Minortonic family: Difference between revisions

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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^(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)^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 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.
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
Line 32: Line 32:


[[Badness]]: 0.025184
[[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 (equating 10/9 with 49/44 and 21/20 with 22/21).
Subgroup: 2.3.5.7.11
Comma list: 441/440, 4375/4374, 16384/16335
Mapping: [{{val|1 -1 -3 6 10}}, {{val|0 17 35 -21 -43}}]
POTE generator: ~10/9 = 182.482
Vals: {{Val list| 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: [{{val|1 -1 -3 6 10 11}}, {{val|0 17 35 -21 -43 -48}}]
POTE generator: ~10/9 = 182.481
Vals: {{Val list| 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: [{{val|1 -1 -3 6 10 11 5}}, {{val|0 17 35 -21 -43 -48 -6}}]
POTE generator: ~10/9 = 182.481
Vals: {{Val list| 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: [{{val|1 -1 -3 6 3}}, {{val|0 17 35 -21 3}}]
POTE generator: ~10/9 = 182.449
Vals: {{Val list| 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: [{{val|1 -1 -3 6 3 11}}, {{val|0 17 35 -21 3 -48}}]
POTE generator: ~10/9 = 182.470
Vals: {{Val list| 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: [{{val|1 -1 -3 6 3 11 5}}, {{val|0 17 35 -21 3 -48 -6}}]
POTE generator: ~10/9 = 182.471
Vals: {{Val list| 46, 125, 171e, 388ee }}
Badness: 0.028423


== Domain ==
== Domain ==

Revision as of 08:15, 4 July 2021

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 temperament

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

See also: Ragismic microtemperaments § 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 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)^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 EDO. The wedgie is now ⟨⟨ 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

Comma list: 4375/4374, 2100875/2097152

Mapping: [⟨1 -1 -3 6], ⟨0 17 35 -21]]

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 (equating 10/9 with 49/44 and 21/20 with 22/21).

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

Vals: 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

Vals: 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

Vals: 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

Vals: 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

Vals: 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

Vals: Template:Val list

Badness: 0.028423

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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