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This family tempers out the minortone comma (also known as "minortonma"), {{Monzo|-16 35 -17}}. The head of this family is five-limit minortone temperament, with generator a minor tone.
{{Technical data page}}
The '''minortone family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[minortone comma]] ({{monzo|legend=1| -16 35 -17 }}).  


== Minortone temperament ==
== Minortone ==
Subgroup: 2.3.5
The head of this family is the 5-limit minortone temperament, a super-accurate microtemperament with generator a minor tone of [[10/9]], seventeen of which make the [[6/1|6th harmonic]], and another eighteen generators make the interval class of the [[5/1|5th harmonic]]. Its [[ploidacot]] is beta-17-cot. The generator should be tuned about 1/16 of a cent flat, with 6<sup>1/17</sup> being 0.06423 cents flat and 40<sup>1/35</sup> being 0.06234 cents flat. [[171edo|171-]], [[559edo|559-]] and [[730edo]] are among the possible edo tunings.


[[Comma]]: {{Monzo|-16 35 -17}}
[[Subgroup]]: 2.3.5


[[Mapping]]: [{{val|1 -1 -3}}, {{val|0 17 35}}]
[[Comma list]]: {{monzo| -16 35 -17 }}


[[POTE generator]]: ~10/9 = 182.466
{{Mapping|legend=1| 1 -1 -3 | 0 17 35 }}
: mapping generators: ~2, ~10/9


{{Val list|legend=1| 46, 125, 171, 388, 559, 730, 1289, 2019, 2749, 4768, 16323, 21091 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0036{{c}}, ~10/9 = 182.4670{{c}}
: [[error map]]: {{val| +0.004 -0.020 +0.020 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 182.4666{{c}}
: error map: {{val| 0.000 -0.023 +0.017 }}


[[Badness]]: 0.029765
{{Optimal ET sequence|legend=1| 46, 125, 171, 388, 559, 730, 1289, 2019, 2749, 4768, 16323, 21091, 25859b }}
 
[[Badness]] (Sintel): 0.698


== Mitonic ==
== Mitonic ==
{{see also|Ragismic microtemperaments #Mitonic}}
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 & 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
 
[[Comma list]]: 4375/4374, 2100875/2097152
 
{{Mapping|legend=1| 1 -1 -3 6 | 0 17 35 -21 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0963{{c}}, ~10/9 = 182.4727{{c}}
: [[error map]]: {{val| +0.096 -0.015 -0.057 -0.176 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~10/9 = 182.4592{{c}}
: error map: {{val| 0.000 -0.149 -0.243 -0.468 }}
 
{{Optimal ET sequence|legend=1| 46, 125, 171, 1927d, 2098d, …, 3637bcdd }}
 
[[Badness]] (Sintel): 0.637
 
=== Mineral ===
Subgroup: 2.3.5.7.11
 
Comma list: 441/440, 4375/4374, 16384/16335
 
Mapping: {{mapping| 1 -1 -3 6 10 | 0 17 35 -21 -43 }}
 
Optimal tuning:
* WE: ~2 = 1200.8759{{c}}, ~10/9 = 182.4631{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 182.4818{{c}}
 
{{Optimal ET sequence|legend=0| 46, 125e, 171, 217 }}
 
Badness (Sintel): 1.95
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 364/363, 441/440, 3584/3575, 4375/4374
 
Mapping: {{mapping| 1 -1 -3 6 10 11 | 0 17 35 -21 -43 -48 }}
 
Optimal tunings:
* WE: ~2 = 1199.8808{{c}}, ~10/9 = 182.4633{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 182.4816{{c}}
 
{{Optimal ET sequence|legend=0| 46, 125e, 171, 217 }}
 
Badness (Sintel): 1.37
 
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 364/363, 441/440, 595/594, 1156/1155, 3584/3575
 
Mapping: {{mapping| 1 -1 -3 6 10 11 5 | 0 17 35 -21 -43 -48 -6 }}
 
Optimal tunings:
* WE: ~2 = 1199.8923{{c}}, ~10/9 = 182.4650{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 182.4816{{c}}
 
