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User:BudjarnLambeth/Equalizer subgroup: Difference between revisions

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'''Disclaimer from the author'''
'''Disclaimer from the author'''


Note that everything written on this page is the authors’ personal opinion based on how they subjectively experience harmony and will not be psychoacoustically true for everyone, and may not even be true for most people.  
Note that everything written on this page is the author’s personal opinion based on how they subjectively experience harmony and will not be psychoacoustically true for everyone, and may not even be true for most people.  


Nothing on this page is science. It's just one of many possible frameworks for looking at things, and it's not even a very well-tested one. So take it all with a grain of salt.
Nothing on this page is science. It's just one of many possible frameworks for looking at things, and it's not even a very well-tested one. So take it all with a grain of salt.
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'''Fifigeist'''
'''Fifigeist'''


Subgroup: 2.3.5.13.17.19
[[Subgroup]]: 2.3.5.13.17.19


Comma list: 153/152, 170/169, 250/247, 289/288, 324/323, 325/323, 325/324, 729/722
[[Comma list]]: 153/152, 170/169, 250/247, 289/288, 324/323, 325/323, 325/324, 729/722


Mapping: {{mapping| 2 2 3 6 7 5 | 0 5 7 6 5 15 }}
[[Mapping]]: {{mapping| 2 2 3 6 7 5 | 0 5 7 6 5 15 }}


Optimal tuning (TE): ~1\2 = 600.1848, ~13/12 = 140.2424
[[Optimal tuning]] ([[TE]]): ~1\2 = 600.1848, ~13/12 = 140.2424


Recommended EDOs: {{EDOs|10, 18, 26, 34, 60, 94}}
Recommended [[EDO]]s: {{EDOs|10, 18, 26, 34, 60, 94}}


TE mistunings (cents): ⟨0.370, -0.373, -4.062, 2.036, -2.450, 7.048]
TE mistunings ([[cents]]): ⟨0.370, -0.373, -4.062, 2.036, -2.450, 7.048]


Adjusted Error (cents): 4.477868
Adjusted [[error]] (cents): 4.477868


Complexity: 2.503140
[[Complexity]]: 2.503140


Related temperaments: [[fifive]]
Related temperaments: [[fifive]]
Line 68: Line 68:
Comma list: 115/114, 153/152, 300/299, 391/390, 1105/1104, 8075/8073
Comma list: 115/114, 153/152, 300/299, 391/390, 1105/1104, 8075/8073


Mapping: {{mapping| 2 2 3 6 7 5 6 | 0 5 7 6 5 15 13 }}
{{Mapping|legend=0| 2 2 3 6 7 5 6 | 0 5 7 6 5 15 13 }}


Optimal tuning (TE): ~1\2 = 600.1820, ~13/12 = 140.3019
Optimal tuning (TE): ~1\2 = 600.1820, ~13/12 = 140.3019
Line 76: Line 76:
TE mistunings (cents): ⟨0.364, -0.082, -3.655, 2.375, -2.172, 7.925, -3.258]
TE mistunings (cents): ⟨0.364, -0.082, -3.655, 2.375, -2.172, 7.925, -3.258]


Adjusted Error (cents): 4.621471
Adjusted error (cents): 4.621471


Complexity: 2.381333
Complexity: 2.381333
Line 90: Line 90:
Comma list: 96/95, 136/135, 153/152, 170/169, 256/255, 289/285, 289/288, 324/323
Comma list: 96/95, 136/135, 153/152, 170/169, 256/255, 289/285, 289/288, 324/323


Mapping: {{mapping| 2 4 3 7 9 11 | 0 -2 4 1 -2 -6 }}
{{Mapping|legend=0| 2 4 3 7 9 11 | 0 -2 4 1 -2 -6 }}


