The Archipelago: Difference between revisions

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~~__FORCETOC__~~

The '''archipelago''' is a rag-tag collection of various regular temperaments of different ranks, including subgroup temperaments, associated with island temperament: the rank five thirteen limit temperament tempering out the island comma, [[676/675]]. Common to all of them is the observation that two intervals of 15/13 are equated with a fourth. Hence a 1-15/13-4/3 chord is a characteristic island chord, and 15/13 tends to be of low complexity. Also characteristic is the barbados triad, the 1-13/10-3/2 triad, as well as its inversion 1-15/13-3/2, the barbados tetrad, 1-13/10-3/2-26/15, plus the tetrads 1-13/10-3/2-8/5 and 1-13/10-3/2-9/5. The [[just intonation subgroup]] generated by 2, 4/3 and 15/13 is 2.3.13/5, and the barbados triad and tetrad are found in that, while the other two tetrads are found in the larger 2.3.5.13 subgroup.

The archipelago is a rag-tag collection of various regular temperaments of different ranks, including subgroup temperaments, associated with island temperament: the rank five thirteen limit temperament tempering out the island comma, [[676/675]]. Common to all of them is the observation that two intervals of 15/13 are equated with a fourth. Hence a 1-15/13-4/3 chord is a characteristic island chord, and 15/13 tends to be of low complexity. Also characteristic is the barbados triad, the 1-13/10-3/2 triad, as well as its inversion 1-15/13-3/2, the barbados tetrad, 1-13/10-3/2-26/15, plus the tetrads 1-13/10-3/2-8/5 and 1-13/10-3/2-9/5. The [[~~Just_intonation_subgroups|~~just intonation subgroup]] generated by 2, 4/3 and 15/13 is 2.3.13/5, and the barbados triad and tetrad are found in that, while the other two tetrads are found in the larger 2.3.5.13 subgroup.

The barbados triad is of particular theoretical interest because, when reduced to lowest terms, it is the 10:13:15 triad. Thus, this triad is only slightly higher in complexity than the 5-limit 10:12:15 minor triad, which means it may be of distinct value as a relatively unexplored musical consonance. It is one of only a few low-complexity triads with a 3/2 on the outer dyad, some others being 4:5:6, 6:7:9, and 10:12:15. It works out to 0-454-702 cents, which means that it is an ''ultramajor'' triad, with a third sharper even than the 9/7 supermajor third.

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Compared to the 7-limit 14:18:21 supermajor triad, 10:13:15 is lower in triadic complexity (10:13:15 vs 14:18:21), but contains dyads that are on average higher in complexity (9/7 vs 13/10 and 7/6 vs 15/13). Its inverse, however, is the ultraminor 26:30:39, which is far more complex than the 7-limit subminor 6:7:9. Temperaments in which 91/90 vanishes equate the two types of triads.

[[~~24edo|~~24edo]] approximates this triad to within an error of four cents, and [[~~29edo|~~29edo]] does even better, getting it to within 1.5 cents; either may be used as a tuning for the barbados temperament discussed below.

[[24edo]] approximates this triad to within an error of four cents, and [[29edo]] does even better, getting it to within 1.5 cents; either may be used as a tuning for the barbados temperament discussed below.

~~= Rank five temperament =~~

~~== Island ==~~

== Rank-5 temperaments ==

=== Island ===

Subgroup: 2.3.5.7.11.13

[[Comma list]]: [[676/675]]

~~Map~~:<br>

[[Mapping]]:<br>

{{val| 1 0 0 0 0 -1 }}<br>

{{val| 0 2 0 0 0 3 }}<br>

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~~Vals:~~ 5, 9, 10, 15, 19, 24, 29, 43, 53, 58, 72, 87, 111, 121, 130, 183, 940

{{Val list|legend=1| 5, 9, 10, 15, 19, 24, 29, 43, 53, 58, 72, 87, 111, 121, 130, 183, 940 }}

[[Optimal patent val]]: [[940edo]]

= Rank ~~four~~ temperaments=

== Rank-4 temperaments ==

== 1001/1000 ==

=== 1001/1000 ===

Commas: 676/675, 1001/1000

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[[Optimal_patent_val|Optimal patent val]]: [[940edo]]

== 49/48 ==

=== 49/48 ===

Commas: 49/48, 91/90

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EDOs: 5, 9, 10, 15, 19, 24

== 1716/1715 ==

=== 1716/1715 ===

Commas: 676/675, 1716/1715

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EDOs: 58, 72, 77, 121, 130, 140, 149, 198, 212, 270

