Minortonic family: Difference between revisions

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.

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

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 [[171edo]]. 21 generators gives a 64/7. MOS of size 20, 33, 46 or 79 notes can be used for mitonic.

[[Mapping]]: [{{val| 1 -1 -3 6 }}, {{val| 0 17 35 -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. 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").

Mapping: [{{val| 1 -1 -3 6 10 }}, {{val| 0 17 35 -21 -43 }}]

Mapping: [{{val| 1 -1 -3 6 10 11 }}, {{val| 0 17 35 -21 -43 -48 }}]

Mapping: [{{val| 1 -1 -3 6 10 11 5 }}, {{val| 0 17 35 -21 -43 -48 -6 }}]

Mapping: [{{val| 1 -1 -3 6 3 }}, {{val| 0 17 35 -21 3 }}]

Mapping: [{{val| 1 -1 -3 6 3 11 }}, {{val| 0 17 35 -21 3 -48 }}]

Mapping: [{{val| 1 -1 -3 6 3 11 5 }}, {{val| 0 17 35 -21 3 -48 -6 }}]

The ''goldmine'' temperament (46&79) is another 13-limit extension of ore, equating [[13/12]] with [[14/13]] and [[16/13]] with two [[10/9]]s.

Mapping: [{{val| 1 -1 -3 6 3 4 }}, {{val| 0 17 35 -21 3 -2 }}]

Mapping: [{{val| 1 -1 -3 6 3 4 5 }}, {{val| 0 17 35 -21 3 -2 -6 }}]

Mapping: [{{val| 2 -2 -6 12 13 }}, {{val| 0 17 35 -21 -20 }}]

''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 [[Chromatic pairs #Terrain|terrain]].

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

 

[[Comma list]]: 9801/9800, 250047/250000, 14348907/14348180

 

[[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}}, ...