Augmented family: Difference between revisions

Revision as of 13:27, 15 November 2024

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 5/4, the classical 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

Subgroup: 2.3.5

Comma list: 128/125

Mapping: [⟨3 0 7], ⟨0 1 0]]

mapping generators: ~5/4, ~3

Optimal tunings:

CTE: ~5/4 = 400.000, ~3/2 = 701.955 (~16/15 = 98.045)
POTE: ~5/4 = 400.000, ~3/2 = 706.638 (~16/15 = 93.362)

Optimal ET sequence: 3, 12, 27, 39, 51c, 90cc

Badness (Smith): 0.022315

Overview to extensions

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 ]]

Optimal tunings:

CTE: ~5/4 = 400.000, ~3/2 = 692.124 (~16/15 = 107.876)
POTE: ~5/4 = 400.000, ~3/2 = 696.011 (~16/15 = 103.989)

Optimal ET sequence: 9, 12, 45cd, 57cd, 69cdd

Badness (Smith): 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]]

Wedgie: ⟨⟨ 3 0 6 6 -7 1 -1 14 14 -4 ]]

Optimal tunings:

CTE: ~5/4 = 400.000, ~3/2 = 687.685 (~16/15 = 112.315)
POTE: ~5/4 = 400.000, ~3/2 = 692.514 (~16/15 = 107.486)

Optimal ET sequence: 9, 12, 21, 33e, 45cde

Badness (Smith): 0.020191

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 27/26, 36/35, 45/44, 56/55

Mapping: [⟨3 0 7 -1 1 -3], ⟨0 1 0 2 2 3]]

Optimal tunings:

CTE: ~5/4 = 400.000, ~3/2 = 685.084 (~16/15 = 114.916)
POTE: ~5/4 = 400.000, ~3/2 = 688.783 (~16/15 = 111.217)

Optimal ET sequence: 9, 12f, 21, 33ef, 54bceeff

Badness (Smith): 0.018448

Augustus

Subgroup: 2.3.5.7.11.13

Comma list: 26/25, 36/35, 45/44, 56/55

Mapping: [⟨3 0 7 -1 1 11], ⟨0 1 0 2 2 0]]

Optimal tunings:

CTE: ~5/4 = 400.000, ~3/2 = 687.685 (~16/15 = 112.315)
POTE: ~5/4 = 400.000, ~3/2 = 685.356 (~16/15 = 114.644)

Optimal ET sequence: 9, 12, 21f

Badness (Smith): 0.022236

Augene

Main article: Augene

Subgroup: 2.3.5.7

Comma list: 64/63, 126/125

Mapping: [⟨3 0 7 18], ⟨0 1 0 -2]]

Wedgie: ⟨⟨ 3 0 -6 -7 -18 -14 ]]

Optimal tunings:

CTE: ~5/4 = 400.000, ~3/2 = 709.595 (~21/20 = 90.405)
POTE: ~5/4 = 400.000, ~3/2 = 709.257 (~21/20 = 90.743)

Optimal ET sequence: 12, 27, 39d, 66cd

Badness (Smith): 0.024816

11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 64/63, 100/99

Mapping: [⟨3 0 7 18 20], ⟨0 1 0 -2 -2]]

Wedgie: ⟨⟨ 3 0 -6 -6 -7 -18 -20 -14 -14 4 ]]

Optimal tunings:

CTE: ~5/4 = 400.000, ~3/2 = 713.570 (~21/20 = 86.430)
POTE: ~5/4 = 400.000, ~3/2 = 711.177 (~21/20 = 88.823)

Optimal ET sequence: 12, 15, 27e

Badness (Smith): 0.019613

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 40/39, 56/55, 64/63, 66/65

Mapping: [⟨3 0 7 18 20 16], ⟨0 1 0 -2 -2 -1]]

Optimal tunings:

CTE: ~5/4 = 400.000, ~3/2 = 716.123 (~21/20 = 83.877)
POTE: ~5/4 = 400.000, ~3/2 = 712.013 (~21/20 = 87.987)

Optimal ET sequence: 12f, 15, 27eff

Badness (Smith): 0.020785

Ogene

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 64/63, 91/90, 100/99

Mapping: [⟨3 0 7 18 20 -8], ⟨0 1 0 -2 -2 4]]

Optimal tunings:

CTE: ~5/4 = 400.000, ~3/2 = 711.902 (~21/20 = 88.098)
POTE: ~5/4 = 400.000, ~3/2 = 712.609 (~21/20 = 87.391)

Optimal ET sequence: 12, 15, 27e, 69bceef

Badness (Smith): 0.022890

Agene

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 64/63, 78/77, 100/99

Mapping: [⟨3 0 7 18 20 35], ⟨0 1 0 -2 -2 -5]]

Optimal tunings:

CTE: ~5/4 = 400.000, ~3/2 = 712.572 (~21/20 = 87.428)
POTE: ~5/4 = 400.000, ~3/2 = 709.677 (~21/20 = 90.323)

Optimal ET sequence: 12f, 27e

Badness (Smith): 0.023113

Eugene

Subgroup: 2.3.5.7.11

Comma list: 55/54, 64/63, 77/75

Mapping: [⟨3 0 7 18 -4], ⟨0 1 0 -2 3]]

Optimal tunings:

CTE: ~5/4 = 400.000, ~3/2 = 713.002 (~21/20 = 86.998)
POTE: ~5/4 = 400.000, ~3/2 = 714.150 (~21/20 = 85.850)

Optimal ET sequence: 12e, 15, 27, 42

Badness (Smith): 0.035554

Inflated

Subgroup: 2.3.5.7

Comma list: 28/27, 128/125

Mapping: [⟨3 0 7 -6], ⟨0 1 0 3]]

Wedgie: ⟨⟨ 3 0 9 -7 6 21 ]]

Optimal tunings:

CTE: ~5/4 = 400.000, ~3/2 = 717.517 (~25/24 = 82.483)
POTE: ~5/4 = 400.000, ~3/2 = 722.719 (~25/24 = 77.281)

