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The '''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 ==
== 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
[[Badness]]: 0.029765


== Mitonic ==
== Mitonic ==
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<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 }}. 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.


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 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)<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.


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 4375/4374, 2100875/2097152
[[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 }}


{{Multival|legend=1| 17 35 -21 16 -81 -147 }}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~10/9 = 182.458


[[POTE generator]]: ~10/9 = 182.458
{{Optimal ET sequence|legend=1| 46, 125, 171, 1927d, 2098d, …, 3637bcdd }}
 
{{Val list|legend=1| 46, 125, 171, 1927d, 2098d, …, 3637bcdd }}


[[Badness]]: 0.025184
[[Badness]]: 0.025184


=== Mineral ===
=== 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").
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
Subgroup: 2.3.5.7.11
Line 40: Line 39:
Comma list: 441/440, 4375/4374, 16384/16335
Comma list: 441/440, 4375/4374, 16384/16335


Mapping: [{{val| 1 -1 -3 6 10 }}, {{val| 0 17 35 -21 -43 }}]
Mapping: {{mapping| 1 -1 -3 6 10 | 0 17 35 -21 -43 }}


POTE generator: ~10/9 = 182.482
Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 182.482


Optimal GPV sequence: {{Val list| 46, 125e, 171, 217, 605ee, 822dee }}
{{Optimal ET sequence|legend=1| 46, 125e, 171, 217, 605ee, 822dee }}


Badness: 0.059060
Badness: 0.059060
Line 53: Line 52:
Comma list: 364/363, 441/440, 3584/3575, 4375/4374
Comma list: 364/363, 441/440, 3584/3575, 4375/4374


Mapping: [{{val| 1 -1 -3 6 10 11 }}, {{val| 0 17 35 -21 -43 -48 }}]
Mapping: {{mapping| 1 -1 -3 6 10 11 | 0 17 35 -21 -43 -48 }}


POTE generator: ~10/9 = 182.481
Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 182.481


Optimal GPV sequence: {{Val list| 46, 125e, 171, 217, 605ee, 822dee }}
{{Optimal ET sequence|legend=1| 46, 125e, 171, 217, 605ee, 822dee }}


Badness: 0.033140
Badness: 0.033140
Line 66: Line 65:
Comma list: 364/363, 441/440, 595/594, 1156/1155, 3584/3575
Comma list: 364/363, 441/440, 595/594, 1156/1155, 3584/3575


Mapping: [{{val| 1 -1 -3 6 10 11 5 }}, {{val| 0 17 35 -21 -43 -48 -6 }}]
Mapping: {{mapping| 1 -1 -3 6 10 11 5 | 0 17 35 -21 -43 -48 -6 }}


POTE generator: ~10/9 = 182.481
Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 182.481


Optimal GPV sequence: {{Val list| 46, 125e, 171, 217, 605ee, 822dee }}
{{Optimal ET sequence|legend=1| 46, 125e, 171, 217, 605ee, 822dee }}


Badness: 0.019792
Badness: 0.019792
Line 81: Line 80:
Comma list: 385/384, 1331/1323, 4375/4374
Comma list: 385/384, 1331/1323, 4375/4374


Mapping: [{{val| 1 -1 -3 6 3 }}, {{val| 0 17 35 -21 3 }}]
Mapping: {{mapping| 1 -1 -3 6 3 | 0 17 35 -21 3 }}


POTE generator: ~10/9 = 182.449
Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 182.449


Optimal GPV sequence: {{Val list| 46, 125, 171e }}
{{Optimal ET sequence|legend=1| 46, 125, 171e }}


Badness: 0.053662
Badness: 0.053662
Line 94: Line 93:
Comma list: 352/351, 385/384, 1331/1323, 3267/3250
Comma list: 352/351, 385/384, 1331/1323, 3267/3250


Mapping: [{{val| 1 -1 -3 6 3 11 }}, {{val| 0 17 35 -21 3 -48 }}]
Mapping: {{mapping| 1 -1 -3 6 3 11 | 0 17 35 -21 3 -48 }}


POTE generator: ~10/9 = 182.470
Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 182.470


Optimal GPV sequence: {{Val list| 46, 125, 171e, 388ee }}
{{Optimal ET sequence|legend=1| 46, 125, 171e, 388ee }}


Badness: 0.046170
Badness: 0.046170
Line 107: Line 106:
Comma list: 352/351, 385/384, 561/560, 715/714, 1452/1445
Comma list: 352/351, 385/384, 561/560, 715/714, 1452/1445


Mapping: [{{val| 1 -1 -3 6 3 11 5 }}, {{val| 0 17 35 -21 3 -48 -6 }}]
Mapping: {{mapping| 1 -1 -3 6 3 11 5 | 0 17 35 -21 3 -48 -6 }}


POTE generator: ~10/9 = 182.471
Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 182.471


Optimal GPV sequence: {{Val list| 46, 125, 171e, 388ee }}
{{Optimal ET sequence|legend=1| 46, 125, 171e, 388ee }}


Badness: 0.028423
Badness: 0.028423


==== Goldmine ====
==== 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.
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
Subgroup: 2.3.5.7.11.13
Line 122: Line 121:
Comma list: 169/168, 325/324, 385/384, 1331/1323
Comma list: 169/168, 325/324, 385/384, 1331/1323


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


POTE generator: ~10/9 = 182.437
Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 182.437


Optimal GPV sequence: {{Val list| 46, 79, 125f, 171ef, 296eff }}
{{Optimal ET sequence|legend=1| 46, 79, 125f, 171ef, 296eff }}


Badness: 0.039302
Badness: 0.039302
Line 135: Line 134:
Comma list: 169/168, 273/272, 325/324, 385/384, 1331/1323
Comma list: 169/168, 273/272, 325/324, 385/384, 1331/1323


Mapping: [{{val| 1 -1 -3 6 3 4 5 }}, {{val| 0 17 35 -21 3 -2 -6 }}]
Mapping: {{mapping| 1 -1 -3 6 3 4 5 | 0 17 35 -21 3 -2 -6 }}


POTE generator: ~10/9 = 182.444
Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 182.444


Optimal GPV sequence: {{Val list| 46, 125f, 171ef }}
{{Optimal ET sequence|legend=1| 46, 125f, 171ef }}


Badness: 0.027440
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 ==
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
[[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
[[Badness]]: 0.013979


[[Category:Regular temperament theory]]
=== Hemidomain ===
Subgroup: 2.3.5.7.11
 
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:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Minortonic family| ]] <!-- main article -->
[[Category:Minortonic family| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
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