Augmented family: Difference between revisions
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The 5-limit parent comma for the '''augmented family''' is 128/125, the [[128/125|diesis]]. Its [[monzo]] is {{Monzo| 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 [[12edo]], 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. | The 5-limit parent comma for the '''augmented family''' is 128/125, the [[128/125|diesis]]. Its [[monzo]] is {{Monzo| 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 [[12edo]], 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 = | == Augmented == | ||
[[Comma]]: 128/125 | [[Comma]]: 128/125 | ||
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[[Badness]]: 0.022315 | [[Badness]]: 0.022315 | ||
== Seven-limit children == | === Seven-limit children === | ||
The second comma of the [[Normal lists|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. | The second comma of the [[Normal lists|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 = | == August == | ||
[[Comma list]]: 36/35, 128/125 | [[Comma list]]: 36/35, 128/125 | ||
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[[Badness]]: 0.026459 | [[Badness]]: 0.026459 | ||
== 11-limit == | === 11-limit === | ||
Comma list: 36/35, 45/44, 56/55 | Comma list: 36/35, 45/44, 56/55 | ||
| Line 41: | Line 39: | ||
Badness: 0.020191 | Badness: 0.020191 | ||
== 13-limit == | === 13-limit === | ||
Comma list: 27/26, 36/35, 45/44, 56/55 | Comma list: 27/26, 36/35, 45/44, 56/55 | ||
| Line 52: | Line 50: | ||
Badness: 0.018448 | Badness: 0.018448 | ||
== Augustus == | === Augustus === | ||
Comma list: 26/25, 36/35, 45/44, 56/55 | Comma list: 26/25, 36/35, 45/44, 56/55 | ||
| Line 63: | Line 61: | ||
Badness: 0.022236 | Badness: 0.022236 | ||
= Augene = | == Augene == | ||
[[Comma list]]: 64/63, 126/125 | [[Comma list]]: 64/63, 126/125 | ||
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Badness: 0.024816 | Badness: 0.024816 | ||
== 11-limit == | === 11-limit === | ||
Comma list: 56/55, 64/63, 100/99 | Comma list: 56/55, 64/63, 100/99 | ||
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Badness: 0.019613 | Badness: 0.019613 | ||
== 13-limit == | === 13-limit === | ||
Comma list: 40/39, 56/55, 64/63, 66/65 | Comma list: 40/39, 56/55, 64/63, 66/65 | ||
| Line 98: | Line 96: | ||
Badness: 0.020785 | Badness: 0.020785 | ||
== Ogene == | === Ogene === | ||
Comma list: 56/55, 64/63, 91/90, 100/99 | Comma list: 56/55, 64/63, 91/90, 100/99 | ||
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Badness: 0.022890 | Badness: 0.022890 | ||
== Agene == | === Agene === | ||
Comma list: 56/55, 64/63, 78/77, 100/99 | Comma list: 56/55, 64/63, 78/77, 100/99 | ||
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Badness: 0.023113 | Badness: 0.023113 | ||
== Eugene == | === Eugene === | ||
Comma list: 55/54, 64/63, 77/75 | Comma list: 55/54, 64/63, 77/75 | ||
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* [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of27sonatina.mp3 Galticeran Sonatina] by [http://soundcloud.com/joelgranttaylor/galticeran_sonatina Joel Taylor] | * [http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of27sonatina.mp3 Galticeran Sonatina] by [http://soundcloud.com/joelgranttaylor/galticeran_sonatina Joel Taylor] | ||
= Inflated = | == Inflated == | ||
Comma list: 28/27, 128/125 | Comma list: 28/27, 128/125 | ||
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Badness: 0.054729 | Badness: 0.054729 | ||
== 11-limit == | === 11-limit === | ||
Comma list: 28/27, 55/54, 128/125 | Comma list: 28/27, 55/54, 128/125 | ||
