The Archipelago: Difference between revisions

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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 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.

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 interval, 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.

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. [[The Biosphere|Temperaments in which 91/90 vanishes]] equate the two types of triads.

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 intervals 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. [[The Biosphere|Temperaments in which 91/90 vanishes]] equate the two types of triads.

[[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.

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{{~~Val list~~|legend=1| 5, 9, 10, 14cf, 15, 19, 24, 29, 34d, 43, 49f, 53, 58, 72, 87, 111, 121, 130, 183, 198, 270, 940, 1210f }}

{{Optimal ET sequence|legend=1| 5, 9, 10, 14cf, 15, 19, 24, 29, 34d, 43, 49f, 53, 58, 72, 87, 111, 121, 130, 183, 198, 270, 940, 1210f }}

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

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[[Mapping]]: [{{val| 1 0 0 0 4 -1 }}, {{val| 0 2 0 0 -3 3 }}, {{val| 0 0 1 0 2 1 }}, {{val| 0 0 0 1 -1 0 }}]

{{~~Val list~~|legend=1| 14cf, 15, 19, 29, 39df, 43, 53, 58, 72, 87, 111, 130, 183, 198, 270, 940, 1210f }}

{{Optimal ET sequence|legend=1| 14cf, 15, 19, 29, 39df, 43, 53, 58, 72, 87, 111, 130, 183, 198, 270, 940, 1210f }}

=== 49/48 ===

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[[Mapping]]: [{{val| 1 0 0 2 0 -1 }}, {{val| 0 2 0 1 0 3 }}, {{val| 0 0 1 0 0 1 }}, {{val| 0 0 0 0 1 0 }}]

{{~~Val list~~|legend=1| 5, 9, 10, 14cf, 15, 19, 24, 29, 38df, 53d, 67cddef, 105cdddeefff }}

{{Optimal ET sequence|legend=1| 5, 9, 10, 14cf, 15, 19, 24, 29, 38df, 53d, 67cddef, 105cdddeefff }}

=== 1716/1715 ===

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[[Mapping]]: [{{val| 1 0 0 0 -1 -1 }}, {{val| 0 2 0 0 -5 3 }}, {{val| 0 0 1 0 0 1 }}, {{val| 0 0 0 1 3 0 }}]

{{~~Val list~~|legend=1| 58, 72, 121, 130, 193, 198, 270, 940, 1210f }}

{{Optimal ET sequence|legend=1| 58, 72, 121, 130, 193, 198, 270, 940, 1210f }}

=== 364/363 ===

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[[Mapping]]: [{{val| 1 0 0 -1 0 -1 }}, {{val| 0 2 0 1 1 3 }}, {{val| 0 0 1 1 1 1 }}, {{val| 0 0 0 2 1 0 }}]

{{~~Val list~~|legend=1| 14cf, 15, 23deff, 24, 29, 34d, 43, 49f, 58, 72, 87, 121, 130, 193, 217, 289, 338e, 410e }}

{{Optimal ET sequence|legend=1| 14cf, 15, 23deff, 24, 29, 34d, 43, 49f, 58, 72, 87, 121, 130, 193, 217, 289, 338e, 410e }}

=== 351/350 ===

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[[Mapping]]: [{{val| 1 0 0 -2 0 -1 }}, {{val| 0 2 0 9 0 3 }}, {{val| 0 0 1 -1 0 1 }}, {{val| 0 0 0 0 1 0 }}]

{{~~Val list~~|legend=1| 14cf, 19, 24, 34d, 53, 58, 72, 111, 130, 183, 313, 462f }}

{{Optimal ET sequence|legend=1| 14cf, 19, 24, 34d, 53, 58, 72, 111, 130, 183, 313, 462f }}

=== 352/351 ===

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[[Mapping]]: [{{val| 1 0 0 0 -6 -1 }}, {{val| 0 2 0 0 9 3 }}, {{val| 0 0 1 0 1 1 }}, {{val| 0 0 0 1 0 0 }}]