{{Optimal ET sequence|legend=0| 46, 125e, 171, 217 }}
 
Badness (Sintel): 1.01
 
=== Ore ===
Subgroup: 2.3.5.7.11
 
Comma list: 385/384, 1331/1323, 4375/4374
 
Mapping: {{mapping| 1 -1 -3 6 3 | 0 17 35 -21 3 }}
 
Optimal tunings:
* WE: ~2 = 1199.3408{{c}}, ~10/9 = 182.5004{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 182.4518{{c}}
 
{{Optimal ET sequence|legend=0| 46, 125, 171e }}
 
Badness (Sintel): 1.77
 
==== Goldmine ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 169/168, 325/324, 385/384, 1331/1323
 
Mapping: {{mapping| 1 -1 -3 6 3 4 | 0 17 35 -21 3 -2 }}
 
Optimal tunings:
* WE: ~2 = 1200.5647{{c}}, ~10/9 = 182.5224{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 182.4409{{c}}
 
{{Optimal ET sequence|legend=0| 46, 79, 125f, 171ef }}
 
Badness (Sintel): 1.62
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 169/168, 273/272, 325/324, 385/384, 1331/1323
 
Mapping: {{mapping| 1 -1 -3 6 3 4 5 | 0 17 35 -21 3 -2 -6 }}
 
Optimal tunings:
* WE: ~2 = 1200.4506{{c}}, ~10/9 = 182.5124{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 182.4465{{c}}
 
{{Optimal ET sequence|legend=0| 46, 125f, 171ef }}
 
Badness (Sintel): 1.40
 
==== Orefield ====
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 385/384, 1331/1323, 3267/3250
 
Mapping: {{mapping| 1 -1 -3 6 3 11 | 0 17 35 -21 3 -48 }}
 
Optimal tunings:
* WE: ~2 = 1200.1737{{c}}, ~10/9 = 182.4966{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 182.4705{{c}}
 
{{Optimal ET sequence|legend=0| 46, 125, 171e }}
 
Badness (Sintel): 1.91
 
===== 17-limit =====
Subgroup: 2.3.5.7.11.13.17


As a 5-limit temperament, mitonic becomes minortonic, a super-accurate microtemperament tempering out the minortone comma, {{Monzo|-16 35 -17}}. Flipping that gives the 5-limit wedgie {{Multival|17 35 16}}, which tells us that 10/9 can be taken as the generator, with 17 of them giving a 6, 18 of them a 20/3, and 35 of them giving a 40. The generator should be tuned about 1/16 of a cent flat, with 6^(1/17) being 0.06423 cents flat and 40^(1/35) being 0.06234 cents flat. 171, 559 and 730 are possible equal temperament tunings.
Comma list: 352/351, 385/384, 561/560, 715/714, 1452/1445


However, as noted before, 32/21 is only a ragisma shy of (10/9)^4, and so a 7-limit interpretation, if not quite so super-accurate, is more or less inevitable. While 559 or 730 are still fine as tunings, the error of the 7-limit is lower by a whisker in [[171edo|171EDO]]. The wedgie is now {{Multival|17 35 -21 16 -81 -147}}, with 21 10/9 generators giving a 64/7. MOS of size 20, 33, 46 or 79 notes can be used for mitonic.
Mapping: {{mapping| 1 -1 -3 6 3 11 5 | 0 17 35 -21 3 -48 -6 }}


Subgroup: 2.3.5.7
Optimal tunings:  
* WE: ~2 = 1200.1403{{c}}, ~10/9 = 182.4924{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~10/9 = 182.4712{{c}}


[[Comma list]]: 4375/4374, 2100875/2097152
{{Optimal ET sequence|legend=0| 46, 125, 171e }}
 