Optimal tuning (TE): ~1\2 = 599.2530, ~15/13 = 247.6643
Optimal tuning (TE): ~1\2 = 599.2530, ~15/13 = 247.6643
Line 98: Line 98:
TE mistunings (cents): ⟨-1.494, -0.272, 2.103, 1.908, -7.007, 8.284]
TE mistunings (cents): ⟨-1.494, -0.272, 2.103, 1.908, -7.007, 8.284]


Adjusted Error (cents): 5.508389
Adjusted error (cents): 5.508389


Complexity: 2.106437
Complexity: 2.106437
Line 112: Line 112:
Comma list: 96/95, 136/135, 153/152, 300/299, 391/390
Comma list: 96/95, 136/135, 153/152, 300/299, 391/390


Mapping: {{mapping| 2 4 3 7 9 11 7 | 0 -2 4 1 -2 -6 5 }}
{{Mapping|legend=0| 2 4 3 7 9 11 7 | 0 -2 4 1 -2 -6 5 }}


Optimal tuning (TE): ~1\2 = 599.1763, ~15/13 = 247.4859
Optimal tuning (TE): ~1\2 = 599.1763, ~15/13 = 247.4859
Line 120: Line 120:
TE mistunings (cents): ⟨-1.647, -0.222, 1.159, 1.192, -7.340, 8.511, 3.389]
TE mistunings (cents): ⟨-1.647, -0.222, 1.159, 1.192, -7.340, 8.511, 3.389]


Adjusted Error (cents): 5.641385
Adjusted error (cents): 5.641385


Complexity: 2.152852
Complexity: 2.152852
Line 134: Line 134:
Comma list: 96/95, 136/135, 153/152, 256/255, 289/285, 289/288, 324/323, 325/323
Comma list: 96/95, 136/135, 153/152, 256/255, 289/285, 289/288, 324/323, 325/323


Mapping: {{mapping| 2 3 5 6 8 8 | 0 1 -2 8 1 3 }}
{{Mapping|legend=0| 2 3 5 6 8 8 | 0 1 -2 8 1 3 }}


Optimal tuning (TE): ~1\2 = 599.3195, ~17/16 = 105.1130
Optimal tuning (TE): ~1\2 = 599.3195, ~17/16 = 105.1130
Line 142: Line 142:
TE mistunings (cents): ⟨-1.361, 1.117, 0.058, -3.706, -5.286, 12.382]
TE mistunings (cents): ⟨-1.361, 1.117, 0.058, -3.706, -5.286, 12.382]


Adjusted Error (cents): 6.376950
Adjusted error (cents): 6.376950


Complexity: 1.825578
Complexity: 1.825578
Line 156: Line 156:
Comma list: 96/95, 115/114, 136/135, 153/152, 325/324, 442/437
Comma list: 96/95, 115/114, 136/135, 153/152, 325/324, 442/437


Mapping: {{mapping| 2 3 5 6 8 8 | 0 1 -2 8 1 3 }}
{{Mapping|legend=0| 2 3 5 6 8 8 | 0 1 -2 8 1 3 }}


Optimal tuning (TE): ~1\2 = 599.3413, ~17/16 = 105.2039
Optimal tuning (TE): ~1\2 = 599.3413, ~17/16 = 105.2039
Line 164: Line 164:
TE mistunings (cents): ⟨-1.317, 1.273, -0.015, -2.849, -5.021, 12.829, -2.320]
TE mistunings (cents): ⟨-1.317, 1.273, -0.015, -2.849, -5.021, 12.829, -2.320]


Adjusted Error (cents): 6.366626
Adjusted error (cents): 6.366626


Complexity: 1.791017
Complexity: 1.791017
Line 178: Line 178:
Comma list: 81/80, 153/152, 170/169, 171/169, 171/170, 289/288, 323/320, 324/323
Comma list: 81/80, 153/152, 170/169, 171/169, 171/170, 289/288, 323/320, 324/323