{{~~Val list~~|legend=1| 10, 19e, 24, 29, 34d, 53, 58, 87, 111, 121, 140, 198, 459b, 517bcdf, 657bdf }}

{{Optimal ET sequence|legend=1| 10, 19e, 24, 29, 34d, 53, 58, 87, 111, 121, 140, 198, 459b, 517bcdf, 657bdf }}

=== 540/539 ===

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[[Mapping]]: [{{val| 1 0 0 0 2 -1 }}, {{val| 0 2 0 0 6 3 }}, {{val| 0 0 1 0 1 1 }}, {{val| 0 0 0 1 -2 0 }}]

{{~~Val list~~|legend=1| 9, 10, 14cf, 19, 33cdff, 39df, 48c, 49f, 53, 58, 72, 111, 121, 130, 183, 251e, 304d, 376, 434de }}

{{Optimal ET sequence|legend=1| 9, 10, 14cf, 19, 33cdff, 39df, 48c, 49f, 53, 58, 72, 111, 121, 130, 183, 251e, 304d, 376, 434de }}

=== 847/845 ===

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[[Mapping]]: [{{val| 1 0 0 0 -1 -1 }}, {{val| 0 2 0 0 3 3 }}, {{val| 0 0 1 0 1 1 }}, {{val| 0 0 0 2 -1 0 }}]

{{~~Val list~~|legend=1| 24d, 29, 38df, 49f, 53, 58, 87, 111, 140, 198, 347, 487e, 545c }}

{{Optimal ET sequence|legend=1| 24d, 29, 38df, 49f, 53, 58, 87, 111, 140, 198, 347, 487e, 545c }}

== Rank-3 temperaments ==

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* [[Madagascar]] → [[Cataharry family #Madagascar|Cataharry family]]

: +351/350, 540/539

* [[Hagrid]] → [[Cataharry family #Hagrid|Cataharry family]]

: +243/242, 351/350

* [[Baffin]] → [[Olympic clan #Baffin|Olympic clan]]

: +1001/1000, 4096/4095

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[[POTE generator]]: ~15/13 = 248.917

[[Badness]]: 0.013475

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[[Mapping]]: [{{val| 3 2 8 16 9 8 }}, {{val| 0 8 -3 -22 4 9 }}]

[[CTE|CTE generator]]: ~13/12 = 137.777

[[POTE generator]]: ~13/12 = 137.777

[[Badness]]: 0.015557

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[[POTE generator]]: ~14/13 = 128.8902

{{~~Val list~~|legend=1| 121, 149, 270, 1741bc, 2011bcf, 2281bcf, 2551bcf, 2821bcf, 3091bcff, 3361bcff }}

{{Optimal ET sequence|legend=1| 121, 149, 270, 1741bc, 2011bcf, 2281bcf, 2551bcf, 2821bcf, 3091bcff, 3361bcff }}

[[Badness]]: 0.019494

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POTE generator: ~14/13 = 128.8912

~~Vals:~~ {{~~Val list~~| 121, 149, 270 }}

Badness: 0.019107

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[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~15/13 = 248.621

{{~~Val list~~|legend=1| 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362 }}

{{Optimal ET sequence|legend=1| 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362 }}

[[Badness]]: 0.002335

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[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~15/13 = 248.868

[[Badness]]: ?

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[[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~15/13 = 249.312

[[Tp tuning#T2 tuning|RMS error]]: 0.3533 cents

==== Pakkanian hemipyth ====

[[Subgroup]]: 2.3.11.13/5.17

[[Comma list]]: 221/220, 243/242, 289/288

[[Optimal tuning]]s:

* [[Tp tuning|subgroup CTE]]: ~17/12 = 1\2, ~26/15 = 950.7656 (~15/13 = 249.2344)

* [[Tp tuning|subgroup CWE]]: ~17/12 = 1\2, ~26/15 = 950.6011 (~15/13 = 249.3989)

