Xenharmonic Wiki

Augmented family: Difference between revisions

← Older editNewer edit →
VisualWikitext
→External links: Marked dead link
+minimal gens
Line 15: Line 15:


[[Optimal tuning]]s:  
[[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 }}
{{Optimal ET sequence|legend=1| 3, 12, 27, 39, 51c, 90cc }}


[[Badness]]: 0.022315
[[Badness]] (Smith): 0.022315


=== Overview to extensions ===
=== Overview to extensions ===
Line 35: Line 35:


[[Optimal tuning]]s:
[[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 }}
{{Optimal ET sequence|legend=1| 9, 12, 45cd, 57cd, 69cdd }}


[[Badness]]: 0.026459
[[Badness]] (Smith): 0.026459


=== 11-limit ===
=== 11-limit ===
Line 49: Line 49:
Mapping: {{mapping| 3 0 7 -1 1 | 0 1 0 2 2 }}
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:
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 ====
==== 13-limit ====
Line 67: Line 67:


Optimal tunings:  
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 ====
==== Augustus ====
Line 82: Line 82:


Optimal tunings:  
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 ==
== Augene ==
Line 101: Line 101:


[[Optimal tuning]]s:  
[[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 }}
{{Optimal ET sequence|legend=1| 12, 27, 39d, 66cd }}


[[Badness]]: 0.024816
[[Badness]] (Smith): 0.024816


=== 11-limit ===
=== 11-limit ===
Line 115: Line 115:
Mapping: {{mapping| 3 0 7 18 20 | 0 1 0 -2 -2 }}
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:  
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 ====
==== 13-limit ====
Line 133: Line 133:


Optimal tunings:  
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 ====
==== Ogene ====
Line 148: Line 148:


Optimal tunings:  
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 ====
==== Agene ====
Line 163: Line 163:


Optimal tunings:  
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 ===
=== Eugene ===
Line 178: Line 178:


Optimal tunings:  
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 ==
== Inflated ==
Line 195: Line 195:


[[Optimal tuning]]s:  
[[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 }}
{{Optimal ET sequence|legend=1| 3d, 12d, 15 }}


[[Badness]]: 0.054729
[[Badness]] (Smith): 0.054729


=== 11-limit ===
=== 11-limit ===
Line 210: Line 210:


Optimal tunings:  
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 ==
== Deflated ==
Line 227: Line 227:


[[Optimal tuning]]s:  
[[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 }}
{{Optimal ET sequence|legend=1| 3, 9 }}


[[Badness]]: 0.059079
[[Badness]] (Smith): 0.059079


=== 11-limit ===
=== 11-limit ===
Line 242: Line 242:


Optimal tunings:  
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 ==
== Hexe ==
Line 261: Line 261:


[[Optimal tuning]]s:  
[[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 }}
{{Optimal ET sequence|legend=1| 6, 12, 30d, 42dd, 54cdd }}


[[Badness]]: 0.057730
[[Badness]] (Smith): 0.057730


=== 11-limit ===
=== 11-limit ===
Line 276: Line 276:


Optimal tunings:  
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 ===
=== 13-limit ===
Line 291: Line 291:


Optimal tunings:  
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 ==
== Triforce ==
Line 312: Line 312:


[[Optimal tuning]]s:  
[[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 }}
{{Optimal ET sequence|legend=1| 6, 9, 15, 24, 39, 63cd, 102ccdd }}


[[Badness]]: 0.054988
[[Badness]] (Smith): 0.054988


=== 11-limit ===
=== 11-limit ===
Line 326: Line 326:
Mapping: {{mapping| 3 0 7 6 8 | 0 2 0 1 1 }}
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:  
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
; Music
Line 348: Line 348:


Optimal tunings:  
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
; Scales
Line 368: Line 368:


Optimal tunings:  
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 ==
== Hemiaug ==
Line 387: Line 387:


[[Optimal tuning]]s:  
[[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 }}
{{Optimal ET sequence|legend=1| 24, 27 }}


[[Badness]]: 0.070463
[[Badness]] (Smith): 0.070463


=== 11-limit ===
=== 11-limit ===
Line 402: Line 402:


Optimal tunings:  
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 ===
=== 13-limit ===
Line 417: Line 417:


Optimal tunings:  
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 ==
== Hemiug ==
Line 436: Line 436:


[[Optimal tuning]]s:  
[[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 }}
{{Optimal ET sequence|legend=1| 21, 24, 45c }}


[[Badness]]: 0.137764
[[Badness]] (Smith): 0.137764


=== 11-limit ===
=== 11-limit ===
Line 451: Line 451:


Optimal tunings:  
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 ===
=== 13-limit ===
Line 466: Line 466:


Optimal tunings:  
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 ==
== Oodako ==
Line 485: Line 485:


[[Optimal tuning]]s:  
[[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 }}
{{Optimal ET sequence|legend=1| 6, 21, 27, 75c, 102ccd, 129bccd }}


[[Badness]]: 0.113209
[[Badness]] (Smith): 0.113209


=== 11-limit ===
=== 11-limit ===
Line 500: Line 500:


Optimal tunings:  
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 ===
=== 13-limit ===
Line 515: Line 515:


Optimal tunings:  
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 ==
== Hemisemiaug ==
Line 534: Line 534:


[[Optimal tuning]]s:  
[[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 }}
{{Optimal ET sequence|legend=1| 18, 24, 42 }}


[[Badness]]: 0.210984
[[Badness]] (Smith): 0.210984


=== 11-limit ===
=== 11-limit ===
Line 549: Line 549:


Optimal tunings:  
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 ==
== Niner ==
Line 570: Line 570:


[[Optimal tuning]]s:  
[[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 }}
{{Optimal ET sequence|legend=1| 9, 18, 27, 63c, 90cc }}


[[Badness]]: 0.067157
[[Badness]] (Smith): 0.067157


=== 11-limit ===
=== 11-limit ===
Line 585: Line 585:


Optimal tunings:  
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 ===
=== 13-limit ===
Line 600: Line 600:


Optimal tunings:  
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 ==
== Trug ==
Line 617: Line 617:


[[Optimal tuning]]s:  
[[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 }}
{{Optimal ET sequence|legend=1| 9bd, 12 }}


[[Badness]]: 0.138279
[[Badness]] (Smith): 0.138279


== Ternary ==
== Ternary ==
Line 636: Line 636:


[[Optimal tuning]]s:  
[[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 }}
{{Optimal ET sequence|legend=1| 3d, 6, 9bd, 15bbd }}


[[Badness]]: 0.028673
[[Badness]] (Smith): 0.028673


=== 11-limit ===
=== 11-limit ===
Line 650: Line 650:


Optimal tunings:  
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 ==
== External links ==
Retrieved from "https://en.xen.wiki/w/Augmented_family"