|
|
| Line 312: |
Line 312: |
| {{Optimal ET sequence|legend=1|3219c, 4884, 8103, 12987}}, ... | | {{Optimal ET sequence|legend=1|3219c, 4884, 8103, 12987}}, ... |
|
| |
|
| == 118th-octave temperaments ==
| |
| [[118edo]] is accurate for harmonics 3 and 5, so various 118th-octave temperaments actually make sense.
| |
|
| |
|
| === Parakleischis ===
| |
| 118edo and its multiples are members of both [[parakleismic]] and [[Schismatic family|schismic]], and from this it derives its name.
| |
|
| |
| [[Subgroup]]: 2.3.5.7
| |
|
| |
| [[Comma list]]: 32805/32768, 1224440064/1220703125
| |
|
| |
| [[Mapping]]: [{{val| 118 187 274 0 }}, {{val| 0 0 0 1 }}]
| |
|
| |
| Mapping generators: ~15625/15552, ~7
| |
|
| |
| [[Optimal tuning]] ([[POTE]]): ~7/4 = 968.7235
| |
|
| |
| {{Optimal ET sequence|legend=1| 118, 236, 354, 472, 2242, 2714b, 3186b, 3658b }}
| |
|
| |
| [[Badness]]: 0.145166
| |
|
| |
| ==== 11-limit ====
| |
| Subgroup: 2.3.5.7.11
| |
|
| |
| Comma list: 9801/9800, 32805/32768, 137781/137500
| |
|
| |
| Mapping: [{{val| 118 187 274 0 77 }}, {{val| 0 0 0 1 1 }}]
| |
|
| |
| Optimal tuning (POTE): ~7/4 = 968.5117
| |
|
| |
| {{Optimal ET sequence|legend=1| 118, 354, 472 }}
| |
|
| |
| Badness: 0.049316
| |
|
| |
| ==== Centenniamajor ====
| |
| Named after the fact that 18 is the age of majority in most countries, and 100 (centennial) + 18 (major) = 118.
| |
|
| |
| Subgroup: 2.3.5.7.11
| |
|
| |
| Comma list: 32805/32768, 151263/151250, 1224440064/1220703125
| |
|
| |
| Mapping: [{{val| 118 187 274 0 -420 }}, {{val| 0 0 0 2 5 }}]
| |
|
| |
| Mapping generators: ~15625/15552, ~405504/153125
| |
|
| |
| Optimal tuning (CTE): ~202752/153125 = 484.4837
| |
|
| |
| {{Optimal ET sequence|legend=1| 354, 944e, 1298 }}
| |
|
| |
| Badness: 0.357
| |
|
| |
| ===== 13-limit =====
| |
| Subgroup: 2.3.5.7.11.13
| |
|
| |
| Comma list: 1716/1715, 32805/32768, 34398/34375, 384912/384475
| |
|
| |
| Mapping: [{{val| 118 187 274 0 -420 271 }}, {{val| 0 0 0 2 5 1 }}]
| |
|
| |
| Optimal tuning (CTE): ~8125/6144 = 484.4867
| |
|
| |
| {{Optimal ET sequence|legend=1| 354, 944e, 1298 }}
| |
|
| |
| Badness: 0.122
| |
|
| |
| === Oganesson ===
| |
| Named after the 118th element. In the 13-limit, the period corresponds to [[169/168]], and in the 17-limit, it corresponds also to [[170/169]], meaning that [[28561/28560]] is tempered out. As opposed to being an extension of parakleischis, this has the generator that splits the third harmonic into three equal parts.
| |
|
| |
| In the 7-limit and 11-limit, the period corresponds to [[bronzisma]].
| |
|
| |
| [[Subgroup]]: 2.3.5.7
| |
|
| |
| [[Comma list]]: {{monzo| 30 10 -27 6 }}, {{monzo| 77 -20 -5 -12 }}
| |
|
| |
| [[Mapping]]: [{{val| 118 0 274 643 }}, {{val| 0 3 0 -5 }}]
| |
|
| |
| Mapping generators: ~2097152/2083725, ~1953125/1354752
| |
|
| |
| [[Optimal tuning]] ([[CTE]]): ~1953125/1354752 = 634.0068
| |
|
| |
| {{Optimal ET sequence|legend=1| 354, 2360, 2714, 3068, 3442, 3776, 7198cd, 10974bccdd }}
| |
|
| |
| [[Badness]]: 2.66
| |
|
| |
| ==== 11-limit ====
| |
| Subgroup: 2.3.5.7.11
| |
|
| |
| Comma list: 9801/9800, {{monzo| 13 -1 4 -16 7 }}, {{monzo| 55 -7 -15 -2 -1 }}
| |
|
| |
| Mapping: [{{val| 118 0 274 643 1094 }}, {{val| 0 3 0 -5 -11 }}]
| |
|
| |
| Optimal tuning (CTE): ~1953125/1354752 = 634.0085
| |
|
| |
| {{Optimal ET sequence|legend=1| 354, 3068e, 3442, 3776, 11682ccdde }}
| |
|
| |
| Badness: 0.568
| |
|
| |
| ==== 13-limit ====
| |
| Subgroup: 2.3.5.7.11.13
| |
|
| |
| Comma list: 4096/4095, 9801/9800, 537403776/537109375, 453874312332/453857421875
| |
|
| |
| Mapping: [{{val| 118 0 274 643 1094 499}}, {{val| 0 3 0 -5 -11 -1}}]
| |
|
| |
| Mapping generators: ~169/168, ~1124864/779625
| |
|
| |
| Optimal tuning (CTE): ~1124864/779625 = 634.0087
| |
|
| |
| {{Optimal ET sequence|legend=1| 354, 3068e, 3422, 3776 }}
| |
|
| |
| Badness: 0.172
| |
|
| |
| ==== 17-limit ====
| |
| Subgroup: 2.3.5.7.11.13.17
| |
|
| |
| Comma list: 4096/4095, 9801/9800, 34391/34375, 361250/361179, 562432/562275
| |
|
| |
| Mapping: [{{val| 118 0 274 643 1094 499 607 }}, {{val| 0 3 0 -5 -11 -1 2 }}]
| |
|
| |
| Mapping generators: ~170/169, ~238/165
| |
|
| |
| Optimal tuning (CTE): ~238/165 = 634.0080
| |
|
| |
| {{Optimal ET sequence|legend=1| 354, 3068e, 3422, 3776 }}
| |
|
| |
| Badness: 0.105
| |
|
| |
|
| [[Category:Temperament collections]] | | [[Category:Temperament collections]] |
| [[Category:Rank 2]] | | [[Category:Rank 2]] |
| [[Category:Lists of temperaments]] | | [[Category:Lists of temperaments]] |