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Kleismic family: Difference between revisions

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== Keemun ==
== Keemun ==
{{Main| Keemun }}
{{Main| Keemun }}
Keemun tempers out [[49/48]] and [[126/125]] and may be described as the 15 & 19 temperament.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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==== 13-limit ====
==== 13-limit ====
This extension was known as ''kema'' in earlier materials such as the [https://x31eq.com/temper/ Temperament Finder]. It is considered the canonical extension here because of the naturalness of mapping 15/13 to the hemifourth.
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 49/48, 56/55, 65/64, 100/99
Comma list: 49/48, 56/55, 91/90, 100/99


{{Mapping|legend=0| 1 0 1 2 4 5 | 0 6 5 3 -2 -5 }}
{{Mapping|legend=0| 1 0 1 2 4 0 | 0 6 5 3 -2 14 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1201.8360{{c}}, ~6/5 = 317.0958{{c}}
* WE: ~2 = 1199.7816{{c}}, ~6/5 = 317.3653{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.6829{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.4070{{c}}


Tuning ranges:  
Tuning ranges:  
* 13- and 15-odd-limit diamond monotone: ~6/5 = 315.789 (5\19)
* 13-odd-limit diamond monotone: ~6/5 = [315.789, 320.000] (5\19 to 4\15)
* 13- and 15-odd-limit diamond tradeoff: ~6/5 = [303.597, 324.341]
* 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 }}


{{Optimal ET sequence|legend=0| 4, 15f, 19 }}
Badness (Sintel): 0.940


Badness (Sintel): 1.23
==== Kimah ====
This extension used to be considered the canonical 13-limit keemun, decanonized in 2026 in favor of the above one.  


==== Kema ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 49/48, 56/55, 91/90, 100/99
Comma list: 49/48, 56/55, 65/64, 100/99


{{Mapping|legend=0| 1 0 1 2 4 0 | 0 6 5 3 -2 14 }}
{{Mapping|legend=0| 1 0 1 2 4 5 | 0 6 5 3 -2 -5 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.7816{{c}}, ~6/5 = 317.3653{{c}}
* WE: ~2 = 1201.8360{{c}}, ~6/5 = 317.0958{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 317.4070{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~6/5 = 316.6829{{c}}


Tuning ranges:  
Tuning ranges:  
* 13-odd-limit diamond monotone: ~6/5 = [315.789, 320.000] (5\19 to 4\15)
* 13- and 15-odd-limit diamond monotone: ~6/5 = 315.789 (5\19)
* 15-odd-limit diamond monotone: ~6/5 = 315.789 (5\19)
* 13- and 15-odd-limit diamond tradeoff: ~6/5 = [303.597, 324.341]
* 13- and 15-odd-limit diamond tradeoff: ~6/5 = [308.744, 324.341]


{{Optimal ET sequence|legend=0| 15, 19, 34 }}
{{Optimal ET sequence|legend=0| 4, 15f, 19 }}


Badness (Sintel): 0.940
Badness (Sintel): 1.23


==== Kumbaya ====
==== Kumbaya ====
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