Optimal ET sequence: 3d, 12d, 15

Badness (Smith): 0.054729

11-limit

Subgroup: 2.3.5.7.11

Comma list: 28/27, 55/54, 128/125

Mapping: [⟨3 0 7 -6 -4], ⟨0 1 0 3 3]]

Optimal tunings:

CTE: ~5/4 = 400.000, ~3/2 = 717.382 (~25/24 = 82.618)
POTE: ~5/4 = 400.000, ~3/2 = 722.663 (~25/24 = 77.337)

Optimal ET sequence: 3de, 12de, 15

Badness (Smith): 0.031171

Deflated

Subgroup: 2.3.5.7

Comma list: 21/20, 128/125

Mapping: [⟨3 0 7 13], ⟨0 1 0 -1]]

Wedgie: ⟨⟨ 3 0 -3 -7 -13 -7 ]]

Optimal tunings:

CTE: ~5/4 = 400.000, ~3/2 = 684.847 (~16/15 = 115.153)
POTE: ~5/4 = 400.000, ~3/2 = 681.629 (~16/15 = 118.371)

Optimal ET sequence: 3, 9

Badness (Smith): 0.059079

11-limit

Subgroup: 2.3.5.7.11

Comma list: 21/20, 33/32, 128/125

Mapping: [⟨3 0 7 13 15], ⟨0 1 0 -1 -1]]

Optimal tunings:

CTE: ~5/4 = 400.000, ~3/2 = 679.881 (~16/15 = 120.119)
POTE: ~5/4 = 400.000, ~3/2 = 680.042 (~16/15 = 119.958)

Optimal ET sequence: 3, 9

Badness (Smith): 0.037183

Hexe

Subgroup: 2.3.5.7

Comma list: 50/49, 128/125

Mapping: [⟨6 0 14 17], ⟨0 1 0 0]]

mapping generators: ~28/25, ~3

Wedgie: ⟨⟨ 6 0 0 -14 -17 0 ]]

Optimal tunings:

CTE: ~28/25 = 200.000, ~3/2 = 701.955 (~16/15 = 98.045)
POTE: ~28/25 = 200.000, ~3/2 = 710.963 (~16/15 = 89.037)

Optimal ET sequence: 6, 12, 30d, 42dd, 54cdd

Badness (Smith): 0.057730

11-limit

Subgroup: 2.3.5.7.11

Comma list: 50/49, 56/55, 125/121

Mapping: [⟨6 0 14 17 21], ⟨0 1 0 0 0]]

Optimal tunings:

CTE: ~28/25 = 200.000, ~3/2 = 701.955 (~16/15 = 98.045)
POTE: ~28/25 = 200.000, ~3/2 = 714.304 (~16/15 = 85.696)

Optimal ET sequence: 6, 12, 30dee, 42ddeee

Badness (Smith): 0.038412

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 50/49, 56/55, 66/65, 105/104

Mapping: [⟨6 0 14 17 21 13], ⟨0 1 0 0 0 1]]

Optimal tunings:

CTE: ~28/25 = 200.000, ~3/2 = 692.433 (~21/20 = 92.433)
POTE: ~28/25 = 200.000, ~3/2 = 710.005 (~16/15 = 89.995)

Optimal ET sequence: 6f, 12f

Badness (Smith): 0.035946

Triforce

Subgroup: 2.3.5.7

Comma list: 49/48, 128/125

Mapping: [⟨3 0 7 6], ⟨0 2 0 1]]

mapping generators: ~5/4, ~7/4

Wedgie: ⟨⟨ 6 0 3 -14 -12 7 ]]

Optimal tunings:

CTE: ~5/4 = 400.000, ~7/4 = 952.295 (~35/32 = 152.295)
POTE: ~5/4 = 400.000, ~7/4 = 952.951 (~35/32 = 152.951)

Optimal ET sequence: 6, 9, 15, 24, 39, 63cd, 102ccdd

Badness (Smith): 0.054988

11-limit

Subgroup: 2.3.5.7.11

Comma list: 49/48, 56/55, 77/75

Mapping: [⟨3 0 7 6 8], ⟨0 2 0 1 1]]

Wedgie: ⟨⟨ 6 0 3 3 -14 -12 -16 7 7 -2 ]]

Optimal tunings:

CTE: ~5/4 = 400.000, ~7/4 = 952.250 (~12/11 = 152.250)
POTE: ~5/4 = 400.000, ~7/4 = 952.932 (~12/11 = 152.932)

Optimal ET sequence: 6, 9, 15, 24, 39

Badness (Smith): 0.026152

Music

The Triforce of Courage (24edo) by Igliashon Jones (2018)
2-2-1-2-2-1-2-2-1 mode of 15 edo play by Chris Vaisvil

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 56/55, 66/65, 77/75

Mapping: [⟨3 0 7 6 8 4], ⟨0 2 0 1 1 3]]

Optimal tunings:

CTE: ~5/4 = 400.000, ~7/4 = 950.805 (~12/11 = 150.805)
POTE: ~5/4 = 400.000, ~7/4 = 951.687 (~12/11 = 151.687)

Optimal ET sequence: 6, 9, 15, 24

Badness (Smith): 0.020248

Scales

triphi, triforce[9] with L:s = phi

Semitriforce

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 56/55, 77/75, 507/500

Mapping: [⟨6 0 14 12 16 27], ⟨0 2 0 1 1 -1]]

mapping generators: ~44/39, ~7/4

Optimal tunings:

CTE: ~44/39 = 200.000, ~7/4 = 952.531 (~40/39 = 47.469)
POTE: ~44/39 = 200.000, ~7/4 = 953.358 (~40/39 = 46.642)

Optimal ET sequence: 6, 18bd, 24

Badness (Smith): 0.059160

Hemiaug

Subgroup: 2.3.5.7

Comma list: 128/125, 245/243

Mapping: [⟨3 1 7 -1], ⟨0 2 0 5]]

mapping generators: ~5/4, ~14/9

Wedgie: ⟨⟨ 6 0 15 -14 7 35 ]]

