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Augmented family: Difference between revisions

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== Triforce ==
== Triforce ==
[[File:triforce9.jpg|alt=triforce9.jpg|triforce9.jpg]]
Subgroup: 2.3.5.7
[[Comma list]]: 49/48, 128/125
[[Comma list]]: 49/48, 128/125


[[POTE generator]]: ~7/6 = 247.049
[[Mapping]]: [{{val|3 0 7 6}}, {{val|0 2 0 1}}]


Map: [<3 0 7 6|, <0 2 0 1|]
{{Multival|legend=1|6 0 3 -14 -12 7}}


Wedgie: <<6 0 3 -14 -12 7||
[[POTE generator]]: ~7/6 = 247.049


EDOs: 6, 9, [[15edo|15]], [[24edo|24]], [[39edo|39]], [[63edo|63cd]], [[102edo|102cd]]
{{Val list|legend=1| 6, 9, 15, 24, 39, 63cd, 102ccdd }}


Badness: 0.054988
[[Badness]]: 0.054988


[[File:triforce9.jpg|alt=triforce9.jpg|triforce9.jpg]]
=== 11-limit ===
Subgroup: 2.3.5.7.11


=== 11-limit ===
Comma list: 56/55, 77/75, 128/125
Comma list: 56/55, 77/75, 128/125


[[POTE_tuning|POTE generator]]: ~7/6 = 247.068
Mapping: [{{val|3 0 7 6 8}}, {{val|0 2 0 1 1}}]


Map: [<3 0 7 6 8|, <0 2 0 1 1|]
POTE generator: ~7/6 = 247.068


EDOs: 6, 9, 15, 24, 39, [[63edo|63cd]], [[102edo|102cd]]
Vals: {{Val list| 6, 9, 15, 24, 39, 63cd, 102ccdd }}


Badness: 0.026152
Badness: 0.026152
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=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 49/48, 56/55, 66/65, 77/75
Comma list: 49/48, 56/55, 66/65, 77/75


[[POTE_tuning|POTE generator]]: ~7/6 = 248.313
Mapping: [{{val|3 0 7 6 8 4}}, {{val|0 2 0 1 1 3}}]


Map: [<3 0 7 6 8 4|, <0 2 0 1 1 3|]
POTE generator: ~7/6 = 248.313


EDOs: 6, 9, 15, 24, [[63edo|63cdf]], [[87edo|87cdf]]
Vals: {{Val list| 6, 9, 15, 24, 63cdf, 87cddff }}


Badness: 0.020248
Badness: 0.020248


; Scales
[[triphi|Triforce(9) with L:s = phi]]
[[triphi|Triforce(9) with L:s = phi]]


=== Semitriforce ===
=== Semitriforce ===
Subgroup: 2.3.5.7.13
Comma list: 49/48, 56/55, 77/75, 507/500
Comma list: 49/48, 56/55, 77/75, 507/500
Mapping: [{{val|6 0 14 12 16 27}}, {{val|0 2 0 1 1 -1}}]


POTE generator: ~7/6 = 246.642
POTE generator: ~7/6 = 246.642


Map: [<6 0 14 12 16 27|, <0 2 0 1 1 -1|]
Vals: {{Val list| 6, 24, 54cd, 78cd, 102ccddf }}


EDOs: 6, 24, 54cd, 78cd, 102cdf
Badness: 0.059160


Badness: 0.059160
== Hemiug ==
Subgroup: 2.3.5.7


[[Comma list]]: 128/125, 1323/1250


== Hemiug  ==
[[Mapping]]: [{{val|3 1 7 14}}, {{val|0 2 0 -3}}]
Comma list: 128/125, 1323/1250


POTE generator: ~21/20 = 52.093
{{Multival|legend=1|6 0 -9 -14 -31 -21}}


Map: [<3 1 7 14|, <0 2 0 -3|]
[[POTE generator]]: ~21/20 = 52.093


Wedgie: <<6 0 -9 -14 -31 -21||
{{Val list|legend=1| 21, 24, 45c, 69cd }}


EDOs: 21, 24, 45c, 69cd
[[Badness]]: 0.137764


Badness: 0.137764
=== 11-limit ===
Subgroup: 2.3.5.7.11


=== 11-limit  ===
Comma list: 56/55, 128/125, 1323/1250
Comma list: 56/55, 128/125, 1323/1250
Mapping: [{{val|3 1 7 14 16}}, {{val|0 2 0 -3 -3}}]


POTE generator: ~21/20 = 51.655
POTE generator: ~21/20 = 51.655


Map: [<3 1 7 14 16|, <0 2 0 -3 -3|]
Vals: {{Val list| 21, 24, 69cd, 93ccdd }}


EDOs: 21, 24, 69cd, 93cd
Badness: 0.068062


Badness: 0.068062
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


=== 13-limit  ===
Comma list: 56/55, 66/65, 105/104, 507/500
Comma list: 56/55, 66/65, 105/104, 507/500
Mapping: [{{val|3 1 7 14 16 13}}, {{val|0 2 0 -3 -3 -1}}]


