Island chords: Difference between revisions

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An '''island chord''' is an [[15-odd-limit]] [[essentially tempered dyadic chord]] which is tempered by [[676/675]] (island comma).  
'''Island chords''' are [[Dyadic chord|essentially tempered chords]] tempered by the island comma, [[676/675]].


The island tetrad under [[octave reduction]] consists of three [[15/13]] intervals followed by a [[13/10]], which closes on the octave since the island comma, [[676/675]], is [[tempering out|tempered out]]; in other words an island tempered version of
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.
* 1-15/13-[[4/3]]-[[20/13]], with steps 15/13-15/13-15/13-13/10.  


Contained within it are a '''barbados triad''',
For triads, there are one palindromic chord and four pairs of chords in inverse relationship.
* 1-13/10-[[3/2]], with steps 13/10-15/13-4/3,


its inversion,  
The palindromic triad consists of two [[semifourth]]s and one [[perfect fifth]], equating two [[15/13]]s and one [[4/3]]:
* 1-15/13-3/2, with steps 15/13-13/10-4/3,
* 1-15/13-4/3 with steps of 15/13-15/13-3/2.


and an '''island triad''',  
The inversely related pairs of chords are
* 1-15/13-4/3, with steps 15/13-15/13-3/2,  
* 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.


which in another position is the 1-3/2-[[26/15]] chord.  
For tetrads, there are seven palindromic chords and fifteen pairs of chords in inverse relationship.


Another island tetrad of interest is a 26/15 over a major triad,  
One of the palindromic tetrads consists of three semifourths and one [[semisixth]],
* 1-5/4-3/2-26/15, with steps 5/4-6/5-15/13-15/13,
* 1-13/10-3/2-26/15 with steps of 13/10-15/13-15/13-15/13.


and its inversion,  
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),
* 1-6/5-3/2-26/15, with steps 6/5-5/4-15/13-15/13.  
* 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
* 1-15/13-4/3-16/9 with steps of 15/13-15/13-4/3-9/8;
* 1-5/4-4/3-13/9 with steps of 5/4-16/15-13/12-18/13, and its inverse
* 1-13/12-15/13-13/9 with steps of 13/12-16/15-5/4-18/13;
* 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;
* 1-15/13-4/3-13/9 with steps of 15/13-15/13-13/12-18/13, and its inverse
* 1-13/12-5/4-13/9 with steps of 13/12-15/13-15/13-18/13;
* 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;
* 1-15/13-5/4-4/3-20/13 with steps of 15/13-13/12-16/15-15/13-13/10, and its inverse
* 1-15/13-16/13-4/3-20/13 with steps of 15/13-16/15-13/12-15/13-13/10;
* 1-15/13-4/3-13/9-20/13 with steps of 15/13-15/13-13/12-16/15-13/10, and its inverse
* 1-16/15-15/13-4/3-20/13 with steps of 16/15-13/12-15/13-15/13-13/10;
* 1-15/13-5/4-13/9-20/13 with steps of 15/13-13/12-15/13-16/15-13/10, and its inverse
* 1-16/15-16/13-4/3-20/13 with steps of 16/15-15/13-13/12-15/13-13/10;
* 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 {{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 29: Line 173:
[[Category:Triads]]
[[Category:Triads]]
[[Category:Tetrads]]
[[Category:Tetrads]]
[[Category:Pentads]]
[[Category:Hexads]]
[[Category:Heptads]]
[[Category:Island]]
[[Category:Island]]

Revision as of 10:53, 11 November 2023

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, equating two 15/13s and one 4/3:

  • 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
  • 1-15/13-4/3-16/9 with steps of 15/13-15/13-4/3-9/8;
  • 1-5/4-4/3-13/9 with steps of 5/4-16/15-13/12-18/13, and its inverse
  • 1-13/12-15/13-13/9 with steps of 13/12-16/15-5/4-18/13;
  • 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;
  • 1-15/13-4/3-13/9 with steps of 15/13-15/13-13/12-18/13, and its inverse
  • 1-13/12-5/4-13/9 with steps of 13/12-15/13-15/13-18/13;
  • 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;
  • 1-15/13-5/4-4/3-20/13 with steps of 15/13-13/12-16/15-15/13-13/10, and its inverse
  • 1-15/13-16/13-4/3-20/13 with steps of 15/13-16/15-13/12-15/13-13/10;
  • 1-15/13-4/3-13/9-20/13 with steps of 15/13-15/13-13/12-16/15-13/10, and its inverse
  • 1-16/15-15/13-4/3-20/13 with steps of 16/15-13/12-15/13-15/13-13/10;
  • 1-15/13-5/4-13/9-20/13 with steps of 15/13-13/12-15/13-16/15-13/10, and its inverse
  • 1-16/15-16/13-4/3-20/13 with steps of 16/15-15/13-13/12-15/13-13/10;
  • 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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