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Kleismic family: Difference between revisions

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| de = Hanson Kleismisch
| en = Kleismic family
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[[6144/6125]], the porwell comma, gives hemikleismic. [[245/243]], sensamagic, gives clyde. [[1029/1024]], the gamelisma, gives tritikleismic. [[2401/2400]] the breedsma, gives quadritikleismic. Hemikleismic splits the 6/5 in half to get a neutral second generator of 35/32, and clyde similarly splits the 5/3 in half to get a 9/7 generator. Finally, tritikleismic has a 1/3-octave period with minor third generator, and quadritikleismic a 1/4-octave period with the minor third generator.
[[6144/6125]], the porwell comma, gives hemikleismic. [[245/243]], sensamagic, gives clyde. [[1029/1024]], the gamelisma, gives tritikleismic. [[2401/2400]] the breedsma, gives quadritikleismic. Hemikleismic splits the 6/5 in half to get a neutral second generator of 35/32, and clyde similarly splits the 5/3 in half to get a 9/7 generator. Finally, tritikleismic has a 1/3-octave period with minor third generator, and quadritikleismic a 1/4-octave period with the minor third generator.
== Keemun ==
{{Main| Keemun }}
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 49/48, 126/125
{{Mapping|legend=1| 1 0 1 2 | 0 6 5 3 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~6/5 = 316.473{{c}}
[[Tuning ranges]]:
* 7-odd-limit [[diamond monotone]]: ~6/5 = [300.000, 327.273] (1\4 to 3\11)
* 9-odd-limit diamond monotone: ~6/5 = [315.789, 320.000] (5\19 to 4\15)
* 7- and 9-odd-limit [[diamond tradeoff]]: ~6/5 = [308.744, 322.942]
{{Optimal ET sequence|legend=1| 15, 19, 53d, 72dd, 91dd }}
[[Badness]] (Smith): 0.027408
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 49/48, 56/55, 100/99
Mapping: {{mapping| 1 0 1 2 4 | 0 6 5 3 -2 }}
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 317.576{{c}}
Tuning ranges:
* 11-odd-limit diamond monotone: ~6/5 = [315.789, 320.000] (5\19 to 4\15)
* 11-odd-limit diamond tradeoff: ~6/5 = [308.744, 324.341]
{{Optimal ET sequence|legend=0| 4, 15, 19, 34 }}
Badness (Smith): 0.027410
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 49/48, 56/55, 65/64, 100/99
Mapping: {{mapping| 1 0 1 2 4 5 | 0 6 5 3 -2 -5 }}
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 316.611{{c}}
Tuning ranges:
* 13- and 15-odd-limit diamond monotone: ~6/5 = 315.789 (5\19)
* 13- and 15-odd-limit diamond tradeoff: ~6/5 = [303.597, 324.341]
{{Optimal ET sequence|legend=0| 4, 15f, 19, 53def, 72def }}
Badness (Smith): 0.029749
==== Kema ====
Subgroup: 2.3.5.7.11.13
Comma list: 49/48, 56/55, 91/90, 100/99
Mapping: {{mapping| 1 0 1 2 4 0 | 0 6 5 3 -2 14 }}
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 317.423{{c}}
Tuning ranges:
* 13-odd-limit diamond monotone: ~6/5 = [315.789, 320.000] (5\19 to 4\15)
* 15-odd-limit diamond monotone: ~6/5 = 315.789 (5\19)
* 13- and 15-odd-limit diamond tradeoff: ~6/5 = [308.744, 324.341]
{{Optimal ET sequence|legend=0| 15, 19, 34, 87ddee }}
Badness (Smith): 0.022749
==== Kumbaya ====
Subgroup: 2.3.5.7.11.13
Comma list: 40/39, 49/48, 56/55, 66/65
Mapping: {{mapping| 1 0 1 2 4 4 | 0 6 5 3 -2 -1 }}
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 318.595{{c}}
{{Optimal ET sequence|legend=0| 4, 15, 19f, 34ff }}
Badness (Smith): 0.031633
=== Qeema ===
Subgroup: 2.3.5.7.11
Comma list: 45/44, 49/48, 126/125