Mapping: {{mapping| 2 3 4 7 8 8 | 0 2 8 5 2 6 }}
{{Mapping|legend=0| 2 3 4 7 8 8 | 0 2 8 5 2 6 }}


Optimal tuning (TE): ~1\2 = 600.7546, ~27/26 = 48.0949
Optimal tuning (TE): ~1\2 = 600.7546, ~27/26 = 48.0949
Line 186: Line 186:
TE mistunings (cents): ⟨1.509, -3.502, 1.464, 5.229, -2.729, -2.907]
TE mistunings (cents): ⟨1.509, -3.502, 1.464, 5.229, -2.729, -2.907]


Adjusted Error (cents): 5.610433
Adjusted error (cents): 5.610433


Complexity: 2.151739
Complexity: 2.151739
Line 200: Line 200:
Comma list: 153/152, 171/170, 300/299, 391/390, 741/736, 1105/1104, 1311/1300, 9747/9568
Comma list: 153/152, 171/170, 300/299, 391/390, 741/736, 1105/1104, 1311/1300, 9747/9568


Mapping: {{mapping| 2 3 4 7 8 8 8 | 0 2 8 5 2 6 13 }}
{{Mapping|legend=0| 2 3 4 7 8 8 8 | 0 2 8 5 2 6 13 }}


Optimal tuning (TE): ~1\2 = 600.7827, ~27/26 = 47.9997
Optimal tuning (TE): ~1\2 = 600.7827, ~27/26 = 47.9997
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TE mistunings (cents): ⟨1.565, -3.608, 0.815, 4.950, -2.695, -3.253, 1.983]
TE mistunings (cents): ⟨1.565, -3.608, 0.815, 4.950, -2.695, -3.253, 1.983]


Adjusted Error (cents): 5.607494
Adjusted error (cents): 5.607494


Complexity: 2.266610
Complexity: 2.266610


Related temperaments: [[decimal]]
Related temperaments: [[decimal]]
==== Scales ====
==== Scales ====


Line 235: Line 236:
Comma list: 171/170, 256/255, 324/323, 361/360, 460/459, 513/512, 576/575, 874/867
Comma list: 171/170, 256/255, 324/323, 361/360, 460/459, 513/512, 576/575, 874/867


Mapping: {{mapping| 1 2 -1 7 3 12 | 0 -1 8 -7 3 -18 }}
{{Mapping|legend=0| 1 2 -1 7 3 12 | 0 -1 8 -7 3 -18 }}


Optimal tuning (TE): ~2/1 = 1199.8084, ~4/3 = 498.3709
Optimal tuning (TE): ~2/1 = 1199.8084, ~4/3 = 498.3709
Line 243: Line 244:
TE mistunings (cents): ⟨-0.192, -0.709, 0.845, 5.107, -2.975, -1.249]
TE mistunings (cents): ⟨-0.192, -0.709, 0.845, 5.107, -2.975, -1.249]


Adjusted Error (cents): 2.918377
Adjusted error (cents): 2.918377


Complexity: 2.264161
Complexity: 2.264161
Line 304: Line 305:
Comma list: 324/323, 513/512, 2187/2176, 2197/2176, 2197/2187, 6144/6137
Comma list: 324/323, 513/512, 2187/2176, 2197/2176, 2197/2187, 6144/6137


Mapping: {{mapping| 1 3 7 14 0 | 0 -3 -7 -21 9 }}
{{Mapping|legend=0| 1 3 7 14 0 | 0 -3 -7 -21 9 }}


Optimal tuning (TE): ~2/1 = 1200.3681, ~18/13 = 566.5519
Optimal tuning (TE): ~2/1 = 1200.3681, ~18/13 = 566.5519
Line 312: Line 313:
TE mistunings (cents): ⟨0.368, -0.506, -3.814, 2.609, 1.454]
TE mistunings (cents): ⟨0.368, -0.506, -3.814, 2.609, 1.454]


