Marvel temperaments: Difference between revisions
→Decic: make it explicit that it's a 10th-octave temp |
Move hendeca here |
||
| Line 26: | Line 26: | ||
* ''[[Amavil]]'' (+17496/16807) → [[Mabila family #Amavil|Mabila family]] | * ''[[Amavil]]'' (+17496/16807) → [[Mabila family #Amavil|Mabila family]] | ||
* ''[[Betic]]'' (+1071875/1062882) → [[Sycamore family #Betic|Sycamore family]] | * ''[[Betic]]'' (+1071875/1062882) → [[Sycamore family #Betic|Sycamore family]] | ||
* [[Compton]] (+250047/250000) → [[Compton family #Compton|Compton family]] | * [[Compton]] (+250047/250000) → [[Compton family #Compton|Compton family]] | ||
* ''[[Raccoon]]'' (+41943040/40353607) → [[Vavoom family #Raccoon|Vavoom family]] | * ''[[Raccoon]]'' (+41943040/40353607) → [[Vavoom family #Raccoon|Vavoom family]] | ||
| Line 32: | Line 31: | ||
* ''[[Gammy]]'' (+94143178827/91913281250) → [[Gammic family #Gammy|Gammic family]] | * ''[[Gammy]]'' (+94143178827/91913281250) → [[Gammic family #Gammy|Gammic family]] | ||
Considered below are wizard, tritonic, septimin, merman, slender, triton, marvolo, decic, enneaportent, gracecordial, alphorn, misneb, untriton, naiadical, quintannic, gwazy, and tertiosec, in the order of increasing [[badness]]. | Considered below are wizard, tritonic, septimin, merman, slender, triton, marvolo, decic, enneaportent, gracecordial, alphorn, misneb, untriton, naiadical, quintannic, hendeca, gwazy, and tertiosec, in the order of increasing [[badness]]. | ||
Since {{nowrap|(5/4)<sup>2</sup> {{=}} (225/224)⋅(14/9)}}, these temperaments tend to have a relatively small complexity for 5/4. They also possess a version of the augmented triad where each third approximates either 5/4 or 9/7. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before 12edo established itself as the standard tuning, it is arguably more authentic to tune it as two stacked major thirds and a diminished fourth, which is what it is in meantone, than as the modern version of three stacked very sharp major thirds. | Since {{nowrap|(5/4)<sup>2</sup> {{=}} (225/224)⋅(14/9)}}, these temperaments tend to have a relatively small complexity for 5/4. They also possess a version of the augmented triad where each third approximates either 5/4 or 9/7. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before 12edo established itself as the standard tuning, it is arguably more authentic to tune it as two stacked major thirds and a diminished fourth, which is what it is in meantone, than as the modern version of three stacked very sharp major thirds. | ||
| Line 1,273: | Line 1,272: | ||
Badness (Sintel): 1.17 | Badness (Sintel): 1.17 | ||
== Hendeca == | |||
{{Distinguish| Hendec }} | |||
Hendeca has a period of 1/11 octave, nine of them represent [[7/4]] in the 7-limit, the same as [[keemic temperaments #Undeka|undeka]]. It can be tuned to [[22edo]] or [[33edo]] using the [[patent val]], or [[55edo]] using the 55d val. | |||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: 225/224, 122880/117649 | |||
{{Mapping|legend=1| 11 0 43 31 | 0 1 -1 0 }} | |||
: mapping generators: ~16/15, ~3 | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~16/15 = 108.9526{{c}}, ~3/2 = 702.4215{{c}} | |||
: [[error map]]: {{val| -1.521 -1.055 -2.252 +8.705 }} | |||
* [[CWE]]: ~16/15 = 109.0909{{c}}, ~3/2 = 703.2071{{c}} | |||
: error map: {{val| 0.000 +1.252 +1.388 +12.992 }} | |||
{{Optimal ET sequence|legend=1| 22, 55d, 77d, 99dd }} | |||
[[Badness]] (Sintel): 4.37 | |||
=== 11-limit === | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 121/120, 225/224, 352/343 | |||
Mapping: {{mapping| 11 0 43 31 38 | 0 1 -1 0 0 }} | |||
Optimal tunings: | |||
* WE: ~16/15 = 109.0109{{c}}, ~3/2 = 702.5746{{c}} | |||
* CWE: ~16/15 = 109.0909{{c}}, ~3/2 = 703.0395{{c}} | |||
{{Optimal ET sequence|legend=0| 22, 55d }} | |||
Badness (Sintel): 2.46 | |||
== Gwazy == | == Gwazy == | ||