Mapping: {{mapping| 1 0 1 2 -1 | 0 6 5 3 17 }}
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 314.730{{c}}
{{Optimal ET sequence|legend=0| 4e, 19, 42bcd, 61bcdd }}
Badness (Smith): 0.040056
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 45/44, 49/48, 78/77, 126/125
Mapping: {{mapping| 1 0 1 2 -1 0 | 0 6 5 3 17 14 }}
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 315.044{{c}}
{{Optimal ET sequence|legend=0| 4ef, 19 }}
Badness (Smith): 0.029419
=== Darjeeling ===
Subgroup: 2.3.5.7.11
Comma list: 49/48, 55/54, 77/75
Mapping: {{mapping| 1 0 1 2 0 | 0 6 5 3 13 }}
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 317.656{{c}}
{{Optimal ET sequence|legend=0| 15, 19e, 34e }}
Badness (Smith): 0.027648
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 49/48, 55/54, 66/65, 77/75
Mapping: {{mapping| 1 0 1 2 0 0 | 0 6 5 3 13 14 }}
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 317.298{{c}}
{{Optimal ET sequence|legend=0| 15, 19e, 34e, 53dee }}
Badness (Smith): 0.021445
== Catalan ==
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 64/63, 15625/15552
{{Mapping|legend=1| 1 0 1 6 | 0 6 5 -12 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~6/5 = 318.267{{c}}
[[Tuning ranges]]:
* 7- and 9-odd-limit [[diamond monotone]]: ~6/5 = [317.647, 320.000] (9\34 to 4\15)
* 7- and 9-odd-limit [[diamond tradeoff]]: ~6/5 = [315.641, 319.265]
{{Optimal ET sequence|legend=1| 15, 34d, 49, 132bcdd, 181bbcddd }}
[[Badness]] (Smith): 0.094872
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 64/63, 100/99, 1331/1323
Mapping: {{mapping| 1 0 1 6 4 | 0 6 5 -12 -2 }}
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 318.282{{c}}
Tuning ranges:
* 11-odd-limit diamond monotone: ~6/5 = [317.647, 320.000] (9\34 to 4\15)
* 11-odd-limit diamond tradeoff: ~6/5 = [315.641, 324.341]
{{Optimal ET sequence|legend=0| 15, 34d, 49, 181bbcdddeee }}
Badness (Smith): 0.036894
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 64/63, 100/99, 144/143, 275/273
Mapping: {{mapping| 1 0 1 6 4 0 | 0 6 5 -12 -2 14 }}
Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~6/5 = 317.9159{{c}}
{{Optimal ET sequence|legend=0| 15, 34d, 49f, 83def, 132bcddeefff }}
Badness (Smith): 0.0263


== Catakleismic ==
== Catakleismic ==
Line 420: Line 236:


Badness (Smith): 0.015771
Badness (Smith): 0.015771
== Keemun ==
{{Main| Keemun }}
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 49/48, 126/125
{{Mapping|legend=1| 1 0 1 2 | 0 6 5 3 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~6/5 = 316.473{{c}}
[[Tuning ranges]]:
* 7-odd-limit [[diamond monotone]]: ~6/5 = [300.000, 327.273] (1\4 to 3\11)
* 9-odd-limit diamond monotone: ~6/5 = [315.789, 320.000] (5\19 to 4\15)
* 7- and 9-odd-limit [[diamond tradeoff]]: ~6/5 = [308.744, 322.942]
{{Optimal ET sequence|legend=1| 15, 19, 53d, 72dd, 91dd }}
[[Badness]] (Smith): 0.027408
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 49/48, 56/55, 100/99
Mapping: {{mapping| 1 0 1 2 4 | 0 6 5 3 -2 }}
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 317.576{{c}}
Tuning ranges:
* 11-odd-limit diamond monotone: ~6/5 = [315.789, 320.000] (5\19 to 4\15)
* 11-odd-limit diamond tradeoff: ~6/5 = [308.744, 324.341]
{{Optimal ET sequence|legend=0| 4, 15, 19, 34 }}
Badness (Smith): 0.027410
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 49/48, 56/55, 65/64, 100/99
Mapping: {{mapping| 1 0 1 2 4 5 | 0 6 5 3 -2 -5 }}