{{Optimal ET sequence|legend=1| 10, 14, 24, 106, 130, 154, 178*, 202* }}

<nowiki>*</nowiki> wart for 13/5

=== Cata ===

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[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 317.076

{{Optimal ET sequence|legend=1| 15, 19, 34, 53, 87, 140, 193, 246 }}

[[Badness]]: 0.~~394~~

[[Badness]]: 0.00394

=== Taylor ===

Taylor is the "reduction" of [[hemischis]] to the 2.3.5.13 subgroup, tempering out the [[schisma]] in addition to 676/675. It can be reasonably extended to ~~the 2.3.5.13.~~19 ~~subgroup~~ like ~~all~~ schismic temperaments, dubbed ''dakota'' (not to be confused with [[595/594 #Temperaments|dakotismic and/or dakotic]]).

Taylor is the "reduction" of [[hemischis]] to the 2.3.5.13 subgroup, tempering out the [[schisma]] in addition to 676/675. It can be reasonably extended to include harmonic 19 like most schismic temperaments, but even better, the hemifourth may be interpreted as an octave-reduced harmonic 37 ([[37/32]]). The extension is dubbed ''dakota'' (not to be confused with [[595/594 #Temperaments|dakotismic and/or dakotic]]).

[[Subgroup]]: 2.3.5.13

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[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~26/15 = 950.8331

{{Optimal ET sequence|legend=1| 24, 53, 130, 183, 236, 525f, 761ff }}

[[Badness]]: 0.0100

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[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~26/15 = 950.8199

{{Optimal ET sequence|legend=1| 24, 29, 53, 130, 183, 236h, 289h }}

[[Badness]]: 0.00575

===== 2.3.5.13.19.37 subgroup =====

[[Subgroup]]: 2.3.5.13.19.37

[[Comma list]]: 361/360, 481/480, 513/512, 676/675

[[Sval]] [[mapping]]: [{{val| 1 0 15 14 9 6 }}, {{val| 0 2 -16 -13 -6 -1 }}]

[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~26/15 = 950.8187

{{Optimal ET sequence|legend=1| 24, 29, 53, 183, 236h, 289hl, 631fhhll }}

[[Badness]]: 0.00357

=== Parizekmic ===

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[[Sval]] [[mapping]]: [{{val| 1 0 0 -1 }}, {{val| 0 2 0 3 }}, {{val| 0 0 1 1 }}]

{{~~Val list~~|legend=1| 5, 9, 10, 15, 19, 34, 53, 130, 140, 164, 183, 217, 270 }}

{{Optimal ET sequence|legend=1| 5, 9, 10, 15, 19, 34, 53, 130, 140, 164, 183, 217, 270 }}

