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Trisedodge family: Difference between revisions

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=== Overview to extensions ===
=== Overview to extensions ===
The second comma of the comma list defines which 7-limit family member we are looking at. Among these are septimal trisedodge (65d & 80), which adds [[4000/3993]], and coblack (50 & 65), which adds [[126/125]]. Remarkably, septimal trisedodge admits an extension to the full [[29-limit]], which, except for prime 13, is obvious and simple a way to extend the 11-limit representation.  
The second comma of the comma list defines which 7-limit family member we are looking at. Among these are septimal trisedodge (65d & 80), which adds [[4000/3993]], coblack (50 & 65), which adds [[126/125]], and decistearn (50 & 80), which adds [[3136/3125]] and splits the period in two. Remarkably, septimal trisedodge admits an extension to the full [[29-limit]], which, except for prime 13, is obvious and simple a way to extend the 11-limit representation.  


Temperaments discussed elsewhere include [[15th-octave temperaments #Quindecic|quindecic]] and [[Stearnsmic clan #Decistearn|decistearn]]. Considered below are trisedodge and coblack.  
Temperaments discussed elsewhere include [[15th-octave temperaments #Quindecic|quindecic]]. The rest are considered below.  


Septimal trisedodge and coblack have the common [[2.3.5.11-subgroup|2.3.5.11 subgroup]] [[restriction]], called countdown, considered immediately below. In this temperament, the generator can be taken to be ~11/10, reached as a period minus 25/24, that is, (55/48)/(25/24) = 11/10. Therefore, since a period plus a gen is 6/5 and a period minus a gen is 11/10, we reach 12/11 in 2 generator steps.
Septimal trisedodge and coblack have the common [[2.3.5.11-subgroup|2.3.5.11 subgroup]] [[restriction]], called countdown, considered immediately below. In this temperament, the generator can be taken to be ~11/10, reached as a period minus 25/24, that is, (55/48)/(25/24) = 11/10. Therefore, since a period plus a gen is 6/5 and a period minus a gen is 11/10, we reach 12/11 in 2 generator steps.
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Badness (Sintel): 1.49
Badness (Sintel): 1.49
== Decistearn ==
Named by [[Xenllium]] in 2021, the decistearn temperament has a period of 1/10-octave and tempers out 3136/3125, the [[hemimean comma]], 118098/117649, the [[stearnsma]], as well as the [[linus comma]]. It may be described as 80 & 130.
By equating 16/13 with 69/56 and 85/69 (enabling 56:69:85 chords), we find [[23/16]] at 2 generators underneath 6 periods (which is the [[5edo]] fifth, [[10edo|6\10]]).
[[Subgroup]]: 2.3.5.7
[[Comma list]]: 3136/3125, 118098/117649
{{Mapping|legend=1| 10 2 14 5 | 0 3 2 5 }}
: mapping generators: ~15/14, ~135/98
[[Optimal tuning]]s:
* [[WE]]: ~15/14 = 119.9752{{c}}, ~135/98 = 553.8455{{c}}
: [[error map]]: {{val| -0.248 -0.468 +1.030 +0.277 }}
* [[CWE]]: ~15/14 = 120.0000{{c}}, ~135/98 = 553.8935{{c}}
: error map: {{val| 0.000 -0.275 +1.473 +0.642 }}
{{Optimal ET sequence|legend=1| 50, 80, 130, 470cd, 600cd, 730cd }}
[[Badness]] (Sintel): 2.42
=== 11-limit ===
Subgroup: 2.3.5.7.11
Comma list: 540/539, 3136/3125, 4000/3993
Mapping: {{mapping| 10 2 14 5 30 | 0 3 2 5 1 }}
Optimal tunings:
* WE: ~15/14 = 119.9534{{c}}, ~11/8 = 553.8639{{c}}
* CWE: ~15/14 = 120.0000{{c}}, ~11/8 = 553.9779{{c}}
{{Optimal ET sequence|legend=0| 50, 80, 130, 210e, 340ce, 470cdee, 810ccddeee }}
Badness (Sintel): 1.27
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 351/350, 364/363, 540/539, 3136/3125
Mapping: {{mapping| 10 2 14 5 30 37 | 0 3 2 5 1 0 }}
Optimal tunings:
* WE: ~15/14 = 119.9741{{c}}, ~11/8 = 553.8275{{c}}
* CWE: ~15/14 = 120.0000{{c}}, ~11/8 = 553.9051{{c}}
{{Optimal ET sequence|legend=0| 50, 80, 130 }}
Badness (Sintel): 1.11
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Comma list: 221/220, 289/288, 351/350, 540/539, 1632/1625
Mapping: {{mapping| 10 2 14 5 30 37 27 | 0 3 2 5 1 0 3 }}
Optimal tunings:
* WE: ~15/14 = 119.9899{{c}}, ~11/8 = 553.8840{{c}}
* CWE: ~15/14 = 120.0000{{c}}, ~11/8 = 553.9141{{c}}
{{Optimal ET sequence|legend=0| 50, 80, 130 }}
Badness (Sintel): 1.23
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 221/220, 289/288, 351/350, 361/360, 456/455, 476/475
Mapping: {{mapping| 10 2 14 5 30 37 27 24 | 0 3 2 5 1 0 3 4 }}
Optimal tunings:
* WE: ~15/14 = 119.9949{{c}}, ~11/8 = 553.9253{{c}}
* CWE: ~15/14 = 120.0000{{c}}, ~11/8 = 553.9402{{c}}
{{Optimal ET sequence|legend=0| 50, 80, 130 }}
Badness (Sintel): 1.11
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 221/220, 289/288, 351/350, 361/360, 456/455, 476/475, 897/896
Mapping: {{mapping| 10 2 14 5 30 37 27 24 36 | 0 3 2 5 1 0 3 4 2 }}
Optimal tunings:
* WE: ~15/14 = 119.9973{{c}}, ~11/8 = 553.9247{{c}}
* CWE: ~15/14 = 120.0000{{c}}, ~11/8 = 553.9331{{c}}
{{Optimal ET sequence|legend=0| 50, 80, 130 }}
Badness (Sintel): 1.02


== References ==
== References ==
Retrieved from "https://en.xen.wiki/w/Trisedodge_family"