Badness (Sintel): 1.45
 
=== Seminar ===
Subgroup: 2.3.5.7.11
 
Comma list: 3025/3024, 4375/4374, 2100875/2097152
 
Mapping: {{mapping| 2 -2 -6 12 13 | 0 17 35 -21 -20 }}
 
Optimal tunings:
* WE: ~99/70 = 600.0535{{c}}, ~10/9 = 182.4735{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~10/9 = 182.4578{{c}}
 
{{Optimal ET sequence|legend=0| 46, 250, 296, 342, 3466cddee, 3808bcddeee, …, 5176bccdddeeee }}
 
Badness (Sintel): 0.886
 
==== 2.3.5.7.11.17 subgroup ====
Subgroup: 2.3.5.7.11.17
 
Comma list: 1089/1088, 1225/1224, 1701/1700, 262395/262144


[[Mapping]]: [{{val|1 -1 -3 6}}, {{val|0 17 35 -21}}]
Mapping: {{mapping| 2 -2 -6 12 13 4 | 0 17 35 -21 -20 6 }}


[[POTE generator]]: ~10/9 = 182.458
Optimal tunings:  
* WE: ~99/70 = 600.0371{{c}}, ~10/9 = 182.4699{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~10/9 = 182.4589{{c}}


{{Val list|legend=1| 46, 125, 171 }}
{{Optimal ET sequence|legend=0| 46, 250, 296, 342, 1414de }}


[[Badness]]: 0.025184
Badness (Sintel): 0.831


== Domain ==
== Domain ==
Domain temperament adds the landscape comma, 250047/250000, to the minortone comma, giving a temperament which is perhaps most notable for its inclusion of the remarkable subgroup temperament [[Chromatic pairs #Terrain|terrain]].
Domain adds the [[landscape comma]], 250047/250000, to the minortone comma, giving a temperament which is perhaps most notable for its inclusion of the remarkable subgroup temperament [[terrain]].


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 250047/250000, 645700815/645657712
[[Comma list]]: 250047/250000, 645700815/645657712


[[Mapping]]: [{{val|3 -3 -9 -8}}, {{val|0 17 35 36}}]
{{Mapping|legend=1| 3 -3 -9 -8 | 0 17 35 36 }}
: mapping generators: ~63/50, ~10/9
 
[[Optimal tuning]]s:
* [[WE]]: ~63/50 = 400.0013{{c}}, ~10/9 = 182.4672{{c}}
: [[error map]]: {{val| +0.004 -0.016 +0.028 -0.016 }}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~10/9 = 182.4668{{c}}
: error map: {{val| 0.000 -0.020 +0.024 -0.021 }}
 
{{Optimal ET sequence|legend=1| 171, 1164, 1335, 1506, 1677, 1848, 2019, 11943, 13962, 15981, 18000, 20019, 22038 }}
 
[[Badness]] (Sintel): 0.354
 
=== Semidomain ===
Subgroup: 2.3.5.7.11
 
Comma list: 9801/9800, 151263/151250, 1771561/1771470
 
Mapping: {{mapping| 6 -6 -18 -16 -13 | 0 17 35 36 37 }}
: mapping generators: ~55/49, ~10/9


[[POTE generator]]: ~10/9 = 182.467
Optimal tunings:  
* WE: ~55/49 = 200.0009{{c}}, ~10/9 = 182.4669{{c}}
* CWE: ~55/49 = 200.0000{{c}}, ~10/9 = 182.4675{{c}}


{{Val list|legend=1| 171, 1164, 1335, 1506, 1677, 1848, 2019, 11943, 13962, 15981, 18000, 20019, 22038 }}
{{Optimal ET sequence|legend=0| 342, 1164, 1506, 1848, 4038, 5886, 9924 }}


[[Badness]]: 0.013979
Badness (Sintel): 0.360


[[Category:Theory]]
[[Category:Minortone family| ]] <!-- main article -->
[[Category:Temperament family]]
[[Category:Temperament families]]
[[Category:Minortonic]]
[[Category:Catalogs of rank-2 temperaments]]
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