Minortonic family: Difference between revisions

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-'''Minortonic 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 '''minortonic family''' tempers out the minortone comma (also known as "minortonma"), {{monzo| -16 35 -17 }}. The head of this family is 5-limit minortone temperament, with generator a minor tone.
-== Minortone temperament ==
+== Minortone ==
-Subgroup: 2.3.5
+[[Subgroup]]: 2.3.5
-[[Comma]]: {{Monzo|-16 35 -17}}
+[[Comma list]]: {{monzo| -16 35 -17 }}
-[[Mapping]]: [{{val|1 -1 -3}}, {{val|0 17 35}}]
+{{Mapping|legend=1| 1 -1 -3 | 0 17 35 }}
-[[POTE generator]]: ~10/9 = 182.466
+[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~10/9 = 182.466
-{{Val list|legend=1| 46, 125, 171, 388, 559, 730, 1289, 2019, 2749, 4768, 16323, 21091 }}
+{{Optimal ET sequence|legend=1| 46, 125, 171, 388, 559, 730, 1289, 2019, 2749, 4768, 16323, 21091 }}
 [[Badness]]: 0.029765
 == Mitonic ==
-{{see also|Ragismic microtemperaments #Mitonic}}
+As a 5-limit temperament, mitonic becomes minortonic, a super-accurate microtemperament tempering out the minortone comma, {{monzo| -16 35 -17 }}. 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<sup>1/17</sup> being 0.06423 cents flat and 40<sup>1/35</sup> being 0.06234 cents flat. 171, 559 and 730 are possible equal temperament tunings.
-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.
+However, as noted before, 32/21 is only a ragisma shy of (10/9)<sup>4</sup>, 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]]. 21 generators gives a ~64/7. [[Mos scale]]s of size 20, 33, 46 or 79 notes can be used for mitonic.
-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 [[171&nbsp;EDO]]. 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.
+[[Subgroup]]: 2.3.5.7
-Subgroup: 2.3.5.7
 [[Comma list]]: 4375/4374, 2100875/2097152
-[[Mapping]]: [{{val|1 -1 -3 6}}, {{val|0 17 35 -21}}]
+{{Mapping|legend=1| 1 -1 -3 6 | 0 17 35 -21 }}
-[[POTE generator]]: ~10/9 = 182.458
+[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~10/9 = 182.458
-{{Val list|legend=1| 46, 125, 171 }}
+{{Optimal ET sequence|legend=1| 46, 125, 171, 1927d, 2098d, …, 3637bcdd }}
 [[Badness]]: 0.025184
+=== Mineral ===
+Extending mitonic to the 11-limit is not so simple. There are two mappings that are comparable in complexity and error: ''mineral'' (46 &amp; 171) and ''ore'' (46 &amp; 125). The mineral temperament tempers out 441/440 and 16384/16335 in the 11-limit. 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").
+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 (POTE): ~2 = 1\1, ~10/9 = 182.482
+{{Optimal ET sequence|legend=1| 46, 125e, 171, 217, 605ee, 822dee }}
+Badness: 0.059060
+==== 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 tuning (POTE): ~2 = 1\1, ~10/9 = 182.481
+{{Optimal ET sequence|legend=1| 46, 125e, 171, 217, 605ee, 822dee }}
+Badness: 0.033140
+==== 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 tuning (POTE): ~2 = 1\1, ~10/9 = 182.481
+{{Optimal ET sequence|legend=1| 46, 125e, 171, 217, 605ee, 822dee }}
+Badness: 0.019792
+=== Ore ===
+The ore temperament tempers out 385/384 and 1331/1323 in the 11-limit, and maps [[11/8]] to three generators.
+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 tuning (POTE): ~2 = 1\1, ~10/9 = 182.449
+{{Optimal ET sequence|legend=1| 46, 125, 171e }}
+Badness: 0.053662
+==== 13-limit ====
+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 tuning (POTE): ~2 = 1\1, ~10/9 = 182.470
+{{Optimal ET sequence|legend=1| 46, 125, 171e, 388ee }}
+Badness: 0.046170
+===== 17-limit =====
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 352/351, 385/384, 561/560, 715/714, 1452/1445
+Mapping: {{mapping| 1 -1 -3 6 3 11 5 | 0 17 35 -21 3 -48 -6 }}
+Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 182.471
+{{Optimal ET sequence|legend=1| 46, 125, 171e, 388ee }}
+Badness: 0.028423
+==== Goldmine ====
+The goldmine temperament (46 &amp; 79) is another 13-limit extension of ore, equating [[13/12]] with [[14/13]] and [[16/13]] with two [[10/9]]s.
+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 tuning (POTE): ~2 = 1\1, ~10/9 = 182.437
+{{Optimal ET sequence|legend=1| 46, 79, 125f, 171ef, 296eff }}
+Badness: 0.039302
+===== 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 tuning (POTE): ~2 = 1\1, ~10/9 = 182.444
+{{Optimal ET sequence|legend=1| 46, 125f, 171ef }}
+Badness: 0.027440
+=== 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 tuning (POTE): ~2 = 1\1, ~10/9 = 182.457
+{{Optimal ET sequence|legend=1| 46, 204c, 250, 296, 342 }}
+Badness: 0.026808
 == 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]].
+{{See also| Landscape microtemperaments #Domain }}
+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 [[Subgroup temperaments #Terrain|terrain]].
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
 [[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 }}
-[[POTE generator]]: ~10/9 = 182.467
+[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~10/9 = 182.467
-{{Val list|legend=1| 171, 1164, 1335, 1506, 1677, 1848, 2019, 11943, 13962, 15981, 18000, 20019, 22038 }}
+{{Optimal ET sequence|legend=1| 171, 1164, 1335, 1506, 1677, 1848, 2019, 11943, 13962, 15981, 18000, 20019, 22038 }}
 [[Badness]]: 0.013979
-[[Category:Theory]]
+=== Hemidomain ===
-[[Category:Temperament family]]
+Subgroup: 2.3.5.7.11
-[[Category:Minortonic]]
+Comma list: 9801/9800, 250047/250000, 14348907/14348180
+Mapping: {{mapping| 6 11 17 20 24 | 0 -17 -35 -36 -37 }}
+: mapping generators: ~55/49 = 1\6, ~100/99 = 17.533
+Optimal tuning (CTE): ~100/99 = 17.533
+{{Optimal ET sequence|legend=1| 342, 480, 822, 1164, 1506, 1848, … }}
+[[Category:Temperament families]]
+[[Category:Pages with mostly numerical content]]
+[[Category:Minortonic family| ]] <!-- main article -->
+[[Category:Rank 2]]