== 364/363 ==

=== 364/363 ===

Commas: 364/363, 676/675

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EDOs: 9, 15, 29, 43, 58, 72, 87, 121, 130

== 351/350 ==

=== 351/350 ===

Commas: 351/350, 676/675

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EDOs: 19, 53, 58, 72, 77, 111, 130

== 352/351 ==

=== 352/351 ===

Commas: 352/351, 676/675

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EDOs: 29, 34, 53, 58, 63, 77, 87, 111, 121

== 540/539 ==

=== 540/539 ===

Commas: 540/539, 676/675

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EDOs: 9, 19, 53, 58, 63, 72, 111, 121, 183

== 847/845 ==

=== 847/845 ===

Commas: 676/675, 847/845

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EDOs: 9, 29, 53, 58, 87, 111, 140, 149, 198

= Rank ~~three~~ temperaments =

== Rank-3 temperaments ==

== [[Breed_family|Greenland]] ==

=== [[Breed_family|Greenland]] ===

Commas: 676/675, 1001/1000, 1716/1715

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[[Spectrum_of_a_temperament|Spectrum]]: 15/13, 7/5, 8/7, 7/6, 4/3, 15/14, 5/4, 18/13, 13/12, 14/13, 13/10, 6/5, 16/15, 11/10, 9/7, 9/8, 16/13, 10/9, 14/11, 11/8, 15/11, 12/11, 13/11, 11/9

== [[Werckismic_temperaments|History]] ==

=== [[Werckismic_temperaments|History]] ===

Commas: 364/363, 441/440, 1001/1000

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Spectrum: 11/10, 15/13, 14/11, 4/3, 7/5, 5/4, 11/8, 18/13, 15/11, 13/12, 13/10, 6/5, 8/7, 16/15, 12/11, 13/11, 9/8, 16/13, 15/14, 10/9, 7/6, 11/9, 14/13, 9/7

== Borneo ==

=== Borneo ===

Commas: 676/675, 1001/1000, 3025/3024

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Spectrum: 12/11, 15/13, 11/8, 4/3, 11/10, 18/13, 6/5, 5/4, 13/12, 15/11, 11/9, 13/10, 10/9, 7/5, 16/15, 13/11, 9/8, 16/13, 8/7, 14/11, 15/14, 7/6, 14/13, 9/7

== Sumatra ==

=== Sumatra ===

Commas: 325/324, 385/384, 625/624

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Badness: 0.000680

== [[Cataharry_family|Madagascar]] ==

=== [[Cataharry_family|Madagascar]] ===

Commas: 351/350, 540/539, 676/675

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[[madagascar19]]

== Baffin ==

=== Baffin ===

Commas: 676/675, 1001/1000, 4225/4224

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Spectrum: 15/13, 16/15, 13/12, 4/3, 16/13, 5/4, 18/13, 13/10, 6/5, 9/8, 11/10, 8/7, 7/5, 15/11, 10/9, 13/11, 15/14, 11/8, 7/6, 14/13, 12/11, 9/7, 11/9, 14/11

== Kujuku ==

=== Kujuku ===

Commas: 352/351, 364/363, 676/675

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Spectrum: 15/13, 4/3, 13/10, 9/8, 13/11, 15/11, 12/11, 11/9, 11/8, 14/11, 16/13, 16/15, 11/10, 13/12, 9/7, 5/4, 18/13, 7/6, 6/5, 8/7, 10/9, 14/13, 15/14, 7/5

= Rank ~~two~~ temperaments =

== Rank-2 temperaments ==

It is interesting to note the Graham complexity of 15/13 in these temperaments. This is 18 in hemiennealimmal, 6 in harry, 9 in tritikleismic, 3 in catakleismic, 2 in negri, 2 in buzzard, 12 in quadritikleismic. Catakleismic and buzzard are particularly interesting from an archipelago point of view. Mystery is special case, since the 15/13 part of it belongs to [[29edo|29EDO]] alone.

