Quartismic family: Difference between revisions
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{{Val list|legend=1| 24, 46c, 65d, 89, 135, 159, 224, 383, 472, 696, 1168, 1327, 1551, 2023e }} | {{Val list|legend=1| 24, 46c, 65d, 89, 135, 159, 224, 383, 472, 696, 1168, 1327, 1551, 2023e }} | ||
[[Badness]]: 4.563 × 10<sup>-3</sup> | |||
=== 13-limit === | === 13-limit === | ||
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[[POTE generator]]s: ~11/10 = 166.0628, ~33/32 = 53.4151 | [[POTE generator]]s: ~11/10 = 166.0628, ~33/32 = 53.4151 | ||
[[Badness]]: 16.903 × 10<sup>-3</sup> | |||
== Meanquarter == | == Meanquarter == | ||
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{{Val list|legend=1| 24, 43, 67, 110c }} | {{Val list|legend=1| 24, 43, 67, 110c }} | ||
[[Badness]]: 15.125 × 10<sup>-3</sup> | |||
== Coin == | == Coin == | ||
| Line 94: | Line 100: | ||
{{Val list|legend=1| 19d, 22 }} | {{Val list|legend=1| 19d, 22 }} | ||
[[Badness]]: 70.470 × 10<sup>-3</sup> | |||
== Escapismic == | == Escapismic == | ||
In escapisimic, both the escapade comma and the quartisma are tempered out, thus, it is essentially an [[Escapade family| | In escapisimic, both the escapade comma and the quartisma are tempered out, thus, it is essentially an [[Escapade family|escapade expansion]]. | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 108: | Line 116: | ||
{{Val list|legend=1| 21, 22, 43, 65d, 521d, 543, 564, 586, 629c, 651 }} | {{Val list|legend=1| 21, 22, 43, 65d, 521d, 543, 564, 586, 629c, 651 }} | ||
[[Badness]]: 64.233 × 10<sup>-3</sup> | |||
== Dietismic == | == Dietismic == | ||
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[[POTE generator]]s: ~3/2 = 704.5238, ~33/32 = 53.4408 | [[POTE generator]]s: ~3/2 = 704.5238, ~33/32 = 53.4408 | ||
[[Mapping]]: [{{val| 2 | [[Mapping]]: [{{val| 2 0 11 2 10 }}, {{val| 0 1 -2 1 -1 }}, {{val| 0 0 0 5 1 }}] | ||
{{Val list|legend=1| 22, 46, 68, 114 }} | {{Val list|legend=1| 22, 46, 68, 114 }} | ||
[[Badness]]: 23.250 × 10<sup>-3</sup> | |||
Scales: | Scales: | ||
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{{Val list|legend=1| 68, 91, 159, 246, 337, 405 }} | {{Val list|legend=1| 68, 91, 159, 246, 337, 405 }} | ||
[[Badness]]: 26.882 × 10<sup>-3</sup> | |||
== Doublefour == | == Doublefour == | ||
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{{Val list|legend=1| 48d, 68, 116d, 157c, 225 }} | {{Val list|legend=1| 48d, 68, 116d, 157c, 225 }} | ||
[[Badness]]: 81.083 × 10<sup>-3</sup> | |||
[[Category:Regular temperament theory]] | [[Category:Regular temperament theory]] | ||
Revision as of 14:44, 7 June 2021
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Todo: discuss title This doesn't follow the definition of temperament family |
The quartismic family is built up from temperaments of various ranks that temper out the quartisma- the unnoticeable comma with the ratio 117440512/117406179, and a monzo of [24 -6 0 1 -5⟩. Among the members of this family are quartismatic, altierran, meanquarter, coin, escapismic, dietismic, kleirtismic, and doublefour.
Quartismic
The 11-limit parent comma for the quartismic family is the the quartisma with a ratio of 117440512/117406179 and a monzo of [24 -6 0 1 -5⟩. As the quartisma is an unnoticeable comma, this rank-4 temperament is a microtemperament.
Subgroup: 2.3.5.7.11
Comma list: 117440512/117406179
Mapping: [⟨1 0 0 1 5], ⟨0 1 0 1 -1], ⟨0 0 1 0 0], ⟨0 0 0 5 1]]
Wedgie: ⟨⟨⟨⟨ 5 1 0 -6 -24 ]]]]
POTE generators: ~3/2 = 701.9826, ~5/4 = 386.3427, ~33/32 = 53.3748
Badness: 0.274 × 10-6
Quartismatic
There are some temperaments in the quartismic family in which the quartisma is tempered out, but without any sort of five-limit representation. This particular temperament is the parent temperament of all such no-fives children, and is referred to as Saquinlu-azo temperament in color notation.