[[Badness]]: 0.00811 × 10<sup>-3</sup>

@@ Line 1: / Line 1: @@
+{{Technical data page}}
 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 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.
+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 interval, 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.
-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. [[The Biosphere|Temperaments in which 91/90 vanishes]] equate the two types of triads.
+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 intervals 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. [[The Biosphere|Temperaments in which 91/90 vanishes]] equate the two types of triads.
 [[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.
@@ Line 20: / Line 21: @@
 {{val| 0 0 0 0 1 0 }}
-{{Val list|legend=1| 5, 9, 10, 14cf, 15, 19, 24, 29, 34d, 43, 49f, 53, 58, 72, 87, 111, 121, 130, 183, 198, 270, 940, 1210f }}
+{{Optimal ET sequence|legend=1| 5, 9, 10, 14cf, 15, 19, 24, 29, 34d, 43, 49f, 53, 58, 72, 87, 111, 121, 130, 183, 198, 270, 940, 1210f }}
 [[Optimal patent val]]: [[940edo|940]]
@@ Line 32: / Line 33: @@
 [[Mapping]]: [{{val| 1 0 0 0 4 -1 }}, {{val| 0 2 0 0 -3 3 }}, {{val| 0 0 1 0 2 1 }}, {{val| 0 0 0 1 -1 0 }}]
-{{Val list|legend=1| 14cf, 15, 19, 29, 39df, 43, 53, 58, 72, 87, 111, 130, 183, 198, 270, 940, 1210f }}
+{{Optimal ET sequence|legend=1| 14cf, 15, 19, 29, 39df, 43, 53, 58, 72, 87, 111, 130, 183, 198, 270, 940, 1210f }}
 === 49/48 ===
@@ Line 41: / Line 42: @@
 [[Mapping]]: [{{val| 1 0 0 2 0 -1 }}, {{val| 0 2 0 1 0 3 }}, {{val| 0 0 1 0 0 1 }}, {{val| 0 0 0 0 1 0 }}]
-{{Val list|legend=1| 5, 9, 10, 14cf, 15, 19, 24, 29, 38df, 53d, 67cddef, 105cdddeefff }}
+{{Optimal ET sequence|legend=1| 5, 9, 10, 14cf, 15, 19, 24, 29, 38df, 53d, 67cddef, 105cdddeefff }}
 === 1716/1715 ===
@@ Line 50: / Line 51: @@
 [[Mapping]]: [{{val| 1 0 0 0 -1 -1 }}, {{val| 0 2 0 0 -5 3 }}, {{val| 0 0 1 0 0 1 }}, {{val| 0 0 0 1 3 0 }}]
-{{Val list|legend=1| 58, 72, 121, 130, 193, 198, 270, 940, 1210f }}
+{{Optimal ET sequence|legend=1| 58, 72, 121, 130, 193, 198, 270, 940, 1210f }}
 === 364/363 ===
@@ Line 59: / Line 60: @@
 [[Mapping]]: [{{val| 1 0 0 -1 0 -1 }}, {{val| 0 2 0 1 1 3 }}, {{val| 0 0 1 1 1 1 }}, {{val| 0 0 0 2 1 0 }}]
-{{Val list|legend=1| 14cf, 15, 23deff, 24, 29, 34d, 43, 49f, 58, 72, 87, 121, 130, 193, 217, 289, 338e, 410e }}
+{{Optimal ET sequence|legend=1| 14cf, 15, 23deff, 24, 29, 34d, 43, 49f, 58, 72, 87, 121, 130, 193, 217, 289, 338e, 410e }}
 === 351/350 ===
@@ Line 68: / Line 69: @@
 [[Mapping]]: [{{val| 1 0 0 -2 0 -1 }}, {{val| 0 2 0 9 0 3 }}, {{val| 0 0 1 -1 0 1 }}, {{val| 0 0 0 0 1 0 }}]
-{{Val list|legend=1| 14cf, 19, 24, 34d, 53, 58, 72, 111, 130, 183, 313, 462f }}
+{{Optimal ET sequence|legend=1| 14cf, 19, 24, 34d, 53, 58, 72, 111, 130, 183, 313, 462f }}
 === 352/351 ===
@@ Line 77: / Line 78: @@
