Minortone family

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 minortone family of temperaments tempers out the minortone comma (monzo: [-16 35 -17⟩).

Minortone

The head of this family is the 5-limit minortone temperament, with generator a minor tone.

Subgroup: 2.3.5

Comma list: [-16 35 -17⟩

Mapping: [⟨1 -1 -3], ⟨0 17 35]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~10/9 = 182.466 ¢

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

Badness (Smith): 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 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 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 = 1200.000 ¢, ~10/9 = 182.458 ¢

Optimal ET sequence: 46, 125, 171, 1927d, 2098d, …, 3637bcdd

Badness (Smith): 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 = 1200.000 ¢, ~10/9 = 182.482 ¢

Optimal ET sequence: 46, 125e, 171, 217, 605ee, 822dee

Badness (Smith): 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 = 1200.000 ¢, ~10/9 = 182.481 ¢

Optimal ET sequence: 46, 125e, 171, 217, 605ee, 822dee

Badness (Smith): 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 = 1200.000 ¢, ~10/9 = 182.481 ¢

Optimal ET sequence: 46, 125e, 171, 217, 605ee, 822dee

Badness (Smith): 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 = 1200.000 ¢, ~10/9 = 182.449 ¢

Optimal ET sequence: 46, 125, 171e

Badness (Smith): 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 = 1200.000 ¢, ~10/9 = 182.470 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~10/9 = 182.471 ¢

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

Badness (Smith): 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 = 1200.000 ¢, ~10/9 = 182.437 ¢

Optimal ET sequence: 46, 79, 125f, 171ef, 296eff

Badness (Smith): 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 = 1200.000 ¢, ~10/9 = 182.444 ¢

Optimal ET sequence: 46, 125f, 171ef

Badness (Smith): 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 = 1200.000 ¢, ~10/9 = 182.457 ¢

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

Badness (Smith): 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 = 400.000 ¢, ~10/9 = 182.467 ¢

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

Badness (Smith): 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 = 200.000 ¢, ~100/99 = 17.533 ¢

Optimal tuning (CTE): ~100/99 = 17.533

Optimal ET sequence: 342, 480, 822, 1164, 1506, 1848, …