Minortonic family: Difference between revisions
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== 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^(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. | ||
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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&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). | 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). 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). | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Revision as of 14:03, 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
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 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
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). 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).
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
Badness: 0.013979