User:BudjarnLambeth/Equalizer subgroup: Difference between revisions

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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 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 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 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 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 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 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
Line 236: 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 305: 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 327: 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 349: 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 371: 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 387: Line 387:




'''Semiamulet'''
'''Semihaunt'''


Subgroup: 2.3.13.17.19
Subgroup: 2.3.13.17.19
Line 393: 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
Line 409: Line 409:




'''Semiamulet (+23)'''
'''Semihaunt (+23)'''


Subgroup: 2.3.13.17.19.23
Subgroup: 2.3.13.17.19.23
Line 415: 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 437: 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 459: 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 481: 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 503: 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 546: 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

Latest revision as of 12:33, 24 September 2026

Disclaimer from the author

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.

This article or section contains multiple idiosyncratic terms. Such terms are used by only a few people and are not regularly used within the community.

An equalizer subgroup is a just intonation subgroup which contains two or three structural harmonics, no assertive harmonics, and at least three ethereal harmonics.

Structural vs assertive vs ethereal

Structural harmonics means harmonics 2, 3, 4 and 5. These harmonics provide a solid sense of scale structure and tonal gravity, they also provide a warm, soothing bath of consonance that envelops the whole scale.

Assertive harmonics means harmonics 5 through 12, inclusive. These harmonics have very strong personalities, tending to define and shape the overall flavor of the scale.

Ethereal harmonics means harmonics 13, 14, 15, 16, 17 and so on. These harmonics usually tend to fade into the background, but they subtly wield their influence, adding consonances and dissonances undetectable on their own, but influential together.

5/1 can act as either a structural or an assertive harmonic depending on context.

Aim of equalizer subgroups

The reason for using an equalizer subgroup is to benefit from the stability, structure and powerful consonance of structural harmonics, but to keep the strong personalities of the assertive harmonics out of the way so that the subtler, more shy ethereal harmonics can shine through.

The hope is that there might be subtle harmonies and consonances involving ethereal harmonics that can allow for novel new sounds to be discovered, if we give them the space to show themselves.

The reason for the name is that when music producers or listeners use an equalizer, they often turn down the middle pitches, while boosting the low and high end, which is visually similar to an equalizer subgroup. (These are very much not the same thing though, don’t read too deep into the analogy.)

The sections of this article will showcase temperaments and scales from different equalizer subgroups. Feel free to add more.

2.3.5.n subgroups

2.3.5.13.n subgroups

Examples of valid equalizer subgroups in this category (not exhaustive):

  • 2.3.5.13.14.17
  • 2.3.5.13.14.17.19
  • 2.3.5.13.17.21

Temperaments

Fifigeist

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

Mapping: [⟨2 2 3 6 7 5], ⟨0 5 7 6 5 15]]

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

Recommended EDOs: 10, 18, 26, 34, 60, 94

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

Adjusted error (cents): 4.477868

Complexity: 2.503140

Related temperaments: fifive


Fifigeist (+23)

Subgroup: 2.3.5.13.17.19.23

Comma list: 115/114, 153/152, 300/299, 391/390, 1105/1104, 8075/8073

Mapping: [⟨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

Recommended EDOs: 10, 18, 26, 34, 60, 94

TE mistunings (cents): ⟨0.364, -0.082, -3.655, 2.375, -2.172, 7.925, -3.258]

Adjusted error (cents): 4.621471

Complexity: 2.381333

Related temperaments: fifive


Phantadecimal

Subgroup: 2.3.5.13.17.19

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

Mapping: [⟨2 4 3 7 9 11], ⟨0 -2 4 1 -2 -6]]

Optimal tuning (TE): ~1\2 = 599.2530, ~15/13 = 247.6643

Recommended EDOs: 10, 14, 24, 34, 58, 92

TE mistunings (cents): ⟨-1.494, -0.272, 2.103, 1.908, -7.007, 8.284]

Adjusted error (cents): 5.508389

Complexity: 2.106437

Related temperaments: decimal


Phantadecimal (+23)

Subgroup: 2.3.5.13.17.19.23

Comma list: 96/95, 136/135, 153/152, 300/299, 391/390

Mapping: [⟨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

Recommended EDOs: 10, 14, 24, 34, 58, 92

TE mistunings (cents): ⟨-1.647, -0.222, 1.159, 1.192, -7.340, 8.511, 3.389]

Adjusted error (cents): 5.641385

Complexity: 2.152852

Related temperaments: decimal


Phantasrutal

Subgroup: 2.3.5.13.17.19

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

Mapping: [⟨2 3 5 6 8 8], ⟨0 1 -2 8 1 3]]

