Xenharmonic Wiki

Island chords: Difference between revisions

← Older edit
VisualWikitext
m Remove repeated link
ArrowHead294 (talk | contribs)
14,512 edits
mNo edit summary
 
(6 intermediate revisions by 2 users not shown)
Line 1: Line 1:
The '''island tetrad''' is an [[Dyadic chord#Essentially tempered dyadic chords|essentially tempered dyadic chord]] which 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 a 15/13-15/13-15/13-13/10 chord. It can also be viewed as an island tempered version of 1-15/13-[[4/3]]-[[20/13]]. Contained within it are a [[barbados triad]], the 1-13/10-[[3/2]] chord, and an [[island triad]], the 1-15/13-4/3 chord, which in another position is the 1-3/2-[[26/15]] chord. Another island chord of interest is a 26/15 over a major triad, 1-5/4-3/2-26/15, 5/4-6/5-15/13-15/13 in terms of intervals.
'''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.
 
For triads, there are one palindromic chord and four pairs of chords in inverse relationship.
 
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.
 
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 [[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.
 
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 {{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 ==
* [[Arto and tendo theory]]
* [[The Archipelago]]


* [[Arto and Tendo Theory]]
[[Category:15-odd-limit chords]]
 
[[Category:Essentially tempered chords]]
[[Category:15-limit]]
[[Category:Triads]]
[[Category:Dyadic]]
[[Category:Tetrads]]
[[Category:Pentads]]
[[Category:Hexads]]
[[Category:Heptads]]
[[Category:Island]]
[[Category:Island]]
[[Category:Tetrad]]
Retrieved from "https://en.xen.wiki/w/Island_chords"