Minortonic family: Difference between revisions

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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.
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 ==
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[[Comma list]]: {{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 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~10/9 = 182.466
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~10/9 = 182.466
Line 15: Line 15:


== 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 }}. 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.


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
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[[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 }}
{{Multival|legend=1| 17 35 -21 16 -81 -147 }}
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=== 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 40:
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 }}


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


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


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


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


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


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


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


Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 182.444
Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 182.444
Line 148: Line 148:
Comma list: 3025/3024, 4375/4374, 2100875/2097152
Comma list: 3025/3024, 4375/4374, 2100875/2097152


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


Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 182.457
Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 182.457
Line 159: Line 159:
{{See also| Landscape microtemperaments #Domain }}
{{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 [[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 [[Chromatic pairs #Terrain|terrain]].


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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[[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 }}


[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~10/9 = 182.467
[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~10/9 = 182.467
Line 174: Line 174:


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


[[Subgroup]]: 2.3.5.7.11
Comma list: 9801/9800, 250047/250000, 14348907/14348180
 
[[Comma list]]: 9801/9800, 250047/250000, 14348907/14348180
 
[[Mapping]]: [{{val|6 11 17 20 24}}, {{val|0 -17 -35 -36 -37}}]


[[Mapping]] [[generators]]: ~55/49 = 1\6, ~100/99 = 17.533
Mapping: {{mapping| 6 11 17 20 24 | 0 -17 -35 -36 -37 }}


[[Optimal tuning]] ([[CTE]]): ~100/99 = 17.533
: mapping generators: ~55/49 = 1\6, ~100/99 = 17.533


{{Optimal ET sequence|legend=1|342, 480, 822, 1164, 1506, 1848}}, ...
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:Minortonic family| ]] <!-- main article -->
[[Category:Minortonic family| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Revision as of 11:49, 13 September 2023

The minortonic family tempers out the minortone comma (also known as "minortonma"), [-16 35 -17⟩. The head of this family is 5-limit minortone temperament, with generator a minor tone.

Minortone

Subgroup: 2.3.5

Comma list: [-16 35 -17⟩

Mapping: [⟨1 -1 -3], ⟨0 17 35]]

Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 182.466

Optimal ET sequence: 46, 125, 171, 388, 559, 730, 1289, 2019, 2749, 4768, 16323, 21091

Badness: 0.029765

Mitonic

As a 5-limit temperament, mitonic becomes minortonic, a super-accurate microtemperament tempering out the minortone comma, [-16 35 -17⟩. Flipping that gives the 5-limit wedgie ⟨⟨ 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 61/17 being 0.06423 cents flat and 401/35 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)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. 21 generators gives a ~64/7. Mos scales of size 20, 33, 46 or 79 notes can be used for mitonic.

Subgroup: 2.3.5.7

Comma list: 4375/4374, 2100875/2097152

Mapping: [⟨1 -1 -3 6], ⟨0 17 35 -21]]

Wedgie: ⟨⟨ 17 35 -21 16 -81 -147 ]]

Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 182.458

Optimal ET sequence: 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 & 171) and ore (46 & 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: [⟨1 -1 -3 6 10], ⟨0 17 35 -21 -43]]

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

Optimal ET sequence: 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: [⟨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: 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: [⟨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: 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: [⟨1 -1 -3 6 3], ⟨0 17 35 -21 3]]

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

Optimal ET sequence: 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: [⟨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: 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: [⟨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: 46, 125, 171e, 388ee

Badness: 0.028423

Goldmine

The goldmine temperament (46 & 79) is another 13-limit extension of ore, equating 13/12 with 14/13 and 16/13 with two 10/9s.

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 325/324, 385/384, 1331/1323

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: 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: [⟨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: 46, 125f, 171ef

Badness: 0.027440

Seminar

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4375/4374, 2100875/2097152

Mapping: [⟨2 -2 -6 12 13], ⟨0 17 35 -21 -20]]

Optimal tuning (POTE): ~2 = 1\1, ~10/9 = 182.457

Optimal ET sequence: 46, 204c, 250, 296, 342

Badness: 0.026808

Domain

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 terrain.

Subgroup: 2.3.5.7

Comma list: 250047/250000, 645700815/645657712

Mapping: [⟨3 -3 -9 -8], ⟨0 17 35 36]]

Optimal tuning (POTE): ~63/50 = 1\3, ~10/9 = 182.467

Optimal ET sequence: 171, 1164, 1335, 1506, 1677, 1848, 2019, 11943, 13962, 15981, 18000, 20019, 22038

Badness: 0.013979

Hemidomain

Subgroup: 2.3.5.7.11

Comma list: 9801/9800, 250047/250000, 14348907/14348180

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: 342, 480, 822, 1164, 1506, 1848, …

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