Optimal tuning (TE): ~1\2 = 599.3195, ~17/16 = 105.1130

Recommended EDOs: 10, 12, 22, 34, 46, 80, 114

TE mistunings (cents): ⟨-1.361, 1.117, 0.058, -3.706, -5.286, 12.382]

Adjusted error (cents): 6.376950

Complexity: 1.825578

Related temperaments: srutal


Phantasrutal (+23)

Subgroup: 2.3.5.13.17.19.23

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

Mapping: [⟨2 3 5 6 8 8], ⟨0 1 -2 8 1 3]]

Optimal tuning (TE): ~1\2 = 599.3413, ~17/16 = 105.2039

Recommended EDOs: 10, 12, 22, 34, 46, 80, 114

TE mistunings (cents): ⟨-1.317, 1.273, -0.015, -2.849, -5.021, 12.829, -2.320]

Adjusted error (cents): 6.366626

Complexity: 1.791017

Related temperaments: srutal


Shadecimal

Subgroup: 2.3.5.13.17.19

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

Mapping: [⟨2 3 4 7 8 8], ⟨0 2 8 5 2 6]]

Optimal tuning (TE): ~1\2 = 600.7546, ~27/26 = 48.0949

Recommended EDOs: 24, 26, 28, 48, 50, 52, 74, 76

TE mistunings (cents): ⟨1.509, -3.502, 1.464, 5.229, -2.729, -2.907]

Adjusted error (cents): 5.610433

Complexity: 2.151739

Related temperaments: decimal


Shadecimal (+23)

Subgroup: 2.3.5.13.17.19.23

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

Mapping: [⟨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

Recommended EDOs: 24, 26, 28, 48, 50, 52, 74, 76

TE mistunings (cents): ⟨1.565, -3.608, 0.815, 4.950, -2.695, -3.253, 1.983]

Adjusted error (cents): 5.607494

Complexity: 2.266610

Related temperaments: decimal

Scales

2.3.5.14.n subgroups

Examples of valid equalizer subgroups in this category (not exhaustive):

  • 2.3.5.14.17.19
  • 2.3.5.14.17.19.21
  • 2.3.5.14.19.23

Temperaments

Scales

2.3.5.17.n subgroups

Examples of valid equalizer subgroups in this category (not exhaustive):

  • 2.3.5.17.19.21
  • 2.3.5.17.19.21.22
  • 2.3.5.17.21.26

Temperaments

Ectophotia

Subgroup: 2.3.5.17.19.23

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

Mapping: [⟨1 2 -1 7 3 12], ⟨0 -1 8 -7 3 -18]]

Optimal tuning (TE): ~2/1 = 1199.8084, ~4/3 = 498.3709

Recommended EDOs: 24, 26, 28, 48, 50, 52, 74, 76

TE mistunings (cents): ⟨-0.192, -0.709, 0.845, 5.107, -2.975, -1.249]

Adjusted error (cents): 2.918377

Complexity: 2.264161

Related temperaments: photia

Scales

2.3.5.19.n subgroups

Examples of valid equalizer subgroups in this category (not exhaustive):

  • 2.3.5.19.21.22
  • 2.3.5.19.21.22.23
  • 2.3.5.19.22.26

Temperaments

Scales

2.3.5.21.n subgroups

Examples of valid equalizer subgroups in this category (not exhaustive):

  • 2.3.5.21.22.23
  • 2.3.5.21.22.23.26
  • 2.3.5.21.23.28

Temperaments

Scales

2.3.5.22.n subgroups

Examples of valid equalizer subgroups in this category (not exhaustive):

  • 2.3.5.22.23.26
  • 2.3.5.22.23.26.28
  • 2.3.5.22.26.29

Temperaments

Scales

2.3.5.23.n subgroups

Examples of valid equalizer subgroups in this category (not exhaustive):

  • 2.3.5.23.26.28
  • 2.3.5.23.26.28.29
  • 2.3.5.23.28.31

Temperaments

Scales

Higher subgroups

Examples of valid equalizer subgroups in this category (not exhaustive):

  • 2.3.5.31.33.35
  • 2.3.5.37.41.43
  • 2.3.5.31.39.42.70

Temperaments

Scales

2.3.n subgroups

2.3.13.n subgroups

Examples of valid equalizer subgroups in this category (not exhaustive):

  • 2.3.13.14.17
  • 2.3.13.14.17.19
  • 2.3.13.17.21

Temperaments

Lieerie

Subgroup: 2.3.13.17.19

Comma list: 324/323, 513/512, 2187/2176, 2197/2176, 2197/2187, 6144/6137

Mapping: [⟨1 3 7 14 0], ⟨0 -3 -7 -21 9]]

Optimal tuning (TE): ~2/1 = 1200.3681, ~18/13 = 566.5519

Recommended EDOs: 17, 19, 36, 53

TE mistunings (cents): ⟨0.368, -0.506, -3.814, 2.609, 1.454]