Optimal tunings:

CTE: ~5/4 = 400.000, ~14/9 = 752.834 (~36/35 = 47.166)
POTE: ~5/4 = 400.000, ~14/9 = 754.882 (~36/35 = 45.118)

Optimal ET sequence: 24, 27

Badness (Smith): 0.070463

11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 128/125, 243/242

Mapping: [⟨3 1 7 -1 1], ⟨0 2 0 5 5]]

Optimal tunings:

CTE: ~5/4 = 400.000, ~14/9 = 752.051 (~36/35 = 47.949)
POTE: ~5/4 = 400.000, ~14/9 = 754.212 (~36/35 = 45.788)

Optimal ET sequence: 24, 27e, 51ce

Badness (Smith): 0.038232

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 91/90, 128/125, 245/243

Mapping: [⟨3 1 7 -1 1 13], ⟨0 2 0 5 5 -1]]

Optimal tunings:

CTE: ~5/4 = 400.000, ~14/9 = 752.128 (~36/35 = 47.872)
POTE: ~5/4 = 400.000, ~14/9 = 753.750 (~36/35 = 46.250)

Optimal ET sequence: 24, 27e, 51ce

Badness (Smith): 0.030159

Hemiug

Subgroup: 2.3.5.7

Comma list: 128/125, 1323/1250

Mapping: [⟨3 1 7 14], ⟨0 2 0 -3]]

mapping generators: ~5/4, ~32/21

Wedgie: ⟨⟨ 6 0 -9 -14 -31 -21 ]]

Optimal tunings:

CTE: ~5/4 = 400.000, ~32/21 = 747.948 (~21/20 = 52.052)
POTE: ~5/4 = 400.000, ~32/21 = 747.907 (~21/20 = 52.093)

Optimal ET sequence: 21, 24, 45c

Badness (Smith): 0.137764

11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 128/125, 1323/1250

Mapping: [⟨3 1 7 14 16], ⟨0 2 0 -3 -3]]

Optimal tunings:

CTE: ~5/4 = 400.000, ~32/21 = 748.296 (~33/32 = 51.704)
POTE: ~5/4 = 400.000, ~32/21 = 748.345 (~33/32 = 51.655)

Optimal ET sequence: 21, 24

Badness (Smith): 0.068062

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 66/65, 105/104, 507/500

Mapping: [⟨3 1 7 14 16 13], ⟨0 2 0 -3 -3 -1]]

Optimal tunings:

CTE: ~5/4 = 400.000, ~32/21 = 748.525 (~33/32 = 51.475)
POTE: ~5/4 = 400.000, ~32/21 = 748.452 (~33/32 = 51.548)

Optimal ET sequence: 21, 24

Badness (Smith): 0.042289

Oodako

Subgroup: 2.3.5.7

Comma list: 128/125, 2401/2400

Mapping: [⟨3 3 7 8], ⟨0 4 0 1]]

mapping generators: ~5/4, ~28/25

Wedgie: ⟨⟨ 12 0 3 -28 -29 7 ]]

Optimal tunings:

CTE: ~5/4 = 400.000, ~28/25 = 175.359
POTE: ~5/4 = 400.000, ~28/25 = 176.646

Optimal ET sequence: 6, 21, 27, 75c, 102ccd, 129bccd

Badness (Smith): 0.113209

11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 128/125, 2401/2400

Mapping: [⟨3 3 7 8 10], ⟨0 4 0 1 1]]

Optimal tunings:

CTE: ~5/4 = 400.000, ~11/10 = 175.053
POTE: ~5/4 = 400.000, ~11/10 = 176.981

Optimal ET sequence: 6, 21, 27e

Badness (Smith): 0.059192

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 78/77, 128/125, 507/500

Mapping: [⟨3 3 7 8 10 12], ⟨0 4 0 1 1 -2]]

Optimal tunings:

CTE: ~5/4 = 400.000, ~11/10 = 175.252
POTE: ~5/4 = 400.000, ~11/10 = 176.551

Optimal ET sequence: 6, 21, 27e

Badness (Smith): 0.042323

Hemisemiaug

Subgroup: 2.3.5.7

Comma list: 128/125, 12005/11664

Mapping: [⟨6 1 14 4], ⟨0 2 0 3]]

mapping generators: ~54/49, ~45/28

Wedgie: ⟨⟨ 12 0 18 -28 -5 42 ]]

Optimal tunings:

CTE: ~54/49 = 200.000, ~45/28 = 853.190 (~36/35 = 53.190)
POTE: ~54/49 = 200.000, ~45/28 = 855.485 (~36/35 = 55.485)

Optimal ET sequence: 18, 24, 42

Badness (Smith): 0.210984

11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 128/125, 3773/3645

Mapping: [⟨6 1 14 4 8], ⟨0 2 0 3 3]]

Optimal tunings:

CTE: ~54/49 = 200.000, ~18/11 = 852.597 (~36/35 = 52.597)
POTE: ~54/49 = 200.000, ~18/11 = 855.220 (~36/35 = 55.220)

Optimal ET sequence: 24, 42e, 66ce, 108bccee

Badness (Smith): 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.

