5.13 subgroup 15-odd-limit essentially tempered chords.

For triads, there are one palindromic chord and four pairs of chords in inverse relationship.

The palindromic triad consists of two semifourths and one perfect fifth, splitting a fourth in two:

  • 1–15/13–4/3 with steps of 15/13, 15/13, 3/2.

The inversely related pairs of chords are

  • 1–5/4–13/9 with steps of 5/4, 15/13, 18/13, and its inverse
  • 1–15/13–13/9 with steps of 15/13, 5/4, 18/13;
  • 1–13/10–18/13 with steps of 13/10, 16/15, 13/9, and its inverse
  • 1–16/15–18/13 with steps of 16/15, 13/10, 13/9;
  • 1–15/13–13/10 with steps of 15/13, 9/8, 20/13, and its inverse
  • 1–9/8–13/10 with steps of 9/8, 15/13, 20/13;
  • 1–13/12–15/13 with steps of 13/12, 16/15, 26/15, and its inverse
  • 1–16/15–15/13 with steps of 13/12, 16/15, 26/15.

For tetrads, there are seven palindromic chords and fifteen pairs of chords in inverse relationship.

One of the palindromic tetrads consists of three semifourths and one semisixth,

  • 1–13/10–3/2–26/15 with steps of 13/10, 15/13, 15/13, 15/13.

Aside from above, the following palindromic tetrad also contains a barbados triad (otonal 1–13/10–3/2 chord) and its inversion (utonal 1–15/13–3/2 chord),

  • 1–15/13–13/10–3/2 with steps of 15/13, 9/8, 15/13, 4/3.

The rest five palindromic tetrads are

  • 1–15/13–13/9–5/3 with steps of 15/13, 5/4, 15/13, 6/5;
  • 1–10/9–13/9–20/13 with steps of 10/9, 13/10, 16/15, 13/10;
  • 1–15/13–5/4–13/9 with steps of 15/13, 13/12, 15/13, 18/13;
  • 1–13/12–15/13–5/4 with steps of 13/12, 16/15, 13/12, 8/5;
  • 1–16/15–15/13–16/13 with steps of 16/15, 13/12, 16/15, 13/8.

The inversely related pairs of chords are

  • 1–9/8–13/10–13/8 with steps of 9/8, 15/13, 5/4, 16/13, and its inverse
  • 1–5/4–13/9–13/8 with steps of 5/4, 15/13, 9/8, 16/13;
  • 1–5/4–3/2–26/15 with steps of 5/4, 6/5, 15/13, 15/13, and its inverse
  • 1–6/5–3/2–26/15 with steps of 6/5, 5/4, 15/13, 15/13;
  • 1–5/4–13/9–20/13 with steps of 5/4, 15/13, 16/15, 13/10, and its inverse
  • 1–5/4–13/8–26/15 with steps of 5/4, 13/10, 16/15, 15/13;
  • 1–13/10–3/2–15/8 with steps of 13/10, 15/13, 5/4, 16/15, and its inverse
  • 1–15/13–3/2–8/5 with steps of 15/13, 13/10, 16/15, 5/4;
  • 1–13/10–18/13–3/2 with steps of 13/10, 16/15, 13/12, 4/3, and its inverse
  • 1–13/12–15/13–3/2 with steps of 13/12, 16/15, 13/10, 4/3;
  • 1–15/13–4/3–3/2 with steps of 15/13, 15/13, 9/8, 4/3, and its inverse
  • 1–9/8–13/10–3/2 with steps of 9/8, 15/13, 15/13, 4/3;
  • 1–18/13–3/2–8/5 with steps of 18/13, 13/12, 16/15, 5/4, and its inverse
  • 1–13/12–3/2–15/8 with steps of 13/12, 18/13, 5/4, 16/15;
  • 1–15/13–13/10–13/9 with steps of 15/13, 9/8, 10/9, 18/13, and its inverse
  • 1–10/9–5/4–13/9 with steps of 10/9, 9/8, 15/13, 18/13;
  • 1–18/13–3/2–26/15 with steps of 18/13, 13/12, 15/13, 15/13, and its inverse
  • 1–13/12–3/2–26/15 with steps of 13/12, 18/13, 15/13, 15/13;
  • 1–6/5–13/10–18/13 with steps of 6/5, 13/12, 16/15, 13/9, and its inverse
  • 1–16/15–15/13–18/13 with steps of 16/15, 13/12, 6/5, 13/9;
  • 1–15/13–13/10–18/13 with steps of 15/13, 9/8, 16/15, 13/9, and its inverse
  • 1–16/15–6/5–18/13 with steps of 16/15, 9/8, 15/13, 13/9;
  • 1–9/8–13/10–18/13 with steps of 9/8, 15/13, 16/15, 13/9, and its inverse
  • 1–16/15–16/13–18/13 with steps of 16/15, 15/13, 9/8, 13/9;
  • 1–15/13–16/13–4/3 with steps of 15/13, 16/15, 13/12, 3/2, and its inverse
  • 1–13/12–15/13–4/3 with steps of 13/12, 16/15, 15/13, 3/2;
  • 1–15/13–5/4–4/3 with steps of 15/13, 13/12, 16/15, 3/2, and its inverse
  • 1–16/15–15/13–4/3 with steps of 16/15, 13/12, 15/13, 3/2;
  • 1–9/8–6/5–13/10 with steps of 9/8, 16/15, 13/12, 20/13, and its inverse
  • 1–13/12–15/13–13/10 with steps of 13/12, 16/15, 9/8, 20/13.