Adjusted error (cents): 2.565939

Complexity: 2.398767

Related temperaments: liese


Lieerie (+23)

Subgroup: 2.3.13.17.19.23

Comma list: 208/207, 324/323, 513/512, 741/736, 2187/2176, 2197/2176, 2197/2187, 3159/3128

Mapping: [⟨1 3 7 14 0 5], ⟨0 -3 -7 -21 9 -1]]

Optimal tuning (TE): ~2/1 = 1200.0516, ~18/13 = 566.3531

Recommended EDOs: 17, 19, 36, 53

TE mistunings (cents): ⟨0.052, -0.860, -4.638, 2.351, -0.335, 5.630]

Adjusted error (cents): 3.578280

Complexity: 2.231135

Related temperaments: liese


Poltercompton

Subgroup: 2.3.13.17.19

Comma list: 153/152, 289/288, 324/323, 513/512, 729/722, 1088/1083

Mapping: [⟨12 19 45 49 51], ⟨0 0 -1 0 0]]

Optimal tuning (TE): ~1\12 = 100.0387, ~27/26 = 61.2152

Recommended EDOs: 12, 24, 36, 48, 60, 72, 84, 96

TE mistunings (cents): ⟨0.465, -1.219, 0.000, -3.058, 4.462]

Adjusted error (cents): 2.986065

Complexity: 1.296641

Related temperaments: compton


Poltercompton (+23)

Subgroup: 2.3.13.17.19.23

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

Mapping: [⟨12 19 45 49 51 54], ⟨0 0 -1 0 0 1]]

Optimal tuning (TE): ~1\12 = 100.0323, ~27/26 = 34.0630

Recommended EDOs: 12, 24, 36, 48, 60, 72, 84, 96

TE mistunings (cents): ⟨0.388, -1.341, -5.042, -3.371, 4.136, 7.535]

Adjusted error (cents): 4.929473

Complexity: 1.399563

Related temperaments: compton


Semihaunt

Subgroup: 2.3.13.17.19

Comma list: 324/323, 512/507, 1088/1083, 2888/2873, 4624/4617, 6912/6859

Mapping: [⟨3 5 11 13 13], ⟨0 -2 1 -6 -2]]

Optimal tuning (TE): ~1\3 = 399.6037, ? = 48.1611

Recommended EDOs: 9, 12, 15, 18, 21, 24, 27, 51, 75, 99

TE mistunings (cents): ⟨-1.189, -0.259, 3.274, 0.926, 1.013]

Adjusted error (cents): 2.900547

Complexity: 0.682815

Related temperaments: semiaug


Semihaunt (+23)

Subgroup: 2.3.13.17.19.23

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

Mapping: [⟨3 5 11 13 13 13], ⟨0 -2 1 -6 -2 5]]

Optimal tuning (TE): ~1\3 = 399.4637, ? = 47.6560

Recommended EDOs: 9, 12, 15, 18, 21, 24, 27, 51, 75, 99

TE mistunings (cents): ⟨-1.609, 0.052, 1.229, 2.137, 0.203, 3.034]

Adjusted error (cents): 3.417836

Complexity: 2.661109

Related temperaments: semiaug


Shrutarnatural

Subgroup: 2.3.13.17.19

Comma list: 289/288, 513/512, 2888/2873, 4624/4617, 6137/6084, 16473/16384

Mapping: [⟨2 3 8 8 9], ⟨0 2 -7 2 -6]]

Optimal tuning (TE): ~1\2 = 600.1004, ? = 51.1066

Recommended EDOs: 22, 24, 26, 46, 48, 70, 72, 94, 118

TE mistunings (cents): ⟨0.201, 0.560, 2.530, -1.939, -3.248]

Adjusted error (cents): 2.281355

Complexity: 2.352952

Related temperaments: shrutar


Shrutarnatural (+23)

Subgroup: 2.3.13.17.19.23

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

Mapping: [⟨2 3 8 8 9 10], ⟨0 2 -7 2 -6 -11]]

Optimal tuning (TE): ~1\2 = 600.0341, ? = 51.4873

Recommended EDOs: 22, 24, 26, 46, 48, 70, 72, 94, 118

TE mistunings (cents): ⟨0.068, 1.122, -0.666, -1.708, -6.130, 5.706]

Adjusted error (cents): 3.867737

Complexity: 2.667192

Related temperaments: shrutar


Spectrephore

Subgroup: 2.3.13.17.19

Comma list: 324/323, 513/512, 2187/2176, 2888/2873, 6144/6137, 13718/13689

Mapping: [⟨1 2 1 7 3], ⟨0 -2 13 -14 6]]