 [[Mapping]]: [{{val| 1 0 0 0 -6 -1 }}, {{val| 0 2 0 0 9 3 }}, {{val| 0 0 1 0 1 1 }}, {{val| 0 0 0 1 0 0 }}]
-{{Val list|legend=1| 10, 19e, 24, 29, 34d, 53, 58, 87, 111, 121, 140, 198, 459b, 517bcdf, 657bdf }}
+{{Optimal ET sequence|legend=1| 10, 19e, 24, 29, 34d, 53, 58, 87, 111, 121, 140, 198, 459b, 517bcdf, 657bdf }}
 === 540/539 ===
@@ Line 86: / Line 87: @@
 [[Mapping]]: [{{val| 1 0 0 0 2 -1 }}, {{val| 0 2 0 0 6 3 }}, {{val| 0 0 1 0 1 1 }}, {{val| 0 0 0 1 -2 0 }}]
-{{Val list|legend=1| 9, 10, 14cf, 19, 33cdff, 39df, 48c, 49f, 53, 58, 72, 111, 121, 130, 183, 251e, 304d, 376, 434de }}
+{{Optimal ET sequence|legend=1| 9, 10, 14cf, 19, 33cdff, 39df, 48c, 49f, 53, 58, 72, 111, 121, 130, 183, 251e, 304d, 376, 434de }}
 === 847/845 ===
@@ Line 95: / Line 96: @@
 [[Mapping]]: [{{val| 1 0 0 0 -1 -1 }}, {{val| 0 2 0 0 3 3 }}, {{val| 0 0 1 0 1 1 }}, {{val| 0 0 0 2 -1 0 }}]
-{{Val list|legend=1| 24d, 29, 38df, 49f, 53, 58, 87, 111, 140, 198, 347, 487e, 545c }}
+{{Optimal ET sequence|legend=1| 24d, 29, 38df, 49f, 53, 58, 87, 111, 140, 198, 347, 487e, 545c }}
 == Rank-3 temperaments ==
@@ Line 110: / Line 111: @@
 * [[Madagascar]] → [[Cataharry family #Madagascar|Cataharry family]]
 : +351/350, 540/539
+* [[Hagrid]] → [[Cataharry family #Hagrid|Cataharry family]]
+: +243/242, 351/350
 * [[Baffin]] → [[Olympic clan #Baffin|Olympic clan]]
 : +1001/1000, 4096/4095
@@ Line 131: / Line 134: @@
 [[POTE generator]]: ~15/13 = 248.917
-{{Val list|legend=1| 130, 270, 940, 1210f }}
+{{Optimal ET sequence|legend=1| 130, 270, 940, 1210f }}
 [[Badness]]: 0.013475
@@ Line 143: / Line 146: @@
 [[Mapping]]: [{{val| 3 2 8 16 9 8 }}, {{val| 0 8 -3 -22 4 9 }}]
+[[CTE|CTE generator]]: ~13/12 = 137.777
 [[POTE generator]]: ~13/12 = 137.777
-{{Val list|legend=1| 87, 183, 270 }}
+{{Optimal ET sequence|legend=1| 87, 183, 270 }}
 [[Badness]]: 0.015557
@@ Line 161: / Line 166: @@
 [[POTE generator]]: ~14/13 = 128.8902
-{{Val list|legend=1| 121, 149, 270, 1741bc, 2011bcf, 2281bcf, 2551bcf, 2821bcf, 3091bcff, 3361bcff }}
+{{Optimal ET sequence|legend=1| 121, 149, 270, 1741bc, 2011bcf, 2281bcf, 2551bcf, 2821bcf, 3091bcff, 3361bcff }}
 [[Badness]]: 0.019494
@@ Line 174: / Line 179: @@
 POTE generator: ~14/13 = 128.8912
-Vals: {{Val list| 121, 149, 270 }}
+{{Optimal ET sequence|legend=1| 121, 149, 270 }}
 Badness: 0.019107
@@ Line 190: / Line 195: @@
 [[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~15/13 = 248.621
-{{Val list|legend=1| 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362 }}
+{{Optimal ET sequence|legend=1| 5, 9, 14, 19, 24, 29, 53, 82, 111, 140, 251, 362 }}
 [[Badness]]: 0.002335
@@ Line 210: / Line 215: @@
 [[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~15/13 = 248.868
-{{Val list|legend=1| 5, 24, 29, 53, 82, 111, 135 }}
+{{Optimal ET sequence|legend=1| 5, 24, 29, 53, 82, 111, 135 }}
 [[Badness]]: ?
@@ Line 231: / Line 236: @@
 [[Optimal tuning]] ([[Tp tuning|subgroup POTE]]): ~2 = 1\1, ~15/13 = 249.312