For pentads, there are one palindromic chord and twenty-five pairs of chords in inverse relationship.

The palindromic pentad consists of four semifourths and one whole tone,

  • 1–9/8–13/10–3/2–26/15 with steps of 9/8, 15/13, 15/13, 15/13, 15/13.

The inversely related pairs of chords are

  • 1–6/5–18/13–3/2–26/15 with steps of 6/5, 15/13, 13/12, 15/13, 15/13, and its inverse
  • 1–13/12–5/4–3/2–26/15 with steps of 13/12, 15/13, 6/5, 15/13, 15/13;
  • 1–9/8–5/4–3/2–26/15 with steps of 9/8, 10/9, 6/5, 15/13, 15/13, and its inverse
  • 1–6/5–4/3–3/2–26/15 with steps of 6/5, 10/9, 9/8, 15/13, 15/13;
  • 1–6/5–18/13–3/2–26/15 with steps of 6/5, 13/12, 15/13, 15/13, 15/13, and its inverse
  • 1–15/13–5/4–3/2–26/15 with steps of 15/13, 13/12, 6/5, 15/13, 15/13;
  • 1–9/8–18/13–3/2–26/15 with steps of 9/8, 16/13, 13/12, 15/13, 15/13, and its inverse
  • 1–9/8–13/10–3/2–13/8 with steps of 9/8, 15/13, 15/13, 13/12, 16/13;
  • 1–5/4–3/2–8/5–26/15 with steps of 5/4, 6/5, 16/15, 13/12, 15/13, and its inverse
  • 1–6/5–3/2–26/15–15/8 with steps of 6/5, 5/4, 15/13, 13/12, 16/15;
  • 1–5/4–3/2–13/8–26/15 with steps of 5/4, 6/5, 13/12, 16/15, 15/13, and its inverse
  • 1–6/5–3/2–26/15–24/13 with steps of 6/5, 5/4, 15/13, 16/15, 13/12;
  • 1–15/13–13/9–5/3–15/8 with steps of 15/13, 5/4, 15/13, 9/8, 16/15, and its inverse
  • 1–9/8–13/10–13/8–15/8 with steps of 9/8, 15/13, 5/4, 15/13, 16/15;
  • 1–15/13–18/13–3/2–8/5 with steps of 15/13, 6/5, 13/12, 16/15, 5/4, and its inverse
  • 1–13/12–13/10–3/2–15/8 with steps of 13/12, 6/5, 15/13, 5/4, 16/15;
  • 1–5/4–4/3–3/2–26/15 with steps of 5/4, 16/15, 9/8, 15/13, 15/13, and its inverse
  • 1–9/8–6/5–3/2–26/15 with steps of 9/8, 16/15, 5/4, 15/13, 15/13;
  • 1–15/13–13/10–3/2–15/8 with steps of 15/13, 9/8, 15/13, 5/4, 16/15, and its inverse
  • 1–15/13–13/10–3/2–8/5 with steps of 15/13, 9/8, 15/13, 16/15, 5/4;
  • 1–13/10–18/13–3/2–8/5 with steps of 13/10, 16/15, 13/12, 16/15, 5/4, and its inverse
  • 1–13/12–15/13–3/2–15/8 with steps of 13/12, 16/15, 13/10, 5/4, 16/15;
  • 1–13/10–18/13–3/2–9/5 with steps of 13/10, 16/15, 13/12, 6/5, 10/9, and its inverse
