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
[[Badness]]: 0.029765


== Mitonic ==
== 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 [[171edo|171 EDO]]. The wedgie is now {{Multival|17 35 -21 16 -81 -147}}, with 21 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
[[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
[[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 (equating 10/9 with 49/44 and 21/20 with 22/21).
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


Vals: {{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


Vals: {{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


Vals: {{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


Vals: {{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


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


Badness: 0.046170
Badness: 0.046170


==== 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: 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


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


Badness: 0.028423
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 ==
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: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]]

Latest revision as of 00:42, 24 June 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

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⟩. 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]]

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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