== Decitonic (aka Decoid) ==

=== Decitonic (aka Decoid) ===

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[[Badness]]: 0.013475

== Avicenna ==

=== Avicenna ===

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[[Badness]]: 0.015557

== Tertiathirds ==

=== Tertiathirds ===

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[[Badness]]: 0.019494

=== 17-limit ===

==== 17-limit ====

Comma list: 676/675, 715/714, 1716/1715, 2025/2023, 4225/4224

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Badness: 0.019107

= Subgroup temperaments =

== Subgroup temperaments ==

== Barbados ==

=== Barbados ===

Subgroup: 2.3.13/5

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Badness: 0.002335

~~===~~ Music ~~===~~

; Music

* [http://micro.soonlabel.com/gene_ward_smith/Others/Sevish/Sevish%20-%20Desert%20Island%20Rain.mp3 Desert Island Rain] in 313et tuned Barbados[9], by [https://soundcloud.com/sevish/desert-island-rain Sevish]

== Trinidad ==

=== Trinidad ===

Subgroup: 2.3.5.13

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EDOs: 15, 19, 34, 53, 87, 140, 193, 246

== [[Chromatic pairs #Tobago|Tobago]] ==

=== [[Chromatic pairs #Tobago|Tobago]] ===

Subgroup: 2.3.11.13/5

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EDOs: 10, 14, 24, 58, 82, 130

== Parizekmic ==

=== Parizekmic ===

Subgroup: 2.3.5.13

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EDOs: 5, 9, 10, 15, 19, 34, 53, 130, 140, 164, 183, 217, 270

~~===~~ Music ~~===~~

; Music

* [http://micro.soonlabel.com/petr_parizek/pp_pump_675.mp3 Petr's Pump], a comma pump based ditty in Parizekmic temperament.