Subgroup: 2.3.5.7

Comma list: 128/125, 686/675

Mapping: [⟨9 0 21 11], ⟨0 1 0 1]]

mapping generators: ~49/45, ~3

Wedgie: ⟨⟨ 9 0 9 -21 -11 21 ]]

Optimal tunings:

CTE: ~49/45 = 133.333, ~3/2 = 702.004 (~36/35 = 35.338)
POTE: ~49/45 = 133.333, ~3/2 = 707.167 (~36/35 = 40.501)

Optimal ET sequence: 9, 18, 27, 63c, 90cc

Badness (Smith): 0.067157

11-limit

Subgroup: 2.3.5.7.11

Comma list: 56/55, 128/125, 540/539

Mapping: [⟨9 0 21 11 17], ⟨0 1 0 1 1]]

Optimal tunings:

CTE: ~12/11 = 133.333, ~3/2 = 699.622 (~36/35 = 32.955)
POTE: ~12/11 = 133.333, ~3/2 = 706.726 (~36/35 = 40.059)

Optimal ET sequence: 9, 18e, 27e, 63cee

Badness (Smith): 0.034861

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 56/55, 78/77, 91/90, 128/125

Mapping: [⟨9 0 21 11 17 19], ⟨0 1 0 1 1 1]]

Optimal tunings:

CTE: ~14/13 = 133.333, ~3/2 = 700.433 (~36/35 = 33.766)
POTE: ~14/13 = 133.333, ~3/2 = 706.889 (~36/35 = 40.222)

Optimal ET sequence: 9, 18e, 27e

Badness (Smith): 0.024143

Trug

Subgroup: 2.3.5.7

Comma list: 128/125, 360/343

Mapping: [⟨3 1 7 6], ⟨0 3 0 2]]

mapping generators: ~5/4, ~48/35

Optimal tunings:

CTE: ~5/4 = 400.000, ~48/35 = 498.637 (~15/14 = 98.637)
POTE: ~5/4 = 400.000, ~48/35 = 501.980 (~15/14 = 101.980)

Optimal ET sequence: 9bd, 12

Badness (Smith): 0.138279

Ternary

Subgroup: 2.3.5.7

Comma list: 10/9, 16/15

Mapping: [⟨3 5 7 0], ⟨0 0 0 1]]

mapping generators: ~5/4, ~7

Wedgie: ⟨⟨ 0 0 3 0 5 7 ]]

Optimal tunings:

POTE: ~5/4 = 400.000, ~7/4 = 1034.013 (~15/14 = 165.987)

Optimal ET sequence: 3d, 6, 9bd, 15bbd

Badness (Smith): 0.028673

11-limit

Subgroup: 2.3.5.7.11

Comma list: 10/9, 16/15, 22/21

Mapping: [⟨3 5 7 0 2], ⟨0 0 0 1 1]]

Optimal tunings:

POTE: ~5/4 = 400.000, ~7/4 = 1033.153 (~12/11 = 166.847)