POTE generator: ~21/20 = 51.548
POTE generator: ~21/20 = 51.548


Map: [<3 1 7 14 16 13|, <0 2 0 -3 -3 -1|]
Vals: {{Val list| 21, 24, 69cdf, 93ccddf }}
 
Badness: 0.042289
 
== Trug ==
Subgroup: 2.3.5.7


EDOs: 21, 24, 69cdf, 93cdf
[[Comma list]]: 128/125, 360/343


Badness: 0.042289
[[Mapping]]: [{{val|3 1 7 6}}, {{val|0 3 0 2}}]


== Trug  ==
[[POTE generator]]: ~48/35 = 501.980
Comma list: 128/125, 360/343


POTE generator: ~48/35 = 501.980
{{Val list|legend=1| 9bd, 12 }}


Map: [<3 1 7 6|, <0 3 0 2|]
[[Badness]]: 0.138279


EDOs: 9bd, 12
== Oodako ==
Subgroup: 2.3.5.7


Badness: 0.138279
[[Comma list]]: 128/125, 2401/2400


== Oodako  ==
[[Mapping]]: [{{val|3 3 7 8}}, {{val|0 4 0 1}}]
Comma list: 128/125, 2401/2400


POTE generator: ~8/7 = 223.3540873874507317
{{Multival|legend=1|12 0 3 -28 -29 7}}


Map: [<3 3 7 8|, <0 4 0 1|]
[[POTE generator]]: ~8/7 = 223.354<!--223.3540873874507317-->


Wedgie: &lt;&lt;12 0 3 -28 -29 7||
{{Val list|legend=1| 6, 21, 27, 75c, 102ccd, 129bccd }}


EDOs: 6, 21, 27, 75c, 102cd, 129bcd
[[Badness]]: 0.113209


Badness: 0.113209
=== 11-limit ===
Subgroup: 2.3.5.7.11


=== 11-limit  ===
Comma list: 56/55, 128/125, 3773/3750
Comma list: 56/55, 128/125, 3773/3750
Mapping: [{{val|3 3 7 8 10}}, {{val|0 4 0 1 1}}]


POTE generator: ~8/7 = 223.019
POTE generator: ~8/7 = 223.019


Map: [&lt;3 3 7 8 10|, &lt;0 4 0 1 1|]
Vals: {{Val list| 6, 21, 27e }}


EDOs: 6, 21, 27e
Badness: 0.059192


Badness: 0.059192
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


=== 13-limit  ===
Comma list: 56/55, 78/77, 128/125, 507/500
Comma list: 56/55, 78/77, 128/125, 507/500
Mapping: [{{val|3 3 7 8 10 12}}, {{val|0 4 0 1 1 -2}}]


POTE generator: ~8/7 = 223.449
POTE generator: ~8/7 = 223.449


Map: [&lt;3 3 7 8 10 12|, &lt;0 4 0 1 1 -2|]
Vals: {{Val list| 6, 21, 27e }}
 
EDOs: 6, 21, 27e


Badness: 0.042323
Badness: 0.042323


== Hemisemiaug ==
== Hemisemiaug ==
Comma list: 128/125, 12005/11664
Comma list: 128/125, 12005/11664


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Badness: 0.210984
Badness: 0.210984


=== 11-limit ===
=== 11-limit ===
Comma list: 56/55, 128/125, 3773/3645
Comma list: 56/55, 128/125, 3773/3645


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Badness: 0.080738
Badness: 0.080738


== Niner ==
== Niner ==
Niner gives 9 as the complexity of an otonal tetrad, tying it with augene as a temperament supported by 27edo. Niner[18], therefore, has nine such tetrads.
Niner gives 9 as the complexity of an otonal tetrad, tying it with augene as a temperament supported by 27edo. Niner[18], therefore, has nine such tetrads.


Line 467: Line 489:
Badness: 0.067157
Badness: 0.067157


=== 11-limit ===
=== 11-limit ===
Comma list: 56/55, 128/125, 540/539
Comma list: 56/55, 128/125, 540/539


Line 478: Line 500:
Badness: 0.034861
Badness: 0.034861


=== 13-limit ===
=== 13-limit ===
Comma list: 56/55, 78/77, 91/90, 128/125
Comma list: 56/55, 78/77, 91/90, 128/125


Line 489: Line 511:
Badness: 0.024143
Badness: 0.024143


== Music ==
== Music ==
* [https://www.prismnet.com/~hmiller/music/temp-augmented.html Herman Miller's page about augmented temperament]
* [https://www.prismnet.com/~hmiller/music/temp-augmented.html Herman Miller's page about augmented temperament]


Retrieved from "https://en.xen.wiki/w/Augmented_family"