  • 1–13/12–15/13–3/2–5/3 with steps of 13/12, 16/15, 13/10, 10/9, 6/5;
  • 1–9/8–5/4–13/8–26/15 with steps of 9/8, 10/9, 13/10, 16/15, 15/13, and its inverse
  • 1–15/13–16/13–8/5–16/9 with steps of 15/13, 16/15, 13/10, 10/9, 9/8;
  • 1–13/10–3/2–13/8–26/15 with steps of 13/10, 15/13, 13/12, 16/15, 15/13, and its inverse
  • 1–13/10–3/2–8/5–26/15 with steps of 13/10, 15/13, 16/15, 13/12, 15/13;
  • 1–13/10–3/2–26/15–15/8 with steps of 13/10, 15/13, 15/13, 13/12, 16/15, and its inverse
  • 1–13/10–18/13–3/2–26/15 with steps of 13/10, 16/15, 13/12, 15/13, 15/13;
  • 1–13/10–3/2–13/8–15/8 with steps of 13/10, 15/13, 13/12, 15/13, 16/15, and its inverse
  • 1–15/13–3/2–8/5–24/13 with steps of 15/13, 13/10, 16/15, 15/13, 13/12;
  • 1–6/5–13/10–18/13–3/2 with steps of 6/5, 13/12, 16/15, 13/12, 4/3, and its inverse
  • 1–13/12–15/13–5/4–3/2 with steps of 13/12, 16/15, 13/12, 6/5, 4/3;
  • 1–15/13–5/4–4/3–3/2 with steps of 15/13, 13/12, 16/15, 9/8, 4/3, and its inverse
  • 1–9/8–6/5–13/10–3/2 with steps of 9/8, 16/15, 13/12, 15/13, 4/3;
  • 1–9/8–13/10–18/13–3/2 with steps of 9/8, 15/13, 16/15, 13/12, 4/3, and its inverse
  • 1–13/12–15/13–4/3–3/2 with steps of 13/12, 16/15, 15/13, 9/8, 4/3;
  • 1–15/13–13/10–18/13–3/2 with steps of 15/13, 9/8, 16/15, 13/12, 4/3, and its inverse
  • 1–13/12–15/13–13/10–3/2 with steps of 13/12, 16/15, 9/8, 15/13, 4/3;
  • 1–18/13–3/2–8/5–26/15 with steps of 18/13, 13/12, 16/15, 13/12, 15/13, and its inverse
  • 1–13/12–3/2–26/15–15/8 with steps of 13/12, 18/13, 15/13, 13/12, 16/15;
  • 1–18/13–3/2–8/5–9/5 with steps of 18/13, 13/12, 16/15, 9/8, 10/9, and its inverse
  • 1–13/12–3/2–5/3–15/8 with steps of 13/12, 18/13, 10/9, 9/8, 16/15;
  • 1–13/12–3/2–13/8–15/8 with steps of 13/12, 18/13, 13/12, 15/13, 16/15, and its inverse
  • 1–13/12–3/2–13/8–26/15 with steps of 13/12, 18/13, 13/12, 16/15, 15/13;
  • 1–9/8–6/5–13/10–18/13 with steps of 9/8, 16/15, 13/12, 16/15, 13/9, and its inverse
  • 1–16/15–15/13–16/13–18/13 with steps of 16/15, 13/12, 16/15, 9/8, 13/9;
  • 1–13/12–15/13–5/4–4/3 with steps of 13/12, 16/15, 13/12, 16/15, 3/2, and its inverse
  • 1–16/15–15/13–16/13–4/3 with steps of 16/15, 13/12, 16/15, 13/12, 3/2.