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Badness: 0.031171 | Badness: 0.031171 | ||
= Deflated = | == Deflated == | ||
Comma list: 21/20, 128/125 | Comma list: 21/20, 128/125 | ||
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Badness: 0.059079 | Badness: 0.059079 | ||
== 11-limit == | === 11-limit === | ||
Comma list: 21/20, 33/32, 128/125 | Comma list: 21/20, 33/32, 128/125 | ||
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Badness: 0.037183 | Badness: 0.037183 | ||
= Hexe = | == Hexe == | ||
Comma list: 50/49, 128/125 | Comma list: 50/49, 128/125 | ||
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Badness: 0.057730 | Badness: 0.057730 | ||
== 11-limit == | === 11-limit === | ||
Comma list: 50/49, 56/55, 125/121 | Comma list: 50/49, 56/55, 125/121 | ||
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Badness: 0.038412 | Badness: 0.038412 | ||
== 13-limit == | === 13-limit === | ||
Comma list: 50/49, 56/55, 66/65, 105/104 | Comma list: 50/49, 56/55, 66/65, 105/104 | ||
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Badness: 0.035946 | Badness: 0.035946 | ||
= Hemiaug = | == Hemiaug == | ||
Comma list: 128/125, 245/243 | Comma list: 128/125, 245/243 | ||
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Badness: 0.070463 | Badness: 0.070463 | ||
== 11-limit == | === 11-limit === | ||
Comma list: 56/55, 128/125, 245/243 | Comma list: 56/55, 128/125, 245/243 | ||
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Badness: 0.038232 | Badness: 0.038232 | ||
== 13-limit == | === 13-limit === | ||
Comma list: 56/55, 91/90, 128/125, 245/243 | Comma list: 56/55, 91/90, 128/125, 245/243 | ||
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Badness: 0.030159 | Badness: 0.030159 | ||
= Triforce = | == Triforce == | ||
Comma list: 49/48, 128/125 | Comma list: 49/48, 128/125 | ||
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[[File:triforce9.jpg|alt=triforce9.jpg|triforce9.jpg]] | [[File:triforce9.jpg|alt=triforce9.jpg|triforce9.jpg]] | ||
== 11-limit == | === 11-limit === | ||
Comma list: 56/55, 77/75, 128/125 | Comma list: 56/55, 77/75, 128/125 | ||
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Badness: 0.026152 | Badness: 0.026152 | ||
=== Musical example === | ==== Musical example ==== | ||
* [http://soundcloud.com/cityoftheasleep/the-triforce-of-courage15 The Triforce of Courage (tuned in 15edo)] by Igliashon Jones | * [http://soundcloud.com/cityoftheasleep/the-triforce-of-courage15 The Triforce of Courage (tuned in 15edo)] by Igliashon Jones | ||
== 13-limit == | === 13-limit === | ||
Comma list: 49/48, 56/55, 66/65, 77/75 | Comma list: 49/48, 56/55, 66/65, 77/75 | ||
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[[triphi|Triforce(9) with L:s = phi]] | [[triphi|Triforce(9) with L:s = phi]] | ||
== Semitriforce == | === Semitriforce === | ||
Comma list: 49/48, 56/55, 77/75, 507/500 | Comma list: 49/48, 56/55, 77/75, 507/500 | ||
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= Hemiug = | == Hemiug == | ||
Comma list: 128/125, 1323/1250 | Comma list: 128/125, 1323/1250 | ||
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Badness: 0.137764 | Badness: 0.137764 | ||
== 11-limit == | === 11-limit === | ||
Comma list: 56/55, 128/125, 1323/1250 | Comma list: 56/55, 128/125, 1323/1250 | ||
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Badness: 0.068062 | Badness: 0.068062 | ||
== 13-limit == | === 13-limit === | ||
Comma list: 56/55, 66/65, 105/104, 507/500 | Comma list: 56/55, 66/65, 105/104, 507/500 | ||
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Badness: 0.042289 | Badness: 0.042289 | ||
= Trug = | == Trug == | ||