Adjusted Error (cents): 2.565939
Adjusted error (cents): 2.565939


Complexity: 2.398767
Complexity: 2.398767
Line 326: Line 327:
Comma list: 208/207, 324/323, 513/512, 741/736, 2187/2176, 2197/2176, 2197/2187, 3159/3128
Comma list: 208/207, 324/323, 513/512, 741/736, 2187/2176, 2197/2176, 2197/2187, 3159/3128


Mapping: {{mapping| 1 3 7 14 0 5 | 0 -3 -7 -21 9 -1 }}
{{Mapping|legend=0| 1 3 7 14 0 5 | 0 -3 -7 -21 9 -1 }}


Optimal tuning (TE): ~2/1 = 1200.0516, ~18/13 = 566.3531
Optimal tuning (TE): ~2/1 = 1200.0516, ~18/13 = 566.3531
Line 334: Line 335:
TE mistunings (cents): ⟨0.052, -0.860, -4.638, 2.351, -0.335, 5.630]
TE mistunings (cents): ⟨0.052, -0.860, -4.638, 2.351, -0.335, 5.630]


Adjusted Error (cents): 3.578280
Adjusted error (cents): 3.578280


Complexity: 2.231135
Complexity: 2.231135
Line 348: Line 349:
Comma list: 153/152, 289/288, 324/323, 513/512, 729/722, 1088/1083
Comma list: 153/152, 289/288, 324/323, 513/512, 729/722, 1088/1083


Mapping: {{mapping| 12 19 45 49 51 | 0 0 -1 0 0 }}
{{Mapping|legend=0| 12 19 45 49 51 | 0 0 -1 0 0 }}


Optimal tuning (TE): ~1\12 = 100.0387, ~27/26 = 61.2152
Optimal tuning (TE): ~1\12 = 100.0387, ~27/26 = 61.2152
Line 356: Line 357:
TE mistunings (cents): ⟨0.465, -1.219, 0.000, -3.058, 4.462]
TE mistunings (cents): ⟨0.465, -1.219, 0.000, -3.058, 4.462]


Adjusted Error (cents): 2.986065
Adjusted error (cents): 2.986065


Complexity: 1.296641
Complexity: 1.296641
Line 370: Line 371:
Comma list: 153/152, 208/207, 289/288, 324/323, 442/437, 513/512, 729/722, 741/736
Comma list: 153/152, 208/207, 289/288, 324/323, 442/437, 513/512, 729/722, 741/736


Mapping: {{mapping| 12 19 45 49 51 54 | 0 0 -1 0 0 1 }}
{{Mapping|legend=0| 12 19 45 49 51 54 | 0 0 -1 0 0 1 }}


Optimal tuning (TE): ~1\12 = 100.0323, ~27/26 = 34.0630
Optimal tuning (TE): ~1\12 = 100.0323, ~27/26 = 34.0630
Line 378: Line 379:
TE mistunings (cents): ⟨0.388, -1.341, -5.042, -3.371, 4.136, 7.535]
TE mistunings (cents): ⟨0.388, -1.341, -5.042, -3.371, 4.136, 7.535]


Adjusted Error (cents): 4.929473
Adjusted error (cents): 4.929473


Complexity: 1.399563
Complexity: 1.399563
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'''Semiamulet'''
'''Semihaunt'''


Subgroup: 2.3.13.17.19
Subgroup: 2.3.13.17.19
Line 392: Line 393:
Comma list: 324/323, 512/507, 1088/1083, 2888/2873, 4624/4617, 6912/6859
Comma list: 324/323, 512/507, 1088/1083, 2888/2873, 4624/4617, 6912/6859


Mapping: {{mapping| 3 5 11 13 13 | 0 -2 1 -6 -2 }}
{{Mapping|legend=0| 3 5 11 13 13 | 0 -2 1 -6 -2 }}