-{{Val list|legend=1| 10, 14, 24, 58, 82, 130 }}
+{{Optimal ET sequence|legend=1| 10, 14, 24, 58, 82, 130 }}
 [[Tp tuning#T2 tuning|RMS error]]: 0.3533 cents
+==== Pakkanian hemipyth ====
+[[Subgroup]]: 2.3.11.13/5.17
+[[Comma list]]: 221/220, 243/242, 289/288
+{{Mapping|legend=2| 2 0 -1 -2 5 | 0 2 5 3 2 }}
+[[Optimal tuning]]s:
+* [[Tp tuning|subgroup CTE]]: ~17/12 = 1\2, ~26/15 = 950.7656 (~15/13 = 249.2344)
+* [[Tp tuning|subgroup CWE]]: ~17/12 = 1\2, ~26/15 = 950.6011 (~15/13 = 249.3989)
+{{Optimal ET sequence|legend=1| 10, 14, 24, 106, 130, 154, 178*, 202* }}
+<nowiki>*</nowiki> wart for 13/5
 === Cata ===
@@ Line 249: / Line 270: @@
 [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~6/5 = 317.076
-{{Val list|legend=1| 15, 19, 34, 53, 87, 140, 193, 246 }}
+{{Optimal ET sequence|legend=1| 15, 19, 34, 53, 87, 140, 193, 246 }}
-[[Badness]]: 0.394
+[[Badness]]: 0.00394
 === Taylor ===
-Taylor is the "reduction" of [[hemischis]] to the 2.3.5.13 subgroup, tempering out the [[schisma]] in addition to 676/675. It can be reasonably extended to the 2.3.5.13.19 subgroup like all schismic temperaments, dubbed ''dakota'' (not to be confused with [[595/594 #Temperaments|dakotismic and/or dakotic]]).
+Taylor is the "reduction" of [[hemischis]] to the 2.3.5.13 subgroup, tempering out the [[schisma]] in addition to 676/675. It can be reasonably extended to include harmonic 19 like most schismic temperaments, but even better, the hemifourth may be interpreted as an octave-reduced harmonic 37 ([[37/32]]). The extension is dubbed ''dakota'' (not to be confused with [[595/594 #Temperaments|dakotismic and/or dakotic]]).
 [[Subgroup]]: 2.3.5.13
@@ Line 264: / Line 285: @@
 [[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~26/15 = 950.8331
-{{Val list|legend=1| 24, 53, 130, 183, 236, 525f, 761ff }}
+{{Optimal ET sequence|legend=1| 24, 53, 130, 183, 236, 525f, 761ff }}
 [[Badness]]: 0.0100
@@ Line 277: / Line 298: @@
 [[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~26/15 = 950.8199
-{{Val list|legend=1| 24, 29, 53, 130, 183, 236h, 289h }}
+{{Optimal ET sequence|legend=1| 24, 29, 53, 130, 183, 236h, 289h }}
 [[Badness]]: 0.00575
+===== 2.3.5.13.19.37 subgroup =====
+[[Subgroup]]: 2.3.5.13.19.37
+[[Comma list]]: 361/360, 481/480, 513/512, 676/675
+[[Sval]] [[mapping]]: [{{val| 1 0 15 14 9 6 }}, {{val| 0 2 -16 -13 -6 -1 }}]
+[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~26/15 = 950.8187
+{{Optimal ET sequence|legend=1| 24, 29, 53, 183, 236h, 289hl, 631fhhll }}
+[[Badness]]: 0.00357
 === Parizekmic ===
@@ Line 290: / Line 324: @@
 [[Sval]] [[mapping]]: [{{val| 1 0 0 -1 }}, {{val| 0 2 0 3 }}, {{val| 0 0 1 1 }}]
-{{Val list|legend=1| 5, 9, 10, 15, 19, 34, 53, 130, 140, 164, 183, 217, 270 }}
+{{Optimal ET sequence|legend=1| 5, 9, 10, 15, 19, 34, 53, 130, 140, 164, 183, 217, 270 }}
 [[Badness]]: 0.00811 × 10<sup>-3</sup>