@@ Line 1: / Line 1: @@
-__FORCETOC__
+The '''archipelago''' is a rag-tag collection of various regular temperaments of different ranks, including subgroup temperaments, associated with island temperament: the rank five thirteen limit temperament tempering out the island comma, [[676/675]]. Common to all of them is the observation that two intervals of 15/13 are equated with a fourth. Hence a 1-15/13-4/3 chord is a characteristic island chord, and 15/13 tends to be of low complexity. Also characteristic is the barbados triad, the 1-13/10-3/2 triad, as well as its inversion 1-15/13-3/2, the barbados tetrad, 1-13/10-3/2-26/15, plus the tetrads 1-13/10-3/2-8/5 and 1-13/10-3/2-9/5. The [[just intonation subgroup]] generated by 2, 4/3 and 15/13 is 2.3.13/5, and the barbados triad and tetrad are found in that, while the other two tetrads are found in the larger 2.3.5.13 subgroup.
-The archipelago is a rag-tag collection of various regular temperaments of different ranks, including subgroup temperaments, associated with island temperament: the rank five thirteen limit temperament tempering out the island comma, [[676/675]]. Common to all of them is the observation that two intervals of 15/13 are equated with a fourth. Hence a 1-15/13-4/3 chord is a characteristic island chord, and 15/13 tends to be of low complexity. Also characteristic is the barbados triad, the 1-13/10-3/2 triad, as well as its inversion 1-15/13-3/2, the barbados tetrad, 1-13/10-3/2-26/15, plus the tetrads 1-13/10-3/2-8/5 and 1-13/10-3/2-9/5. The [[Just_intonation_subgroups|just intonation subgroup]] generated by 2, 4/3 and 15/13 is 2.3.13/5, and the barbados triad and tetrad are found in that, while the other two tetrads are found in the larger 2.3.5.13 subgroup.
 The barbados triad is of particular theoretical interest because, when reduced to lowest terms, it is the 10:13:15 triad. Thus, this triad is only slightly higher in complexity than the 5-limit 10:12:15 minor triad, which means it may be of distinct value as a relatively unexplored musical consonance. It is one of only a few low-complexity triads with a 3/2 on the outer dyad, some others being 4:5:6, 6:7:9, and 10:12:15. It works out to 0-454-702 cents, which means that it is an ''ultramajor'' triad, with a third sharper even than the 9/7 supermajor third.
@@ Line 6: / Line 5: @@
 Compared to the 7-limit 14:18:21 supermajor triad, 10:13:15 is lower in triadic complexity (10:13:15 vs 14:18:21), but contains dyads that are on average higher in complexity (9/7 vs 13/10 and 7/6 vs 15/13). Its inverse, however, is the ultraminor 26:30:39, which is far more complex than the 7-limit subminor 6:7:9. Temperaments in which 91/90 vanishes equate the two types of triads.
-[[24edo|24edo]] approximates this triad to within an error of four cents, and [[29edo|29edo]] does even better, getting it to within 1.5 cents; either may be used as a tuning for the barbados temperament discussed below.
+[[24edo]] approximates this triad to within an error of four cents, and [[29edo]] does even better, getting it to within 1.5 cents; either may be used as a tuning for the barbados temperament discussed below.
-= Rank five temperament =
-== Island ==
+== Rank-5 temperaments ==
+=== Island ===
 Subgroup: 2.3.5.7.11.13
 [[Comma list]]: [[676/675]]
-Map:<br>
+[[Mapping]]:<br>
 {{val| 1 0 0 0 0 -1 }}<br>
 {{val| 0 2 0 0 0 3 }}<br>
@@ Line 22: / Line 20: @@
 {{val| 0 0 0 0 1 0 }}
-Vals: 5, 9, 10, 15, 19, 24, 29, 43, 53, 58, 72, 87, 111, 121, 130, 183, 940
+{{Val list|legend=1| 5, 9, 10, 15, 19, 24, 29, 43, 53, 58, 72, 87, 111, 121, 130, 183, 940 }}
 [[Optimal patent val]]: [[940edo]]
-= Rank four temperaments=
+== Rank-4 temperaments ==
-== 1001/1000 ==
+=== 1001/1000 ===
 Commas: 676/675, 1001/1000
@@ Line 36: / Line 34: @@
 [[Optimal_patent_val|Optimal patent val]]: [[940edo]]
-== 49/48 ==
+=== 49/48 ===
 Commas: 49/48, 91/90
@@ Line 43: / Line 41: @@
 EDOs: 5, 9, 10, 15, 19, 24
-== 1716/1715 ==
+=== 1716/1715 ===
 Commas: 676/675, 1716/1715
@@ Line 50: / Line 48: @@
 EDOs: 58, 72, 77, 121, 130, 140, 149, 198, 212, 270
-== 364/363 ==
+=== 364/363 ===
 Commas: 364/363, 676/675
@@ Line 57: / Line 55: @@
 EDOs: 9, 15, 29, 43, 58, 72, 87, 121, 130
-== 351/350 ==
+=== 351/350 ===
 Commas: 351/350, 676/675
@@ Line 64: / Line 62: @@
 EDOs: 19, 53, 58, 72, 77, 111, 130
-== 352/351 ==
+=== 352/351 ===
 Commas: 352/351, 676/675
@@ Line 71: / Line 69: @@