Optimal ET sequence: 3de, 6, 15bbdee

Badness (Smith): 0.025592

External links

Herman Miller's page about augmented temperament ^{[dead link]}

@@ Line 15: / Line 15: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~5/4 = 1\3, ~3/2 = 701.9550
+* [[CTE]]: ~5/4 = 400.000, ~3/2 = 701.955 (~16/15 = 98.045)
-* [[POTE]]: ~5/4 = 1\3, ~3/2 = 706.638
+* [[POTE]]: ~5/4 = 400.000, ~3/2 = 706.638 (~16/15 = 93.362)
 {{Optimal ET sequence|legend=1| 3, 12, 27, 39, 51c, 90cc }}
-[[Badness]]: 0.022315
+[[Badness]] (Smith): 0.022315
 === Overview to extensions ===
@@ Line 35: / Line 35: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~5/4 = 1\3, ~3/2 = 692.1237
+* [[CTE]]: ~5/4 = 400.000, ~3/2 = 692.124 (~16/15 = 107.876)
-* [[POTE]]: ~5/4 = 1\3, ~3/2 = 696.011
+* [[POTE]]: ~5/4 = 400.000, ~3/2 = 696.011 (~16/15 = 103.989)
 {{Optimal ET sequence|legend=1| 9, 12, 45cd, 57cd, 69cdd }}
-[[Badness]]: 0.026459
+[[Badness]] (Smith): 0.026459
 === 11-limit ===
@@ Line 49: / Line 49: @@
 Mapping: {{mapping| 3 0 7 -1 1 | 0 1 0 2 2 }}
-{{Multival|legend=1| 3 0 6 6 -7 1 -1 14 14 -4 }}
+Wedgie: {{multival| 3 0 6 6 -7 1 -1 14 14 -4 }}
 Optimal tunings:
-* CTE: ~5/4 = 1\3, ~3/2 = 687.6853
+* CTE: ~5/4 = 400.000, ~3/2 = 687.685 (~16/15 = 112.315)
-* POTE: ~5/4 = 1\3, ~3/2 = 692.514
+* POTE: ~5/4 = 400.000, ~3/2 = 692.514 (~16/15 = 107.486)
-{{Optimal ET sequence|legend=1| 9, 12, 21, 33e, 45cde }}
+Optimal ET sequence: {{Optimal ET sequence| 9, 12, 21, 33e, 45cde }}
-Badness: 0.020191
+Badness (Smith): 0.020191
 ==== 13-limit ====
@@ Line 67: / Line 67: @@
 Optimal tunings:
-* CTE: ~5/4 = 1\3, ~3/2 = 685.0836
+* CTE: ~5/4 = 400.000, ~3/2 = 685.084 (~16/15 = 114.916)
-* POTE: ~5/4 = 1\3, ~3/2 = 688.783
+* POTE: ~5/4 = 400.000, ~3/2 = 688.783 (~16/15 = 111.217)
-{{Optimal ET sequence|legend=1| 9, 12f, 21, 33ef, 54bceeff }}
+Optimal ET sequence: {{Optimal ET sequence| 9, 12f, 21, 33ef, 54bceeff }}
-Badness: 0.018448
+Badness (Smith): 0.018448
 ==== Augustus ====
@@ Line 82: / Line 82: @@
 Optimal tunings:
-* CTE: ~5/4 = 1\3, ~3/2 = 687.6853
+* CTE: ~5/4 = 400.000, ~3/2 = 687.685 (~16/15 = 112.315)
-* POTE: ~5/4 = 1\3, ~3/2 = 685.356
+* POTE: ~5/4 = 400.000, ~3/2 = 685.356 (~16/15 = 114.644)
-{{Optimal ET sequence|legend=1| 9, 12, 21f }}
+Optimal ET sequence: {{Optimal ET sequence| 9, 12, 21f }}
-Badness: 0.022236
+Badness (Smith): 0.022236
 == Augene ==
@@ Line 101: / Line 101: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~5/4 = 1\3, ~3/2 = 709.5948
+* [[CTE]]: ~5/4 = 400.000, ~3/2 = 709.595 (~21/20 = 90.405)
-* [[POTE]]: ~5/4 = 1\3, ~3/2 = 709.257
+* [[POTE]]: ~5/4 = 400.000, ~3/2 = 709.257 (~21/20 = 90.743)
 {{Optimal ET sequence|legend=1| 12, 27, 39d, 66cd }}
-[[Badness]]: 0.024816
+[[Badness]] (Smith): 0.024816
 === 11-limit ===
@@ Line 115: / Line 115: @@
 Mapping: {{mapping| 3 0 7 18 20 | 0 1 0 -2 -2 }}
-Wedgie: {{Multival| 3 0 -6 -6 -7 -18 -20 -14 -14 4 }}
+Wedgie: {{multival| 3 0 -6 -6 -7 -18 -20 -14 -14 4 }}
 Optimal tunings:
-* CTE: ~5/4 = 1\3, ~3/2 = 713.5701
+* CTE: ~5/4 = 400.000, ~3/2 = 713.570 (~21/20 = 86.430)
-* POTE: ~5/4 = 1\3, ~3/2 = 711.177
+* POTE: ~5/4 = 400.000, ~3/2 = 711.177 (~21/20 = 88.823)
-{{Optimal ET sequence|legend=1| 12, 15, 27e }}
+Optimal ET sequence: {{Optimal ET sequence| 12, 15, 27e }}
-Badness: 0.019613
+Badness (Smith): 0.019613
 ==== 13-limit ====
@@ Line 133: / Line 133: @@
 Optimal tunings:
-* CTE: ~5/4 = 1\3, ~3/2 = 716.1234
+* CTE: ~5/4 = 400.000, ~3/2 = 716.123 (~21/20 = 83.877)
-* POTE: ~5/4 = 1\3, ~3/2 = 712.013
+* POTE: ~5/4 = 400.000, ~3/2 = 712.013 (~21/20 = 87.987)
-{{Optimal ET sequence|legend=1| 12f, 15, 27eff }}
+Optimal ET sequence: {{Optimal ET sequence| 12f, 15, 27eff }}
-Badness: 0.020785
+Badness (Smith): 0.020785
 ==== Ogene ====
@@ Line 148: / Line 148: @@
 Optimal tunings:
-* CTE: ~5/4 = 1\3, ~3/2 = 711.9020
+* CTE: ~5/4 = 400.000, ~3/2 = 711.902 (~21/20 = 88.098)
-* POTE: ~5/4 = 1\3, ~3/2 = 712.609
+* POTE: ~5/4 = 400.000, ~3/2 = 712.609 (~21/20 = 87.391)
-{{Optimal ET sequence|legend=1| 12, 15, 27e, 69bceef }}
+Optimal ET sequence: {{Optimal ET sequence| 12, 15, 27e, 69bceef }}
-Badness: 0.022890
+Badness (Smith): 0.022890
 ==== Agene ====
@@ Line 163: / Line 163: @@
 Optimal tunings:
-* CTE: ~5/4 = 1\3, ~3/2 = 712.5722
