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Minortone family: Difference between revisions

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Switch to Sintel's badness, WE & CWE tunings. Improve intro to each entry
Hemidomain -> semidomain; 13-limit ore -> orefield (see talk)
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Mitonic takes advantage of the fact that [[32/21]] is only a [[ragisma]] shy of (10/9)<sup>4</sup>, and so this 7-limit interpretation where 21 generators gives a ~8/7, if not quite so super-accurate, is more or less inevitable. While [[559edo|559]] or [[730edo|730]] are still fine as tunings, the error of the 7-limit is lower by a whisker in [[171edo]]. Generator chains of 46 or 79 notes can be used for this temperament.
Mitonic takes advantage of the fact that [[32/21]] is only a [[ragisma]] shy of (10/9)<sup>4</sup>, and so this 7-limit interpretation where 21 generators gives a ~8/7, if not quite so super-accurate, is more or less inevitable. While [[559edo|559]] or [[730edo|730]] are still fine as tunings, the error of the 7-limit is lower by a whisker in [[171edo]]. Generator chains of 46 or 79 notes can be used for this temperament.


Extending mitonic to the 11-limit is not so simple. There are two mappings that are comparable in complexity and error: mineral (46 & 171) and ore (46 & 125). Mineral tempers out 441/440 and 16384/16335 in the 11-limit. Ore tempers out 385/384 and 1331/1323 in the 11-limit, and maps [[11/8]] to three generators. In the 17-limit, both mineral and ore temper out 833/832, 1225/1224, 1701/1700, and 4096/4095 (2.3.5.7.13.17 commas). The word ''mineral'' is related to ''mine'' (an excavation from which ore or solid minerals are taken) and ''miner'' (a person who works in a mine, also as a pun on ''minor'').
Extending mitonic to the 11-limit is not so simple. There are two mappings that are comparable in complexity and error: mineral (46 & 171) and ore (46 & 171e). Mineral tempers out 441/440 and 16384/16335 in the 11-limit. Ore tempers out 385/384 and 1331/1323 in the 11-limit, and maps [[11/8]] to three generators. The latter further branches to goldmine (46 & 125f), which equates [[16/13]] with a stack of two [[10/9]]'s and tempers together [[13/12]] and [[14/13]], and orefield (46 & 125), which uses the same mapping for 13 as mineral. All have the same mapping for 17, at -6 generators. The word ''mineral'' is related to ''mine'' and ''miner'', a pun on ''minor''.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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Badness (Sintel): 1.77
Badness (Sintel): 1.77


==== 13-limit ====
==== Goldmine ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 352/351, 385/384, 1331/1323, 3267/3250
Comma list: 169/168, 325/324, 385/384, 1331/1323


Mapping: {{mapping| 1 -1 -3 6 3 11 | 0 17 35 -21 3 -48 }}
Mapping: {{mapping| 1 -1 -3 6 3 4 | 0 17 35 -21 3 -2 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.1737{{c}}, ~10/9 = 182.4966{{c}}
* WE: ~2 = 1200.5647{{c}}, ~10/9 = 182.5224{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 182.4705{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 182.4409{{c}}


{{Optimal ET sequence|legend=0| 46, 125, 171e }}
{{Optimal ET sequence|legend=0| 46, 79, 125f, 171ef }}


Badness (Sintel): 1.91
Badness (Sintel): 1.62


===== 17-limit =====
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 352/351, 385/384, 561/560, 715/714, 1452/1445
Comma list: 169/168, 273/272, 325/324, 385/384, 1331/1323


Mapping: {{mapping| 1 -1 -3 6 3 11 5 | 0 17 35 -21 3 -48 -6 }}
Mapping: {{mapping| 1 -1 -3 6 3 4 5 | 0 17 35 -21 3 -2 -6 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.1403{{c}}, ~10/9 = 182.4924{{c}}
* WE: ~2 = 1200.4506{{c}}, ~10/9 = 182.5124{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 182.4712{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 182.4465{{c}}


{{Optimal ET sequence|legend=0| 46, 125, 171e }}
{{Optimal ET sequence|legend=0| 46, 125f, 171ef }}


Badness (Sintel): 1.45
Badness (Sintel): 1.40
 
==== Goldmine ====
The goldmine temperament (46 & 79) is another 13-limit extension of ore, equating [[13/12]] with [[14/13]] and [[16/13]] with two [[10/9]]s.


==== Orefield ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 169/168, 325/324, 385/384, 1331/1323
Comma list: 352/351, 385/384, 1331/1323, 3267/3250


Mapping: {{mapping| 1 -1 -3 6 3 4 | 0 17 35 -21 3 -2 }}
Mapping: {{mapping| 1 -1 -3 6 3 11 | 0 17 35 -21 3 -48 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.5647{{c}}, ~10/9 = 182.5224{{c}}
* WE: ~2 = 1200.1737{{c}}, ~10/9 = 182.4966{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 182.4409{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 182.4705{{c}}


{{Optimal ET sequence|legend=0| 46, 79, 125f, 171ef }}
{{Optimal ET sequence|legend=0| 46, 125, 171e }}


Badness (Sintel): 1.62
Badness (Sintel): 1.91


===== 17-limit =====
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 169/168, 273/272, 325/324, 385/384, 1331/1323
Comma list: 352/351, 385/384, 561/560, 715/714, 1452/1445


Mapping: {{mapping| 1 -1 -3 6 3 4 5 | 0 17 35 -21 3 -2 -6 }}
Mapping: {{mapping| 1 -1 -3 6 3 11 5 | 0 17 35 -21 3 -48 -6 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1200.4506{{c}}, ~10/9 = 182.5124{{c}}
* WE: ~2 = 1200.1403{{c}}, ~10/9 = 182.4924{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 182.4465{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 182.4712{{c}}


{{Optimal ET sequence|legend=0| 46, 125f, 171ef }}
{{Optimal ET sequence|legend=0| 46, 125, 171e }}


Badness (Sintel): 1.40
Badness (Sintel): 1.45


=== Seminar ===
=== Seminar ===
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[[Badness]] (Sintel): 0.354
[[Badness]] (Sintel): 0.354


=== Hemidomain ===
=== Semidomain ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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