 EDOs: 29, 34, 53, 58, 63, 77, 87, 111, 121
-== 540/539 ==
+=== 540/539 ===
 Commas: 540/539, 676/675
@@ Line 78: / Line 76: @@
 EDOs: 9, 19, 53, 58, 63, 72, 111, 121, 183
-== 847/845 ==
+=== 847/845 ===
 Commas: 676/675, 847/845
@@ Line 85: / Line 83: @@
 EDOs: 9, 29, 53, 58, 87, 111, 140, 149, 198
-= Rank three temperaments =
+== Rank-3 temperaments ==
-== [[Breed_family|Greenland]] ==
+=== [[Breed_family|Greenland]] ===
 Commas: 676/675, 1001/1000, 1716/1715
@@ Line 99: / Line 97: @@
 [[Spectrum_of_a_temperament|Spectrum]]: 15/13, 7/5, 8/7, 7/6, 4/3, 15/14, 5/4, 18/13, 13/12, 14/13, 13/10, 6/5, 16/15, 11/10, 9/7, 9/8, 16/13, 10/9, 14/11, 11/8, 15/11, 12/11, 13/11, 11/9
-== [[Werckismic_temperaments|History]] ==
+=== [[Werckismic_temperaments|History]] ===
 Commas: 364/363, 441/440, 1001/1000
@@ Line 110: / Line 108: @@
 Spectrum: 11/10, 15/13, 14/11, 4/3, 7/5, 5/4, 11/8, 18/13, 15/11, 13/12, 13/10, 6/5, 8/7, 16/15, 12/11, 13/11, 9/8, 16/13, 15/14, 10/9, 7/6, 11/9, 14/13, 9/7
-== Borneo ==
+=== Borneo ===
 Commas: 676/675, 1001/1000, 3025/3024
@@ Line 123: / Line 121: @@
 Spectrum: 12/11, 15/13, 11/8, 4/3, 11/10, 18/13, 6/5, 5/4, 13/12, 15/11, 11/9, 13/10, 10/9, 7/5, 16/15, 13/11, 9/8, 16/13, 8/7, 14/11, 15/14, 7/6, 14/13, 9/7
-== Sumatra ==
+=== Sumatra ===
 Commas: 325/324, 385/384, 625/624
@@ Line 132: / Line 130: @@
 Badness: 0.000680
-== [[Cataharry_family|Madagascar]] ==
+=== [[Cataharry_family|Madagascar]] ===
 Commas: 351/350, 540/539, 676/675
@@ Line 145: / Line 143: @@
 [[madagascar19]]
-== Baffin ==
+=== Baffin ===
 Commas: 676/675, 1001/1000, 4225/4224
@@ Line 158: / Line 156: @@
 Spectrum: 15/13, 16/15, 13/12, 4/3, 16/13, 5/4, 18/13, 13/10, 6/5, 9/8, 11/10, 8/7, 7/5, 15/11, 10/9, 13/11, 15/14, 11/8, 7/6, 14/13, 12/11, 9/7, 11/9, 14/11
-== Kujuku ==
+=== Kujuku ===
 Commas: 352/351, 364/363, 676/675
@@ Line 171: / Line 169: @@
 Spectrum: 15/13, 4/3, 13/10, 9/8, 13/11, 15/11, 12/11, 11/9, 11/8, 14/11, 16/13, 16/15, 11/10, 13/12, 9/7, 5/4, 18/13, 7/6, 6/5, 8/7, 10/9, 14/13, 15/14, 7/5
-= Rank two temperaments =
+== Rank-2 temperaments ==
 Rank two temperaments tempering out 676/675 include the 13-limit versions of [[Ragismic_microtemperaments|hemiennealimmal]], [[Breedsmic_temperaments|harry]], [[Kleismic_family|tritikleismic]], [[Kleismic_family|catakleimsic]], [[Marvel_temperaments|negri]], [[Hemifamity_temperaments|mystery]], [[Hemifamity_temperaments|buzzard]], [[Kleismic_family|quadritikleismic]].
 It is interesting to note the Graham complexity of 15/13 in these temperaments. This is 18 in hemiennealimmal, 6 in harry, 9 in tritikleismic, 3 in catakleismic, 2 in negri, 2 in buzzard, 12 in quadritikleismic. Catakleismic and buzzard are particularly interesting from an archipelago point of view. Mystery is special case, since the 15/13 part of it belongs to [[29edo|29EDO]] alone.
-== Decitonic (aka Decoid) ==
+=== Decitonic (aka Decoid) ===
 {{see also|Breedsmic temperaments#Decoid}}
@@ Line 189: / Line 187: @@
 [[Badness]]: 0.013475
-== Avicenna ==
+=== Avicenna ===
 {{see also|Landscape microtemperaments#Avicenna}}
@@ Line 202: / Line 200: @@
 [[Badness]]: 0.015557
-== Tertiathirds ==
+=== Tertiathirds ===
 {{see also|Wizmic microtemperaments#Tertiathirds}}
@@ Line 215: / Line 213: @@
 [[Badness]]: 0.019494
-=== 17-limit ===
+==== 17-limit ====
 Comma list: 676/675, 715/714, 1716/1715, 2025/2023, 4225/4224
@@ Line 226: / Line 224: @@
 Badness: 0.019107
-= Subgroup temperaments =
+== Subgroup temperaments ==
-== Barbados ==
+=== Barbados ===
 Subgroup: 2.3.13/5
@@ Line 242: / Line 240: @@
 Badness: 0.002335
-=== Music ===
+; Music
 * [http://micro.soonlabel.com/gene_ward_smith/Others/Sevish/Sevish%20-%20Desert%20Island%20Rain.mp3 Desert Island Rain] in 313et tuned Barbados[9], by [https://soundcloud.com/sevish/desert-island-rain Sevish]
-== Trinidad ==
+=== Trinidad ===
 Subgroup: 2.3.5.13
@@ Line 258: / Line 256: @@
 EDOs: 15, 19, 34, 53, 87, 140, 193, 246
-== [[Chromatic pairs #Tobago|Tobago]] ==
+=== [[Chromatic pairs #Tobago|Tobago]] ===
 Subgroup: 2.3.11.13/5
@@ Line 269: / Line 267: @@
 EDOs: 10, 14, 24, 58, 82, 130
-== Parizekmic ==
+=== Parizekmic ===
 Subgroup: 2.3.5.13
@@ Line 280: / Line 278: @@
 EDOs: 5, 9, 10, 15, 19, 34, 53, 130, 140, 164, 183, 217, 270
-=== Music ===
+; Music
 * [http://micro.soonlabel.com/petr_parizek/pp_pump_675.mp3 Petr's Pump], a comma pump based ditty in Parizekmic temperament.