For hexads, there are three palindromic chords and thirteen pairs of chords in inverse relationship. The palindromic chords are

  • 1–13/12–5/4–3/2–26/15–15/8 with steps of 13/12, 15/13, 6/5, 15/13, 13/12, 16/15;
  • 1–13/12–15/13–5/4–3/2–5/3 with steps of 13/12, 16/15, 13/12, 6/5, 10/9, 6/5;
  • 1–13/12–3/2–13/8–26/15–15/8 with steps of 13/12, 18/13, 13/12, 16/15, 13/12, 16/15.

The inversely related pairs of chords are

  • 1–15/13–5/4–4/3–3/2–26/15 with steps of 15/13, 13/12, 16/15, 9/8, 15/13, 15/13, and its inverse
  • 1–9/8–6/5–13/10–3/2–26/15 with steps of 9/8, 16/15, 13/12, 15/13, 15/13, 15/13;
  • 1–9/8–13/10–3/2–13/8–26/15 with steps of 9/8, 15/13, 15/13, 13/12, 16/15, 15/13, and its inverse
  • 1–9/8–13/10–18/13–3/2–26/15 with steps of 9/8, 15/13, 16/15, 13/12, 15/13, 15/13;
  • 1–9/8–6/5–18/13–3/2–26/15 with steps of 9/8, 16/15, 15/13, 13/12, 15/13, 15/13, and its inverse
  • 1–13/12–5/4–4/3–3/2–26/15 with steps of 13/12, 15/13, 16/15, 9/8, 15/13, 15/13;
  • 1–6/5–4/3–3/2–8/5–26/15 with steps of 6/5, 10/9, 9/8, 16/15, 13/12, 15/13, and its inverse
  • 1–9/8–5/4–3/2–26/15–15/8 with steps of 9/8, 10/9, 6/5, 15/13, 13/12, 16/15;
  • 1–6/5–13/10–18/13–3/2–26/15 with steps of 6/5, 13/12, 16/15, 13/12, 15/13, 15/13, and its inverse
  • 1–13/12–15/13–5/4–3/2–26/15 with steps of 13/12, 16/15, 13/12, 6/5, 15/13, 15/13;
  • 1–6/5–18/13–3/2–26/15–24/13 with steps of 6/5, 15/13, 13/12, 15/13, 16/15, 13/12, and its inverse
  • 1–13/12–5/4–3/2–13/8–26/15 with steps of 13/12, 15/13, 6/5, 13/12, 16/15, 15/13;
  • 1–6/5–13/10–3/2–8/5–26/15 with steps of 6/5, 13/12, 15/13, 16/15, 13/12, 15/13, and its inverse
  • 1–15/13–5/4–3/2–26/15–15/8 with steps of 15/13, 13/12, 6/5, 15/13, 13/12, 16/15;
  • 1–6/5–4/3–3/2–26/15–24/13 with steps of 6/5, 10/9, 9/8, 15/13, 16/15, 13/12, and its inverse
  • 1–9/8–5/4–3/2–8/5–26/15 with steps of 9/8, 10/9, 6/5, 13/12, 16/15, 15/13;
  • 1–5/4–3/2–13/8–26/15–15/8 with steps of 5/4, 6/5, 13/12, 16/15, 13/12, 16/15, and its inverse
  • 1–6/5–3/2–8/5–26/15–24/13 with steps of 6/5, 5/4, 16/15, 13/12, 16/15, 13/12;
  • 1–15/13–13/10–18/13–3/2–8/5 with steps of 15/13, 9/8, 16/15, 13/12, 16/15, 5/4, and its inverse
  • 1–13/12–15/13–13/10–3/2–15/8 with steps of 13/12, 16/15, 9/8, 15/13, 5/4, 16/15;
  • 1–13/10–18/13–3/2–8/5–26/15 with steps of 13/10, 16/15, 13/12, 16/15, 13/12, 15/13, and its inverse
  • 1–13/12–15/13–3/2–26/15–15/8 with steps of 13/12, 16/15, 13/10, 15/13, 13/12, 16/15;
  • 1–13/10–18/13–3/2–8/5–9/5 with steps of 13/10, 16/15, 13/12, 16/15, 9/8, 10/9, and its inverse
  • 1–13/12–15/13–3/2–5/3–15/8 with steps of 13/12, 16/15, 13/10, 10/9, 9/8, 16/15;
  • 1–9/8–6/5–13/10–18/13–3/2 with steps of 9/8, 16/15, 13/12, 16/15, 13/12, 4/3, and its inverse
  • 1–13/12–15/13–5/4–4/3–3/2 with steps of 13/12, 16/15, 13/12, 16/15, 9/8, 4/3.

Finally, there are three pairs of heptads in inverse relationship:

  • 1–9/8–6/5–13/10–18/13–3/2–26/15 with steps of 9/8, 16/15, 13/12, 16/15, 13/12, 15/13, 15/13, and its inverse
  • 1–13/12–15/13–5/4–4/3–3/2–26/15 with steps of 13/12, 16/15, 13/12, 16/15, 9/8, 15/13, 15/13;
  • 1–13/12–5/4–3/2–13/8–26/15–15/8 with steps of 13/12, 15/13, 6/5, 13/12, 16/15, 13/12, 16/15, and its inverse
  • 1–13/12–15/13–5/4–3/2–26/15–15/8 with steps of 13/12, 16/15, 13/12, 6/5, 15/13, 13/12, 16/15;
  • 1–13/12–15/13–5/4–4/3–3/2–5/3 with steps of 13/12, 16/15, 13/12, 16/15, 9/8, 10/9, 6/5, and its inverse
  • 1–13/12–15/13–5/4–3/2–5/3–15/8 with steps of 13/12, 16/15, 13/12, 6/5, 10/9, 9/8, 16/15.

Equal temperaments with island chords include 10, 15, 19, 24, 29, 34, 43, 53, 58, 72, 77, 87, 111, 130, 140, 164, 183 and 217.

See also

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