Equalizer subgroup

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

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

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

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

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

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

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

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

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

Semiamulet

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

Semiamulet (+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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Related concepts

• Subgroup temperaments
  • No-twos subgroup temperaments
  • No-threes subgroup temperaments
  • No-fives subgroup temperaments
  • No-sevens subgroup temperaments
  • Half-prime subgroup
• Substitute harmonics
• Dual-fifth temperaments
• Sooty fox scales: many of these tunings are especially great at approximating equalizer subgroups