The Archipelago: Difference between revisions
Cmloegcmluin (talk | contribs) "optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence |
Tristanbay (talk | contribs) →Rank-3 temperaments: Added Hagrid to notable rank-3 temperaments |
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{{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 '''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 | 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 | 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. | [[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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* [[Madagascar]] → [[Cataharry family #Madagascar|Cataharry family]] | * [[Madagascar]] → [[Cataharry family #Madagascar|Cataharry family]] | ||
: +351/350, 540/539 | : +351/350, 540/539 | ||
* [[Hagrid]] → [[Cataharry family #Hagrid|Cataharry family]] | |||
: +243/242, 351/350 | |||
* [[Baffin]] → [[Olympic clan #Baffin|Olympic clan]] | * [[Baffin]] → [[Olympic clan #Baffin|Olympic clan]] | ||
: +1001/1000, 4096/4095 | : +1001/1000, 4096/4095 | ||
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[[Mapping]]: [{{val| 3 2 8 16 9 8 }}, {{val| 0 8 -3 -22 4 9 }}] | [[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 | [[POTE generator]]: ~13/12 = 137.777 | ||
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[[Tp tuning#T2 tuning|RMS error]]: 0.3533 cents | [[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 === | === Cata === | ||
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{{Optimal ET sequence|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. | [[Badness]]: 0.00394 | ||
=== Taylor === | === Taylor === | ||