Optimal tuning (TE): ~1\3 = 399.6037, ? = 48.1611
Optimal tuning (TE): ~1\3 = 399.6037, ? = 48.1611
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TE mistunings (cents): ⟨-1.189, -0.259, 3.274, 0.926, 1.013]
TE mistunings (cents): ⟨-1.189, -0.259, 3.274, 0.926, 1.013]


Adjusted Error (cents): 2.900547
Adjusted error (cents): 2.900547


Complexity: 0.682815
Complexity: 0.682815
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'''Semiamulet (+23)'''
'''Semihaunt (+23)'''


Subgroup: 2.3.13.17.19.23
Subgroup: 2.3.13.17.19.23
Line 414: Line 415:
Comma list: 208/207, 324/323, 512/507, 1088/1083, 2208/2197, 2888/2873, 4624/4617, 4693/4692
Comma list: 208/207, 324/323, 512/507, 1088/1083, 2208/2197, 2888/2873, 4624/4617, 4693/4692


Mapping: {{mapping| 3 5 11 13 13 13 | 0 -2 1 -6 -2 5 }}
{{Mapping|legend=0| 3 5 11 13 13 13 | 0 -2 1 -6 -2 5 }}


Optimal tuning (TE): ~1\3 = 399.4637, ? = 47.6560
Optimal tuning (TE): ~1\3 = 399.4637, ? = 47.6560
Line 422: Line 423:
TE mistunings (cents): ⟨-1.609, 0.052, 1.229, 2.137, 0.203, 3.034]
TE mistunings (cents): ⟨-1.609, 0.052, 1.229, 2.137, 0.203, 3.034]


Adjusted Error (cents): 3.417836
Adjusted error (cents): 3.417836


Complexity: 2.661109
Complexity: 2.661109
Line 436: Line 437:
Comma list: 289/288, 513/512, 2888/2873, 4624/4617, 6137/6084, 16473/16384
Comma list: 289/288, 513/512, 2888/2873, 4624/4617, 6137/6084, 16473/16384


Mapping: {{mapping| 2 3 8 8 9 | 0 2 -7 2 -6 }}
{{Mapping|legend=0| 2 3 8 8 9 | 0 2 -7 2 -6 }}


Optimal tuning (TE): ~1\2 = 600.1004, ? = 51.1066
Optimal tuning (TE): ~1\2 = 600.1004, ? = 51.1066
Line 444: Line 445:
TE mistunings (cents): ⟨0.201, 0.560, 2.530, -1.939, -3.248]
TE mistunings (cents): ⟨0.201, 0.560, 2.530, -1.939, -3.248]


Adjusted Error (cents): 2.281355
Adjusted error (cents): 2.281355


Complexity: 2.352952
Complexity: 2.352952
Line 458: Line 459:
Comma list: 208/207, 289/288, 513/512, 741/736, 2888/2873, 3211/3174, 3757/3726, 4624/4617
Comma list: 208/207, 289/288, 513/512, 741/736, 2888/2873, 3211/3174, 3757/3726, 4624/4617


Mapping: {{mapping| 2 3 8 8 9 10 | 0 2 -7 2 -6 -11 }}
{{Mapping|legend=0| 2 3 8 8 9 10 | 0 2 -7 2 -6 -11 }}


Optimal tuning (TE): ~1\2 = 600.0341, ? = 51.4873
Optimal tuning (TE): ~1\2 = 600.0341, ? = 51.4873
Line 466: Line 467:
TE mistunings (cents): ⟨0.068, 1.122, -0.666, -1.708, -6.130, 5.706]
TE mistunings (cents): ⟨0.068, 1.122, -0.666, -1.708, -6.130, 5.706]


Adjusted Error (cents): 3.867737
Adjusted error (cents): 3.867737


Complexity: 2.667192
Complexity: 2.667192
Line 480: Line 481:
Comma list: 324/323, 513/512, 2187/2176, 2888/2873, 6144/6137, 13718/13689
Comma list: 324/323, 513/512, 2187/2176, 2888/2873, 6144/6137, 13718/13689