Comma list: 128/125, 360/343 | Comma list: 128/125, 360/343 | ||
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Badness: 0.138279 | Badness: 0.138279 | ||
= Oodako = | == Oodako == | ||
Comma list: 128/125, 2401/2400 | Comma list: 128/125, 2401/2400 | ||
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Badness: 0.113209 | Badness: 0.113209 | ||
== 11-limit == | === 11-limit === | ||
Comma list: 56/55, 128/125, 3773/3750 | Comma list: 56/55, 128/125, 3773/3750 | ||
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Badness: 0.059192 | Badness: 0.059192 | ||
== 13-limit == | === 13-limit === | ||
Comma list: 56/55, 78/77, 128/125, 507/500 | Comma list: 56/55, 78/77, 128/125, 507/500 | ||
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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. | ||
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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 | ||
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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 | ||
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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] | ||
Revision as of 16:16, 31 May 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 12edo, 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
Comma: 128/125
POTE generator: ~3/2 = 706.638
Map: [<3 0 7|, <0 1 0|]
Optimal ET sequence: 3, 12, 27, 39, 51c, 90cc
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
Comma list: 36/35, 128/125
POTE generator: ~3/2 = 696.011
Map: [<3 0 7 -1|, <0 1 0 2|]
Wedgie: <<3 0 6 -7 1 14||
Badness: 0.026459
11-limit
Comma list: 36/35, 45/44, 56/55
POTE generator: ~3/2 = 692.514
Map: [<3 0 7 -1 1|, <0 1 0 2 2|]
Badness: 0.020191
13-limit
Comma list: 27/26, 36/35, 45/44, 56/55
POTE generator: ~3/2 = 688.783
Map: [<3 0 7 -1 1 -3|, <0 1 0 2 2 3|]
EDOs: 9, 12f, 21, 33ef, 54bcef
Badness: 0.018448
Augustus
Comma list: 26/25, 36/35, 45/44, 56/55
POTE generator: ~3/2 = 685.356
Map: [<3 0 7 -1 1 11|, <0 1 0 2 2 0|]
EDOs: 9, 12, 21f
Badness: 0.022236
Augene
Comma list: 64/63, 126/125
POTE generator: ~3/2 = 709.257
Map: [<3 0 7 18|, <0 1 0 -2|]
Wedgie: <<3 0 -6 -7 -18 -14||
Badness: 0.024816
11-limit
Comma list: 56/55, 64/63, 100/99
POTE generator: ~3/2 = 711.177
Map: [<3 0 7 18 20|, <0 1 0 -2 -2|]
EDOs: 12, 15, 27e
Badness: 0.019613
13-limit
Comma list: 40/39, 56/55, 64/63, 66/65
POTE generator: ~3/2 = 712.013
Map: [<3 0 7 18 20 16|, <0 1 0 -2 -2 -1|]
EDOs: 12f, 15, 27ef
Badness: 0.020785
Ogene
Comma list: 56/55, 64/63, 91/90, 100/99
POTE generator: ~3/2 = 712.609
Map: [<3 0 7 18 20 -8|, <0 1 0 -2 -2 4|]
EDOs: 12, 15, 27e, 69bcef
Badness: 0.022890
Agene
Comma list: 56/55, 64/63, 78/77, 100/99
POTE generator: ~3/2 = 709.677
Map: [<3 0 7 18 20 35|, <0 1 0 -2 -2 -5|]
EDOs: 12f, 27e
Badness: 0.023113
Eugene
Comma list: 55/54, 64/63, 77/75
POTE generator: ~3/2 = 714.150
Map: [<3 0 7 18 -4|, <0 1 0 -2 3|]
EDOs: 15, 27, 42
Badness: 0.035554
Inflated
Comma list: 28/27, 128/125
POTE generator: ~3/2 = 722.719
Map: [<3 0 7 -6|, <0 1 0 3|]
Wedgie: <<3 0 9 -7 6 21||
EDOs: 3d, 15, 48bc, 63bc, 78bc
Badness: 0.054729
11-limit
Comma list: 28/27, 55/54, 128/125
POTE generator: ~3/2 = 722.663
Map: [<3 0 7 -6 -4|, <0 1 0 3 3|]
EDOs: 3de, 15, 48bce, 63bce, 78bce
Badness: 0.031171
Deflated
Comma list: 21/20, 128/125
POTE generator: ~3/2 = 681.629
Map: [<3 0 7 13|, <0 1 0 -1|]
Wedgie: <<3 0 -3 -7 -13 -7||
EDOs: 3, 9, 21d, 30bd
Badness: 0.059079
11-limit
Comma list: 21/20, 33/32, 128/125
POTE generator: ~3/2 = 680.042
Map: [<3 0 7 13 15|, <0 1 0 -1 -1|]
EDOs: 3, 9, 21de, 30bde
Badness: 0.037183
Hexe
Comma list: 50/49, 128/125