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 316.611{{c}}
Tuning ranges:
* 13- and 15-odd-limit diamond monotone: ~6/5 = 315.789 (5\19)
* 13- and 15-odd-limit diamond tradeoff: ~6/5 = [303.597, 324.341]
{{Optimal ET sequence|legend=0| 4, 15f, 19, 53def, 72def }}
Badness (Smith): 0.029749
==== Kema ====
Subgroup: 2.3.5.7.11.13
Comma list: 49/48, 56/55, 91/90, 100/99
Mapping: {{mapping| 1 0 1 2 4 0 | 0 6 5 3 -2 14 }}
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 317.423{{c}}
Tuning ranges:
* 13-odd-limit diamond monotone: ~6/5 = [315.789, 320.000] (5\19 to 4\15)
* 15-odd-limit diamond monotone: ~6/5 = 315.789 (5\19)
* 13- and 15-odd-limit diamond tradeoff: ~6/5 = [308.744, 324.341]
{{Optimal ET sequence|legend=0| 15, 19, 34, 87ddee }}
Badness (Smith): 0.022749
==== Kumbaya ====
Subgroup: 2.3.5.7.11.13
Comma list: 40/39, 49/48, 56/55, 66/65
Mapping: {{mapping| 1 0 1 2 4 4 | 0 6 5 3 -2 -1 }}
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 318.595{{c}}
{{Optimal ET sequence|legend=0| 4, 15, 19f, 34ff }}
Badness (Smith): 0.031633
=== Qeema ===
Subgroup: 2.3.5.7.11
Comma list: 45/44, 49/48, 126/125
Mapping: {{mapping| 1 0 1 2 -1 | 0 6 5 3 17 }}
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 314.730{{c}}
{{Optimal ET sequence|legend=0| 4e, 19, 42bcd, 61bcdd }}
Badness (Smith): 0.040056
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 45/44, 49/48, 78/77, 126/125
Mapping: {{mapping| 1 0 1 2 -1 0 | 0 6 5 3 17 14 }}
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 315.044{{c}}
{{Optimal ET sequence|legend=0| 4ef, 19 }}
Badness (Smith): 0.029419
=== Darjeeling ===
Subgroup: 2.3.5.7.11
Comma list: 49/48, 55/54, 77/75
Mapping: {{mapping| 1 0 1 2 0 | 0 6 5 3 13 }}
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 317.656{{c}}
{{Optimal ET sequence|legend=0| 15, 19e, 34e }}
Badness (Smith): 0.027648
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Comma list: 49/48, 55/54, 66/65, 77/75
Mapping: {{mapping| 1 0 1 2 0 0 | 0 6 5 3 13 14 }}
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 317.298{{c}}
{{Optimal ET sequence|legend=0| 15, 19e, 34e, 53dee }}
Badness (Smith): 0.021445
== Catalan ==
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 64/63, 15625/15552
{{Mapping|legend=1| 1 0 1 6 | 0 6 5 -12 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~6/5 = 318.267{{c}}
[[Tuning ranges]]:
* 7- and 9-odd-limit [[diamond monotone]]: ~6/5 = [317.647, 320.000] (9\34 to 4\15)
* 7- and 9-odd-limit [[diamond tradeoff]]: ~6/5 = [315.641, 319.265]
{{Optimal ET sequence|legend=1| 15, 34d, 49, 132bcdd, 181bbcddd }}
[[Badness]] (Smith): 0.094872
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 64/63, 100/99, 1331/1323
Mapping: {{mapping| 1 0 1 6 4 | 0 6 5 -12 -2 }}
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~6/5 = 318.282{{c}}
Tuning ranges:
* 11-odd-limit diamond monotone: ~6/5 = [317.647, 320.000] (9\34 to 4\15)
* 11-odd-limit diamond tradeoff: ~6/5 = [315.641, 324.341]
{{Optimal ET sequence|legend=0| 15, 34d, 49, 181bbcdddeee }}
Badness (Smith): 0.036894
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 64/63, 100/99, 144/143, 275/273
Mapping: {{mapping| 1 0 1 6 4 0 | 0 6 5 -12 -2 14 }}
Optimal tuning (CTE): ~2 = 1200.0000{{c}}, ~6/5 = 317.9159{{c}}
{{Optimal ET sequence|legend=0| 15, 34d, 49f, 83def, 132bcddeefff }}
Badness (Smith): 0.0263


== Countercata ==
== Countercata ==
Retrieved from "https://en.xen.wiki/w/Kleismic_family"