+* CTE: ~5/4 = 400.000, ~3/2 = 712.572 (~21/20 = 87.428)
-* POTE: ~5/4 = 1\3, ~3/2 = 709.677
+* POTE: ~5/4 = 400.000, ~3/2 = 709.677 (~21/20 = 90.323)
-{{Optimal ET sequence|legend=1| 12f, 27e }}
+Optimal ET sequence: {{Optimal ET sequence| 12f, 27e }}
-Badness: 0.023113
+Badness (Smith): 0.023113
 === Eugene ===
@@ Line 178: / Line 178: @@
 Optimal tunings:
-* CTE: ~5/4 = 1\3, ~3/2 = 713.0025
+* CTE: ~5/4 = 400.000, ~3/2 = 713.002 (~21/20 = 86.998)
-* POTE: ~5/4 = 1\3, ~3/2 = 714.150
+* POTE: ~5/4 = 400.000, ~3/2 = 714.150 (~21/20 = 85.850)
-{{Optimal ET sequence|legend=1| 12e, 15, 27, 42 }}
+Optimal ET sequence: {{Optimal ET sequence| 12e, 15, 27, 42 }}
-Badness: 0.035554
+Badness (Smith): 0.035554
 == Inflated ==
@@ Line 195: / Line 195: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~5/4 = 1\3, ~3/2 = 717.5172
+* [[CTE]]: ~5/4 = 400.000, ~3/2 = 717.517 (~25/24 = 82.483)
-* [[POTE]]: ~5/4 = 1\3, ~3/2 = 722.719
+* [[POTE]]: ~5/4 = 400.000, ~3/2 = 722.719 (~25/24 = 77.281)
 {{Optimal ET sequence|legend=1| 3d, 12d, 15 }}
-[[Badness]]: 0.054729
+[[Badness]] (Smith): 0.054729
 === 11-limit ===
@@ Line 210: / Line 210: @@
 Optimal tunings:
-* CTE: ~5/4 = 1\3, ~3/2 = 717.5172
+* CTE: ~5/4 = 400.000, ~3/2 = 717.382 (~25/24 = 82.618)
-* POTE: ~5/4 = 1\3, ~3/2 = 722.663
+* POTE: ~5/4 = 400.000, ~3/2 = 722.663 (~25/24 = 77.337)
-{{Optimal ET sequence|legend=1| 3de, 12de, 15 }}
+Optimal ET sequence: {{Optimal ET sequence| 3de, 12de, 15 }}
-Badness: 0.031171
+Badness (Smith): 0.031171
 == Deflated ==
@@ Line 227: / Line 227: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~5/4 = 1\3, ~3/2 = 684.8473
+* [[CTE]]: ~5/4 = 400.000, ~3/2 = 684.847 (~16/15 = 115.153)
-* [[POTE]]: ~5/4 = 1\3, ~3/2 = 681.629
+* [[POTE]]: ~5/4 = 400.000, ~3/2 = 681.629 (~16/15 = 118.371)
 {{Optimal ET sequence|legend=1| 3, 9 }}
-[[Badness]]: 0.059079
+[[Badness]] (Smith): 0.059079
 === 11-limit ===
@@ Line 242: / Line 242: @@
 Optimal tunings:
-* CTE: ~5/4 = 1\3, ~3/2 = 679.881
+* CTE: ~5/4 = 400.000, ~3/2 = 679.881 (~16/15 = 120.119)
-* POTE: ~5/4 = 1\3, ~3/2 = 680.042
+* POTE: ~5/4 = 400.000, ~3/2 = 680.042 (~16/15 = 119.958)
-{{Optimal ET sequence|legend=1| 3, 9 }}
+Optimal ET sequence: {{Optimal ET sequence| 3, 9 }}
-Badness: 0.037183
+Badness (Smith): 0.037183
 == Hexe ==
@@ Line 261: / Line 261: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~28/25 = 1\6, ~3/2 = 701.9550
+* [[CTE]]: ~28/25 = 200.000, ~3/2 = 701.955 (~16/15 = 98.045)
-* [[POTE]]: ~28/25 = 1\6, ~3/2 = 710.963
+* [[POTE]]: ~28/25 = 200.000, ~3/2 = 710.963 (~16/15 = 89.037)
 {{Optimal ET sequence|legend=1| 6, 12, 30d, 42dd, 54cdd }}
-[[Badness]]: 0.057730
+[[Badness]] (Smith): 0.057730
 === 11-limit ===
@@ Line 276: / Line 276: @@
 Optimal tunings:
-* CTE: ~28/25 = 1\6, ~3/2 = 701.9550
+* CTE: ~28/25 = 200.000, ~3/2 = 701.955 (~16/15 = 98.045)
-* POTE: ~28/25 = 1\6, ~3/2 = 714.304
+* POTE: ~28/25 = 200.000, ~3/2 = 714.304 (~16/15 = 85.696)
-{{Optimal ET sequence|legend=1| 6, 12, 30dee, 42ddeee }}
+Optimal ET sequence: {{Optimal ET sequence| 6, 12, 30dee, 42ddeee }}
-Badness: 0.038412
+Badness (Smith): 0.038412
 === 13-limit ===
@@ Line 291: / Line 291: @@
 Optimal tunings:
-* CTE: ~28/25 = 1\6, ~3/2 = 692.4327
+* CTE: ~28/25 = 200.000, ~3/2 = 692.433 (~21/20 = 92.433)
-* POTE: ~28/25 = 1\6, ~3/2 = 710.005
+* POTE: ~28/25 = 200.000, ~3/2 = 710.005 (~16/15 = 89.995)
-{{Optimal ET sequence|legend=1| 6f, 12f }}
+Optimal ET sequence: {{Optimal ET sequence| 6f, 12f }}
-Badness: 0.035946
+Badness (Smith): 0.035946
 == Triforce ==
@@ Line 312: / Line 312: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~5/4 = 1\3, ~7/4 = 952.2948
+* [[CTE]]: ~5/4 = 400.000, ~7/4 = 952.295 (~35/32 = 152.295)
-* [[POTE]]: ~5/4 = 1\3, ~7/4 = 952.951
+* [[POTE]]: ~5/4 = 400.000, ~7/4 = 952.951 (~35/32 = 152.951)
 {{Optimal ET sequence|legend=1| 6, 9, 15, 24, 39, 63cd, 102ccdd }}
-[[Badness]]: 0.054988
+[[Badness]] (Smith): 0.054988
 === 11-limit ===
@@ Line 326: / Line 326: @@
 Mapping: {{mapping| 3 0 7 6 8 | 0 2 0 1 1 }}
-{{Multival|legend=1| 6 0 3 3 -14 -12 -16 7 7 -2 }}
+Wedgie: {{multival| 6 0 3 3 -14 -12 -16 7 7 -2 }}
 Optimal tunings:
-* CTE: ~5/4 = 1\3, ~7/4 = 952.2495
+* CTE: ~5/4 = 400.000, ~7/4 = 952.250 (~12/11 = 152.250)
