Augmented family: Difference between revisions

Revision as of 11:07, 7 June 2021

The 5-limit parent comma for the augmented family is 128/125, the diesis. Its monzo is [7 0 -3⟩, and flipping that yields ⟨⟨ 3 0 -7 ]] for the wedgie. Hence the period is 1/3 octave, and this is what is used for a major third. The generator can be taken as a fifth or a semitone, and 12 EDO, with its excellent fifth, is an obvious tuning for 5-limit augmented, though a sharper fifth might be preferred to go with the sharp third.

Augmented

Subgroup: 2.3.5

Comma: 128/125

Mapping: [⟨3 0 7], ⟨0 1 0]]

POTE generator: ~3/2 = 706.638

Template:Val list

Badness: 0.022315

Seven-limit children

The second comma of the normal comma list defines which 7-limit family member we are looking at. August adds 36/35, augene 64/63, hexe 256/245, hemiaug 245/243, and triforce 49/48. Hexe splits the period to 1/6 octave, and hemiaug the generator, giving quartertones instead of semitones.

August

Subgroup: 2.3.5.7

Comma list: 36/35, 128/125

Mapping: [⟨3 0 7 -1], ⟨0 1 0 2]]

Wedgie: ⟨⟨ 3 0 6 -7 1 14 ]]

POTE generator: ~3/2 = 696.011

Template:Val list

Badness: 0.026459

11-limit

Subgroup: 2.3.5.7.11

Comma list: 36/35, 45/44, 56/55

Mapping: [⟨3 0 7 -1 1], ⟨0 1 0 2 2]]

POTE generator: ~3/2 = 692.514

@@ Line 294: / Line 294: @@
 == Triforce ==
+[[File:triforce9.jpg|alt=triforce9.jpg|triforce9.jpg]]
+Subgroup: 2.3.5.7
 [[Comma list]]: 49/48, 128/125
-[[POTE generator]]: ~7/6 = 247.049
+[[Mapping]]: [{{val|3 0 7 6}}, {{val|0 2 0 1}}]
-Map: [&lt;3 0 7 6|, &lt;0 2 0 1|]
+{{Multival|legend=1|6 0 3 -14 -12 7}}
-Wedgie: &lt;&lt;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
-[[POTE_tuning|POTE generator]]: ~7/6 = 247.068
+Mapping: [{{val|3 0 7 6 8}}, {{val|0 2 0 1 1}}]
-Map: [&lt;3 0 7 6 8|, &lt;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
@@ Line 323: / Line 327: @@
 === 13-limit ===
+Subgroup: 2.3.5.7.11.13
 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: [&lt;3 0 7 6 8 4|, &lt;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
+; Scales
 [[triphi|Triforce(9) with L:s = phi]]
 === Semitriforce ===
+Subgroup: 2.3.5.7.13
 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
-Map: [&lt;6 0 14 12 16 27|, &lt;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: [&lt;3 1 7 14|, &lt;0 2 0 -3|]
+[[POTE generator]]: ~21/20 = 52.093
-Wedgie: &lt;&lt;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
+Mapping: [{{val|3 1 7 14 16}}, {{val|0 2 0 -3 -3}}]
 POTE generator: ~21/20 = 51.655
-Map: [&lt;3 1 7 14 16|, &lt;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
+Mapping: [{{val|3 1 7 14 16 13}}, {{val|0 2 0 -3 -3 -1}}]
 POTE generator: ~21/20 = 51.548
-Map: [&lt;3 1 7 14 16 13|, &lt;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: [&lt;3 1 7 6|, &lt;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: [&lt;3 3 7 8|, &lt;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
+Mapping: [{{val|3 3 7 8 10}}, {{val|0 4 0 1 1}}]
 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
+Mapping: [{{val|3 3 7 8 10 12}}, {{val|0 4 0 1 1 -2}}]
 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
-== Hemisemiaug  ==
+== Hemisemiaug ==
 Comma list: 128/125, 12005/11664
@@ Line 441: / Line 463: @@
 Badness: 0.210984
-=== 11-limit  ===
+=== 11-limit ===
 Comma list: 56/55, 128/125, 3773/3645
@@ Line 452: / Line 474: @@
 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.
@@ Line 467: / Line 489: @@
 Badness: 0.067157
-=== 11-limit  ===
+=== 11-limit ===
 Comma list: 56/55, 128/125, 540/539
@@ Line 478: / Line 500: @@
 Badness: 0.034861
-=== 13-limit  ===
+=== 13-limit ===
 Comma list: 56/55, 78/77, 91/90, 128/125
@@ Line 489: / Line 511: @@
 Badness: 0.024143
-== Music  ==
+== Music ==
 * [https://www.prismnet.com/~hmiller/music/temp-augmented.html Herman Miller's page about augmented temperament]

Augmented family: Difference between revisions

Revision as of 11:07, 7 June 2021

Augmented

Seven-limit children

August

11-limit

13-limit

Augustus

Augene

11-limit

13-limit

Ogene

Agene

Eugene

Inflated

11-limit

Deflated

11-limit

Hexe

11-limit

13-limit

Hemiaug

11-limit

13-limit

Triforce

11-limit

13-limit

Semitriforce

Hemiug

11-limit

13-limit

Trug

Oodako

11-limit

13-limit

Hemisemiaug

11-limit

Niner

11-limit

13-limit

Music

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