Mapping: {{mapping| 1 2 1 7 3 | 0 -2 13 -14 6 }}
{{Mapping|legend=0| 1 2 1 7 3 | 0 -2 13 -14 6 }}


Optimal tuning (TE): ~2/1 = 1199.8867, ~? = 249.4160
Optimal tuning (TE): ~2/1 = 1199.8867, ~? = 249.4160
Line 488: Line 489:
TE mistunings (cents): ⟨-0.113, -1.014, 1.767, 2.427, -1.357]
TE mistunings (cents): ⟨-0.113, -1.014, 1.767, 2.427, -1.357]


Adjusted Error (cents): 1.996594
Adjusted error (cents): 1.996594


Complexity: 2.351780
Complexity: 2.351780
Line 502: Line 503:
Comma list: 324/323, 513/512, 2187/2176, 2208/2197, 2888/2873, 4693/4692, 6144/6137, 6656/6647
Comma list: 324/323, 513/512, 2187/2176, 2208/2197, 2888/2873, 4693/4692, 6144/6137, 6656/6647


Mapping: {{mapping| 1 2 1 7 3 -4 | 0 -2 13 -14 6 41 }}
{{Mapping|legend=0| 1 2 1 7 3 -4 | 0 -2 13 -14 6 41 }}


Optimal tuning (TE): ~2/1 = 1199.9040, ~? = 249.4504
Optimal tuning (TE): ~2/1 = 1199.9040, ~? = 249.4504
Line 510: Line 511:
TE mistunings (cents): ⟨-0.096, -1.048, 2.231, 2.068, -1.099, -0.426]
TE mistunings (cents): ⟨-0.096, -1.048, 2.231, 2.068, -1.099, -0.426]


Adjusted Error (cents): 1.972909
Adjusted error (cents): 1.972909


Complexity: 3.987557
Complexity: 3.987557
Line 545: Line 546:
Comma list: 153/152, 289/288, 324/323, 513/512, 729/722, 1088/1083
Comma list: 153/152, 289/288, 324/323, 513/512, 729/722, 1088/1083


Mapping: {{mapping| 12 19 49 51 54 | 0 0 0 0 0 }}
{{Mapping|legend=0| 12 19 49 51 54 | 0 0 0 0 0 }}


Optimal tuning (TE): ~1\12 = 100.0387, ~? = 26.1830
Optimal tuning (TE): ~1\12 = 100.0387, ~? = 26.1830
Line 553: Line 554:
TE mistunings (cents): ⟨0.465, -1.219, -3.058, 4.462, -0.000]
TE mistunings (cents): ⟨0.465, -1.219, -3.058, 4.462, -0.000]


Adjusted Error (cents): 3.179821
Adjusted error (cents): 3.179821


Complexity: 1.060700
Complexity: 1.060700
Line 649: Line 650:


== Related concepts ==
== Related concepts ==
* [[No-twos subgroup temperaments]]
* [[Subgroup temperaments]]
* [[No-threes subgroup temperaments]]
** [[No-twos subgroup temperaments]]
* [[No-fives subgroup temperaments]]
** [[No-threes subgroup temperaments]]
* [[No-sevens subgroup temperaments]]
** [[No-fives subgroup temperaments]]
** [[No-sevens subgroup temperaments]]
** [[Half-prime subgroup]]
* [[Substitute harmonic]]s
* [[Substitute harmonic]]s
* [[Dual-fifth temperaments]]
* [[Sooty fox scale]]s: many of these tunings are especially great at approximating equalizer subgroups


[[Category:Subgroup]][[Category:Subgroup temperaments]]
[[Category:Subgroup]][[Category:Subgroup temperaments]]
Retrieved from "https://en.xen.wiki/w/User:BudjarnLambeth/Equalizer_subgroup"