-* POTE: ~5/4 = 1\3, ~7/4 = 952.932
+* POTE: ~5/4 = 400.000, ~7/4 = 952.932 (~12/11 = 152.932)
-{{Optimal ET sequence|legend=1| 6, 9, 15, 24, 39 }}
+Optimal ET sequence: {{Optimal ET sequence| 6, 9, 15, 24, 39 }}
-Badness: 0.026152
+Badness (Smith): 0.026152
 ; Music
@@ Line 348: / Line 348: @@
 Optimal tunings:
-* CTE: ~5/4 = 1\3, ~7/4 = 950.8049
+* CTE: ~5/4 = 400.000, ~7/4 = 950.805 (~12/11 = 150.805)
-* POTE: ~5/4 = 1\3, ~7/4 = 951.687
+* POTE: ~5/4 = 400.000, ~7/4 = 951.687 (~12/11 = 151.687)
-{{Optimal ET sequence|legend=1| 6, 9, 15, 24 }}
+Optimal ET sequence: {{Optimal ET sequence| 6, 9, 15, 24 }}
-Badness: 0.020248
+Badness (Smith): 0.020248
 ; Scales
@@ Line 368: / Line 368: @@
 Optimal tunings:
-* CTE: ~44/39 = 1\6, ~7/4 = 952.5307
+* CTE: ~44/39 = 200.000, ~7/4 = 952.531 (~40/39 = 47.469)
-* POTE: ~44/39 = 1\6, ~7/4 = 953.358
+* POTE: ~44/39 = 200.000, ~7/4 = 953.358 (~40/39 = 46.642)
-{{Optimal ET sequence|legend=1| 6, 18bd, 24 }}
+Optimal ET sequence: {{Optimal ET sequence| 6, 18bd, 24 }}
-Badness: 0.059160
+Badness (Smith): 0.059160
 == Hemiaug ==
@@ Line 387: / Line 387: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~5/4 = 1\3, ~14/9 = 752.8335 (~36/35 = 47.1665)
+* [[CTE]]: ~5/4 = 400.000, ~14/9 = 752.834 (~36/35 = 47.166)
-* [[POTE]]: ~5/4 = 1\3, ~14/9 = 754.882 (~36/35 = 45.118)
+* [[POTE]]: ~5/4 = 400.000, ~14/9 = 754.882 (~36/35 = 45.118)
 {{Optimal ET sequence|legend=1| 24, 27 }}
-[[Badness]]: 0.070463
+[[Badness]] (Smith): 0.070463
 === 11-limit ===
@@ Line 402: / Line 402: @@
 Optimal tunings:
-* CTE: ~5/4 = 1\3, ~14/9 = 752.0512 (~36/35 = 47.9488)
+* CTE: ~5/4 = 400.000, ~14/9 = 752.051 (~36/35 = 47.949)
-* POTE: ~5/4 = 1\3, ~14/9 = 754.212 (~36/35 = 45.788)
+* POTE: ~5/4 = 400.000, ~14/9 = 754.212 (~36/35 = 45.788)
-{{Optimal ET sequence|legend=1| 24, 27e, 51ce }}
+Optimal ET sequence: {{Optimal ET sequence| 24, 27e, 51ce }}
-Badness: 0.038232
+Badness (Smith): 0.038232
 === 13-limit ===
@@ Line 417: / Line 417: @@
 Optimal tunings:
-* CTE: ~5/4 = 1\3, ~14/9 = 752.1284 (~36/35 = 47.8716)
+* CTE: ~5/4 = 400.000, ~14/9 = 752.128 (~36/35 = 47.872)
-* POTE: ~5/4 = 1\3, ~14/9 = 753.750 (~36/35 = 46.250)
+* POTE: ~5/4 = 400.000, ~14/9 = 753.750 (~36/35 = 46.250)
-{{Optimal ET sequence|legend=1| 24, 27e, 51ce }}
+Optimal ET sequence: {{Optimal ET sequence| 24, 27e, 51ce }}
-Badness: 0.030159
+Badness (Smith): 0.030159
 == Hemiug ==
@@ Line 436: / Line 436: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~5/4 = 1\3, ~32/21 = 747.9484 (~21/20 = 52.0516)
+* [[CTE]]: ~5/4 = 400.000, ~32/21 = 747.948 (~21/20 = 52.052)
-* [[POTE]]: ~5/4 = 1\3, ~32/21 = 747.907 (~21/20 = 52.093)
+* [[POTE]]: ~5/4 = 400.000, ~32/21 = 747.907 (~21/20 = 52.093)
 {{Optimal ET sequence|legend=1| 21, 24, 45c }}
-[[Badness]]: 0.137764
+[[Badness]] (Smith): 0.137764
 === 11-limit ===
@@ Line 451: / Line 451: @@
 Optimal tunings:
-* CTE: ~5/4 = 1\3, ~32/21 = 748.2962 (~33/32 = 51.7038)
+* CTE: ~5/4 = 400.000, ~32/21 = 748.296 (~33/32 = 51.704)
-* POTE: ~5/4 = 1\3, ~32/21 = 748.345 (~33/32 = 51.655)
+* POTE: ~5/4 = 400.000, ~32/21 = 748.345 (~33/32 = 51.655)
-{{Optimal ET sequence|legend=1| 21, 24 }}
+Optimal ET sequence: {{Optimal ET sequence| 21, 24 }}
-Badness: 0.068062
+Badness (Smith): 0.068062
 === 13-limit ===
@@ Line 466: / Line 466: @@
 Optimal tunings:
-* CTE: ~5/4 = 1\3, ~32/21 = 748.5255 (~33/32 = 51.4745)
+* CTE: ~5/4 = 400.000, ~32/21 = 748.525 (~33/32 = 51.475)
-* POTE: ~5/4 = 1\3, ~32/21 = 748.452 (~33/32 = 51.548)
+* POTE: ~5/4 = 400.000, ~32/21 = 748.452 (~33/32 = 51.548)
-{{Optimal ET sequence|legend=1| 21, 24 }}
+Optimal ET sequence: {{Optimal ET sequence| 21, 24 }}
-Badness: 0.042289
+Badness (Smith): 0.042289
 == Oodako ==
@@ Line 485: / Line 485: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~5/4 = 1\3, ~28/25 = 175.3586
+* [[CTE]]: ~5/4 = 400.000, ~28/25 = 175.359
-* [[POTE]]: ~5/4 = 1\3, ~28/25 = 176.646
+* [[POTE]]: ~5/4 = 400.000, ~28/25 = 176.646
 {{Optimal ET sequence|legend=1| 6, 21, 27, 75c, 102ccd, 129bccd }}
-[[Badness]]: 0.113209
+[[Badness]] (Smith): 0.113209
 === 11-limit ===
@@ Line 500: / Line 500: @@
 Optimal tunings:
-* CTE: ~5/4 = 1\3, ~11/10 = 175.0533
+* CTE: ~5/4 = 400.000, ~11/10 = 175.053