Optimal tuning (TE): ~2/1 = 1199.8867, ~? = 249.4160

Recommended EDOs: 9, 14, 19, 24, 29, 53, 82, 111

TE mistunings (cents): ⟨-0.113, -1.014, 1.767, 2.427, -1.357]

Adjusted error (cents): 1.996594

Complexity: 2.351780

Related temperaments: semaphore


Spectrephore (+23)

Subgroup: 2.3.13.17.19.23

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

Mapping: [⟨1 2 1 7 3 -4], ⟨0 -2 13 -14 6 41]]

Optimal tuning (TE): ~2/1 = 1199.9040, ~? = 249.4504

Recommended EDOs: 9, 14, 19, 24, 29, 53, 82, 111

TE mistunings (cents): ⟨-0.096, -1.048, 2.231, 2.068, -1.099, -0.426]

Adjusted error (cents): 1.972909

Complexity: 3.987557

Related temperaments: semaphore

Scales

2.3.14.n subgroups

Examples of valid equalizer subgroups in this category (not exhaustive):

  • 2.3.14.17.19
  • 2.3.14.17.19.21
  • 2.3.14.19.23

Temperaments

Scales

2.3.15.n subgroups

Examples of valid equalizer subgroups in this category (not exhaustive):

  • 2.3.15.17.19
  • 2.3.15.17.19.21
  • 2.3.15.19.23

Temperaments

Scales

2.3.17.n subgroups

Examples of valid equalizer subgroups in this category (not exhaustive):

  • 2.3.17.19.21
  • 2.3.17.19.21.22
  • 2.3.17.21.26

Temperaments

Werecatler

Subgroup: 2.3.17.19.23

Comma list: 153/152, 289/288, 324/323, 513/512, 729/722, 1088/1083

Mapping: [⟨12 19 49 51 54], ⟨0 0 0 0 0]]

Optimal tuning (TE): ~1\12 = 100.0387, ~? = 26.1830

Recommended EDOs: 12, 24, 36, 48, 60, 72, 84, 96

TE mistunings (cents): ⟨0.465, -1.219, -3.058, 4.462, -0.000]

Adjusted error (cents): 3.179821

Complexity: 1.060700

Related temperaments: catler

Scales

2.3.19.n subgroups

Examples of valid equalizer subgroups in this category (not exhaustive):

  • 2.3.19.21.22
  • 2.3.19.21.22.23
  • 2.3.19.22.26

Temperaments

Scales

2.3.20.n subgroups

Examples of valid equalizer subgroups in this category (not exhaustive):

  • 2.3.20.21.22
  • 2.3.20.21.22.23
  • 2.3.20.22.26

Temperaments

Scales

2.3.21.n subgroups

Examples of valid equalizer subgroups in this category (not exhaustive):

  • 2.3.21.22.23
  • 2.3.21.22.23.26
  • 2.3.21.23.28

Temperaments

Scales

2.3.22.n subgroups

Examples of valid equalizer subgroups in this category (not exhaustive):

  • 2.3.22.23.26
  • 2.3.22.23.26.28
  • 2.3.22.26.29

Temperaments

Scales

2.3.23.n subgroups

Examples of valid equalizer subgroups in this category (not exhaustive):

  • 2.3.23.26.28
  • 2.3.23.26.28.29
  • 2.3.23.28.31

Temperaments

Scales

Higher subgroups

Examples of valid equalizer subgroups in this category (not exhaustive):

  • 2.3.25.29.31
  • 2.3.31.37.41
  • 2.3.25.37.42.70

Temperaments

Scales

2.5.n subgroups

Temperaments

Examples of valid equalizer subgroups in this category (not exhaustive):

  • 2.5.13.14.15
  • 2.5.13.17.19.21
  • 2.5.17.30.38

Scales

3.5.n subgroups

Temperaments

Examples of valid equalizer subgroups in this category (not exhaustive):

  • 3.5.13.14.16
  • 3.5.13.17.19.21
  • 3.5.17.30.38

Scales

3.4.5.n subgroups

Examples of valid equalizer subgroups in this category (not exhaustive):

  • 3.4.5.13.14.16
  • 3.4.5.13.17.19.21
  • 3.4.5.17.30.38

Temperaments

Scales

3.4.n subgroups

Examples of valid equalizer subgroups in this category (not exhaustive):

  • 3.4.13.14.15
  • 3.4.13.17.19.21
  • 3.4.17.30.38

Temperaments

Scales

4.5.n subgroups

Examples of valid equalizer subgroups in this category (not exhaustive):

  • 4.5.13.14.15
  • 4.5.13.17.19.21
  • 4.5.17.30.38

Temperaments

Scales

Related concepts

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