Minortonic family: Difference between revisions
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This tempers out the minortone comma, |-16 35 -17 | This 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 temperament == | |||
Subgroup: 2.3.5 | |||
[[ | [[Comma]]: {{Monzo|-16 35 -17}} | ||
[[ | [[Mapping]]: [{{val|1 -1 -3}}, {{val|0 17 35}}] | ||
[[ | [[POTE generator]]: ~10/9 = 182.466 | ||
== Mitonic | {{Val list|legend=1| 46, 125, 171, 388, 559, 730, 1289, 2019, 2749, 4768, 16323, 21091 }} | ||
[[Badness]]: 0.029765 | |||
== Mitonic == | |||
{{see also|Ragismic microtemperaments #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 | 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|171EDO]]. 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. | |||
Subgroup: 2.3.5.7 | |||
[[Comma | [[Comma list]]: 4375/4374, 2100875/2097152 | ||
[[ | [[Mapping]]: [{{val|1 -1 -3 6}}, {{val|0 17 35 -21}}] | ||
[[ | [[POTE generator]]: ~10/9 = 182.458 | ||
{{Val list|legend=1| 46, 125, 171 }} | |||
[[ | [[Badness]]: 0.025184 | ||
== Domain == | == 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 [[ | 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 [[Chromatic pairs #Terrain|terrain]]. | ||
Subgroup: 2.3.5.7 | |||
[[ | [[Comma list]]: 250047/250000, 645700815/645657712 | ||
[[ | [[Mapping]]: [{{val|3 -3 -9 -8}}, {{val|0 17 35 36}}] | ||
[[ | [[POTE generator]]: ~10/9 = 182.467 | ||
[[ | {{Val list|legend=1| 171, 1164, 1335, 1506, 1677, 1848, 2019, 11943, 13962, 15981, 18000, 20019, 22038 }} | ||
[[Badness]]: 0.013979 | |||
[[Category:Theory]] | [[Category:Theory]] | ||
[[Category:Temperament family]] | [[Category:Temperament family]] | ||
[[Category:Minortonic]] | [[Category:Minortonic]] | ||
Revision as of 12:44, 11 June 2021
This 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 171EDO. The wedgie is now ⟨⟨ 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.
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
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