Subgroup: 2.3.7.11
Comma list: 117440512/117406179
Sval mapping: [⟨1 0 1 5], ⟨0 1 1 -1], ⟨0 0 5 1]]
POTE generators: ~3/2 = 701.9826, ~33/32 = 53.3748
The following unnamed rank-2 quartismic temperament MOS scales have been found
- Rank 2 scale (106.71461627796054, 1200.0), 5|5
- The following scale tree has been found: 1200-106.71461627796054-12-11 Scale Tree
Altierran
In altierran, both the schisma and the quartisma are tempered out.
Subgroup: 2.3.5.7.11
Comma list: 32805/32768, 161280/161051
Mapping: [⟨1 0 15 1 5], ⟨0 1 -8 1 -1], ⟨0 0 0 5 1]]
Wedgie: ⟨⟨⟨ -102 24 -15 75 6 -8 40 1 -5 0 ]]]
POTE generators: ~3/2 = 701.7299, ~33/32 = 53.3889
Badness: 4.563 × 10-3
13-limit
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Todo: rename Not an immediate extension, must be renamed |
Subgroup: 2.3.5.7.11.13
Comma list: 10985/10976, 32805/32768, 161280/161051
Mapping: [⟨1 2 -1 3 3 5], ⟨0 -3 24 -3 3 -11], ⟨0 0 0 5 1 5]]
POTE generators: ~11/10 = 166.0628, ~33/32 = 53.4151
Badness: 16.903 × 10-3
Meanquarter
In meanquarter, both the meantone comma and the quartisma are tempered out.
Subgroup: 2.3.5.7.11
Comma list: 81/80, 4128768/4026275
Mapping: [⟨1 0 -4 1 5], ⟨0 1 4 1 -1], ⟨0 0 5 1]]
POTE generators: ~3/2 = 697.3325, ~33/32 = 54.1064
Badness: 15.125 × 10-3
Coin
In coin, both the magic comma and the quartisma are tempered out.
Subgroup: 2.3.5.7.11
Comma list: 3125/3072, 117440512/117406179
Mapping: [⟨1 0 2 1 5], ⟨0 5 1 0 -6], ⟨0 0 0 5 1]]
POTE generators: ~5/4 = 380.3623, ~9/7 = 433.3120
Badness: 70.470 × 10-3
Escapismic
In escapisimic, both the escapade comma and the quartisma are tempered out, thus, it is essentially an escapade expansion.
Subgroup: 2.3.5.7.11
Comma list: 117440512/117406179, 4294967296/4271484375
Mapping: [⟨1 2 2 3 3], ⟨0 -9 7 -4 10], ⟨0 0 0 5 1]]
POTE generators: ~33/32 = 55.3538
Badness: 64.233 × 10-3
Dietismic
In dietismic, both the diaschisma and the quartisma are tempered out. Dietismic can easily be further tempered to shrutar, and in fact, it is rather unusual to find a different tempering option.
Subgroup: 2.3.5.7.11
Comma list: 2048/2025, 117440512/117406179
POTE generators: ~3/2 = 704.5238, ~33/32 = 53.4408
Mapping: [⟨2 0 11 2 10], ⟨0 1 -2 1 -1], ⟨0 0 0 5 1]]
Badness: 23.250 × 10-3
Scales:
Rank 2 scale (52.6800, 2/1), 13|9
Rank 2 scale (53.3742, 2/1), 13|9
Kleirtismic
In kleirtismic, both the kleisma and the quartisma are tempered out. The "kleir-" in "kleirtismic" is pronounced the same as "Clair".
Subgroup: 2.3.5.7.11
Comma list: 15625/15552, 117440512/117406179
Mapping: [⟨1 0 1 1 5], ⟨0 6 5 1 -7], ⟨0 0 0 5 1]]
POTE generators: ~6/5 = 317.0291, ~68/55 = 370.2940
Badness: 26.882 × 10-3
Doublefour
In doublefour, both the tetracot comma and the quartisma are tempered out.
Subgroup: 2.3.5.7.11
Comma list: 20000/19683, 100656875/99090432
Mapping: [⟨1 1 1 2 4], ⟨0 4 9 4 -4], ⟨0 0 0 5 1]]
POTE generators: ~425/384 = 175.9566, ~33/32 = 52.9708
Badness: 81.083 × 10-3