Minortone 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.
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.


== Minortone ==
== Minortone ==
Subgroup: 2.3.5
Subgroup: 2.3.5


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


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


[[POTE generator]]: ~10/9 = 182.466
[[POTE generator]]: ~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^(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.
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)^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.
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.


Subgroup: 2.3.5.7
Subgroup: 2.3.5.7
Line 23: Line 23:
[[Comma list]]: 4375/4374, 2100875/2097152
[[Comma list]]: 4375/4374, 2100875/2097152


[[Mapping]]: [{{val|1 -1 -3 6}}, {{val|0 17 35 -21}}]
[[Mapping]]: [{{val| 1 -1 -3 6 }}, {{val| 0 17 35 -21 }}]
 
{{Multival|legend=1| 17 35 -21 16 -81 -147 }}


[[POTE generator]]: ~10/9 = 182.458
[[POTE generator]]: ~10/9 = 182.458


{{Val list|legend=1| 46, 125, 171 }}
{{Val list|legend=1| 46, 125, 171, 1927d, 2098d, …, 3637bcdd }}


[[Badness]]: 0.025184
[[Badness]]: 0.025184
Line 38: 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: [{{val| 1 -1 -3 6 10 }}, {{val| 0 17 35 -21 -43 }}]


POTE generator: ~10/9 = 182.482
POTE generator: ~10/9 = 182.482
Line 51: Line 53:
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: [{{val| 1 -1 -3 6 10 11 }}, {{val| 0 17 35 -21 -43 -48 }}]


POTE generator: ~10/9 = 182.481
POTE generator: ~10/9 = 182.481
Line 64: Line 66:
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: [{{val| 1 -1 -3 6 10 11 5 }}, {{val| 0 17 35 -21 -43 -48 -6 }}]


POTE generator: ~10/9 = 182.481
POTE generator: ~10/9 = 182.481
Line 79: Line 81:
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: [{{val| 1 -1 -3 6 3 }}, {{val| 0 17 35 -21 3 }}]


POTE generator: ~10/9 = 182.449
POTE generator: ~10/9 = 182.449
Line 92: Line 94:
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: [{{val| 1 -1 -3 6 3 11 }}, {{val| 0 17 35 -21 3 -48 }}]


POTE generator: ~10/9 = 182.470
POTE generator: ~10/9 = 182.470
Line 105: Line 107:
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: [{{val| 1 -1 -3 6 3 11 5 }}, {{val| 0 17 35 -21 3 -48 -6 }}]


POTE generator: ~10/9 = 182.471
POTE generator: ~10/9 = 182.471
Line 120: Line 122:
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: [{{val| 1 -1 -3 6 3 4 }}, {{val| 0 17 35 -21 3 -2 }}]


POTE generator: ~10/9 = 182.437
POTE generator: ~10/9 = 182.437
Line 133: Line 135:
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: [{{val| 1 -1 -3 6 3 4 5 }}, {{val| 0 17 35 -21 3 -2 -6 }}]


POTE generator: ~10/9 = 182.444
POTE generator: ~10/9 = 182.444
Line 148: Line 150:
[[Comma list]]: 250047/250000, 645700815/645657712
[[Comma list]]: 250047/250000, 645700815/645657712


[[Mapping]]: [{{val|3 -3 -9 -8}}, {{val|0 17 35 36}}]
[[Mapping]]: [{{val| 3 -3 -9 -8 }}, {{val| 0 17 35 36 }}]


[[POTE generator]]: ~10/9 = 182.467
[[POTE generator]]: ~10/9 = 182.467
Line 156: Line 158:
[[Badness]]: 0.013979
[[Badness]]: 0.013979


[[Category:Theory]]
[[Category:Regular temperament theory]]
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Minortonic]]
[[Category:Minortonic family| ]] <!-- main article -->
[[Category:Rank 2]]

Revision as of 12:40, 1 March 2022

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

Minortone

Subgroup: 2.3.5

Comma: [-16 35 -17⟩

Mapping: [⟨1 -1 -3], ⟨0 17 35]]

POTE generator: ~10/9 = 182.466

Template:Val list

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 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]]

⟨⟨17 35 -21 16 -81 -147]]

POTE generator: ~10/9 = 182.458

Template:Val list

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]]

POTE generator: ~10/9 = 182.482

Optimal GPV sequence: Template:Val list

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]]

POTE generator: ~10/9 = 182.481

Optimal GPV sequence: Template:Val list

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]]

POTE generator: ~10/9 = 182.481

Optimal GPV sequence: Template:Val list

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]]

POTE generator: ~10/9 = 182.449

Optimal GPV sequence: Template:Val list

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]]

POTE generator: ~10/9 = 182.470

Optimal GPV sequence: Template:Val list

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]]

POTE generator: ~10/9 = 182.471

Optimal GPV sequence: Template:Val list

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]]

POTE generator: ~10/9 = 182.437

Optimal GPV sequence: Template:Val list

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]]

POTE generator: ~10/9 = 182.444

Optimal GPV sequence: Template:Val list

Badness: 0.027440

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 terrain.

Subgroup: 2.3.5.7

Comma list: 250047/250000, 645700815/645657712

Mapping: [⟨3 -3 -9 -8], ⟨0 17 35 36]]

POTE generator: ~10/9 = 182.467

Template:Val list

Badness: 0.013979

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