-* POTE: ~5/4 = 1\3, ~11/10 = 176.981
+* POTE: ~5/4 = 400.000, ~11/10 = 176.981
-{{Optimal ET sequence|legend=1| 6, 21, 27e }}
+Optimal ET sequence: {{Optimal ET sequence| 6, 21, 27e }}
-Badness: 0.059192
+Badness (Smith): 0.059192
 === 13-limit ===
@@ Line 515: / Line 515: @@
 Optimal tunings:
-* CTE: ~5/4 = 1\3, ~11/10 = 175.2524
+* CTE: ~5/4 = 400.000, ~11/10 = 175.252
-* POTE: ~5/4 = 1\3, ~11/10 = 176.551
+* POTE: ~5/4 = 400.000, ~11/10 = 176.551
-{{Optimal ET sequence|legend=1| 6, 21, 27e }}
+Optimal ET sequence: {{Optimal ET sequence| 6, 21, 27e }}
-Badness: 0.042323
+Badness (Smith): 0.042323
 == Hemisemiaug ==
@@ Line 534: / Line 534: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~54/49 = 1\6, ~45/28 = 853.1901 (~36/35 = 53.1901)
+* [[CTE]]: ~54/49 = 200.000, ~45/28 = 853.190 (~36/35 = 53.190)
-* [[POTE]]: ~54/49 = 1\6, ~45/28 = 855.485 (~36/35 = 55.485)
+* [[POTE]]: ~54/49 = 200.000, ~45/28 = 855.485 (~36/35 = 55.485)
 {{Optimal ET sequence|legend=1| 18, 24, 42 }}
-[[Badness]]: 0.210984
+[[Badness]] (Smith): 0.210984
 === 11-limit ===
@@ Line 549: / Line 549: @@
 Optimal tunings:
-* CTE: ~54/49 = 1\6, ~18/11 = 852.5968 (~36/35 = 52.5968)
+* CTE: ~54/49 = 200.000, ~18/11 = 852.597 (~36/35 = 52.597)
-* POTE: ~54/49 = 1\6, ~18/11 = 855.220 (~36/35 = 55.220)
+* POTE: ~54/49 = 200.000, ~18/11 = 855.220 (~36/35 = 55.220)
-{{Optimal ET sequence|legend=1| 24, 42e, 66ce, 108bccee }}
+Optimal ET sequence: {{Optimal ET sequence| 24, 42e, 66ce, 108bccee }}
-Badness: 0.080738
+Badness (Smith): 0.080738
 == Niner ==
@@ Line 570: / Line 570: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~49/45 = 1\9, ~3/2 = 702.0044
+* [[CTE]]: ~49/45 = 133.333, ~3/2 = 702.004 (~36/35 = 35.338)
-* [[POTE]]: ~49/45 = 1\9, ~3/2 = 707.167
+* [[POTE]]: ~49/45 = 133.333, ~3/2 = 707.167 (~36/35 = 40.501)
 {{Optimal ET sequence|legend=1| 9, 18, 27, 63c, 90cc }}
-[[Badness]]: 0.067157
+[[Badness]] (Smith): 0.067157
 === 11-limit ===
@@ Line 585: / Line 585: @@
 Optimal tunings:
-* CTE: ~12/11 = 1\9, ~3/2 = 699.6216
+* CTE: ~12/11 = 133.333, ~3/2 = 699.622 (~36/35 = 32.955)
-* POTE: ~12/11 = 1\9, ~3/2 = 706.726
+* POTE: ~12/11 = 133.333, ~3/2 = 706.726 (~36/35 = 40.059)
-{{Optimal ET sequence|legend=1| 9, 18e, 27e, 63cee }}
+Optimal ET sequence: {{Optimal ET sequence| 9, 18e, 27e, 63cee }}
-Badness: 0.034861
+Badness (Smith): 0.034861
 === 13-limit ===
@@ Line 600: / Line 600: @@
 Optimal tunings:
-* CTE: ~14/13 = 1\9, ~3/2 = 700.4330
+* CTE: ~14/13 = 133.333, ~3/2 = 700.433 (~36/35 = 33.766)
-* POTE: ~14/13 = 1\9, ~3/2 = 706.889
+* POTE: ~14/13 = 133.333, ~3/2 = 706.889 (~36/35 = 40.222)
-{{Optimal ET sequence|legend=1| 9, 18e, 27e }}
+Optimal ET sequence: {{Optimal ET sequence| 9, 18e, 27e }}
-Badness: 0.024143
+Badness (Smith): 0.024143
 == Trug ==
@@ Line 617: / Line 617: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~5/4 = 1\3, ~48/35 = 498.6367
+* [[CTE]]: ~5/4 = 400.000, ~48/35 = 498.637 (~15/14 = 98.637)
-* [[POTE]]: ~5/4 = 1\3, ~48/35 = 501.980
+* [[POTE]]: ~5/4 = 400.000, ~48/35 = 501.980 (~15/14 = 101.980)
 {{Optimal ET sequence|legend=1| 9bd, 12 }}
-[[Badness]]: 0.138279
+[[Badness]] (Smith): 0.138279
 == Ternary ==
@@ Line 636: / Line 636: @@
 [[Optimal tuning]]s:
-* [[POTE]]: ~5/4 = 1\3, ~7/4 = 1034.013
+* [[POTE]]: ~5/4 = 400.000, ~7/4 = 1034.013 (~15/14 = 165.987)
 {{Optimal ET sequence|legend=1| 3d, 6, 9bd, 15bbd }}
-[[Badness]]: 0.028673
+[[Badness]] (Smith): 0.028673
 === 11-limit ===
@@ Line 650: / Line 650: @@
 Optimal tunings:
-* POTE: ~5/4 = 1\3, ~7/4 = 1033.153
+* POTE: ~5/4 = 400.000, ~7/4 = 1033.153 (~12/11 = 166.847)
-{{Optimal ET sequence|legend=1| 3de, 6, 15bbdee }}
+Optimal ET sequence: {{Optimal ET sequence| 3de, 6, 15bbdee }}
-Badness: 0.025592
+Badness (Smith): 0.025592
 == External links ==

Augmented family: Difference between revisions

Revision as of 13:27, 15 November 2024

Augmented

Overview to extensions

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

Triforce

11-limit

13-limit

Semitriforce

Hemiaug

11-limit

13-limit

Hemiug

11-limit

13-limit

Oodako

11-limit

13-limit

Hemisemiaug

11-limit

Niner

11-limit

13-limit

Trug

Ternary

11-limit

External links

Navigation menu

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