POTE generator: ~3/2 = 710.963
Map: [<6 0 14 17|, <0 1 0 0|]
Wedgie: <<6 0 0 -14 -17 0||
Badness: 0.057730
11-limit
Comma list: 50/49, 56/55, 125/121
POTE generator: ~3/2 = 714.304
Map: [<6 0 14 17 21|, <0 1 0 0 0|]
EDOs: 6, 12, 30de
Badness: 0.038412
13-limit
Comma list: 50/49, 56/55, 66/65, 105/104
POTE generator: ~3/2 = 710.005
Map: [<6 0 14 17 21 13|, <0 1 0 0 0 1|]
EDOs: 6f, 12f
Badness: 0.035946
Hemiaug
Comma list: 128/125, 245/243
POTE generator: ~28/27 = 45.118
Map: [<3 1 7 -1|, <0 2 0 5|]
Wedgie: <<6 0 15 -14 7 35||
Badness: 0.070463
11-limit
Comma list: 56/55, 128/125, 245/243
POTE generator: ~28/27 = 45.788
Map: [<3 1 7 -1 1|, <0 2 0 5 5|]
EDOs: 24, 27e, 51ce, 78ce
Badness: 0.038232
13-limit
Comma list: 56/55, 91/90, 128/125, 245/243
POTE generator: ~28/27 = 46.250
Map: [<3 1 7 -1 1 13|, <0 2 0 5 5 -1|]
EDOs: 24, 27e, 51ce, 78ce
Badness: 0.030159
Triforce
Comma list: 49/48, 128/125
POTE generator: ~7/6 = 247.049
Map: [<3 0 7 6|, <0 2 0 1|]
Wedgie: <<6 0 3 -14 -12 7||
EDOs: 6, 9, 15, 24, 39, 63cd, 102cd
Badness: 0.054988
11-limit
Comma list: 56/55, 77/75, 128/125
POTE generator: ~7/6 = 247.068
Map: [<3 0 7 6 8|, <0 2 0 1 1|]
EDOs: 6, 9, 15, 24, 39, 63cd, 102cd
Badness: 0.026152
Musical example
- The Triforce of Courage (tuned in 15edo) by Igliashon Jones
13-limit
Comma list: 49/48, 56/55, 66/65, 77/75
POTE generator: ~7/6 = 248.313
Map: [<3 0 7 6 8 4|, <0 2 0 1 1 3|]
EDOs: 6, 9, 15, 24, 63cdf, 87cdf
Badness: 0.020248
Semitriforce
Comma list: 49/48, 56/55, 77/75, 507/500
POTE generator: ~7/6 = 246.642
Map: [<6 0 14 12 16 27|, <0 2 0 1 1 -1|]
EDOs: 6, 24, 54cd, 78cd, 102cdf
Badness: 0.059160
Hemiug
Comma list: 128/125, 1323/1250
POTE generator: ~21/20 = 52.093
Map: [<3 1 7 14|, <0 2 0 -3|]
Wedgie: <<6 0 -9 -14 -31 -21||
EDOs: 21, 24, 45c, 69cd
Badness: 0.137764
11-limit
Comma list: 56/55, 128/125, 1323/1250
POTE generator: ~21/20 = 51.655
Map: [<3 1 7 14 16|, <0 2 0 -3 -3|]
EDOs: 21, 24, 69cd, 93cd
Badness: 0.068062
13-limit
Comma list: 56/55, 66/65, 105/104, 507/500
POTE generator: ~21/20 = 51.548
Map: [<3 1 7 14 16 13|, <0 2 0 -3 -3 -1|]
EDOs: 21, 24, 69cdf, 93cdf
Badness: 0.042289
Trug
Comma list: 128/125, 360/343
POTE generator: ~48/35 = 501.980
Map: [<3 1 7 6|, <0 3 0 2|]
EDOs: 9bd, 12
Badness: 0.138279
Oodako
Comma list: 128/125, 2401/2400
POTE generator: ~8/7 = 223.3540873874507317
Map: [<3 3 7 8|, <0 4 0 1|]
Wedgie: <<12 0 3 -28 -29 7||
EDOs: 6, 21, 27, 75c, 102cd, 129bcd
Badness: 0.113209
11-limit
Comma list: 56/55, 128/125, 3773/3750
POTE generator: ~8/7 = 223.019
Map: [<3 3 7 8 10|, <0 4 0 1 1|]
EDOs: 6, 21, 27e
Badness: 0.059192
13-limit
Comma list: 56/55, 78/77, 128/125, 507/500
POTE generator: ~8/7 = 223.449
Map: [<3 3 7 8 10 12|, <0 4 0 1 1 -2|]
EDOs: 6, 21, 27e
Badness: 0.042323
Hemisemiaug
Comma list: 128/125, 12005/11664
POTE generator: ~15/14 = 144.515
Map: [<6 1 14 4|, <0 2 0 3|]
Wedgie: <<12 0 18 -28 -5 42||
Edos: 18, 24, 42, 66c, 108bc
Badness: 0.210984
11-limit
Comma list: 56/55, 128/125, 3773/3645
POTE generator: ~15/14 = 144.780
Map: [<6 1 14 4 8|, <0 2 0 3 3|]
EDOs: 24, 42e, 66ce, 108bce
Badness: 0.080738
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.
Comma list: 128/125, 686/675
POTE generator: ~3/2 = 707.167
Map: [<9 0 21 11|, <0 1 0 1|]
Wedgie: <<9 0 9 -21 -11 21||
EDOs: 9, 18, 27, 63c, 90c
Badness: 0.067157
11-limit
Comma list: 56/55, 128/125, 540/539
POTE generator: ~3/2 = 706.726
Map: [<9 0 21 11 17|, <0 1 0 1 1|]
EDOs: 9, 27e, 36, 63ce
Badness: 0.034861
13-limit
Comma list: 56/55, 78/77, 91/90, 128/125
POTE generator: ~3/2 = 706.889
Map: [<9 0 21 11 17 19|, <0 1 0 1 1 1|]
EDOs: 9, 27e, 36, 63ce
Badness: 0.024143