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User:BudjarnLambeth/Equalizer subgroup: Difference between revisions

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Document an idea that occurred to me and some temperaments generated with it
 
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<div style="border: 1px green solid; background-color: #efe; padding: 0.5em 1em; margin-bottom: 1em">
<div style="border: 1px green solid; background-color: #efe; padding: 0.5em 1em; margin-bottom: 1em">
__NEWSECTIONLINK__
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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 even most people.  
'''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.
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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'''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.
'''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, bit influential together.  
'''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.
[[5/1]] can act as either a structural or an assertive harmonic depending on context.


== Aim of equalizer subgroups ==
== Aim of equalizer subgroups ==
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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 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 (not ready audibly) similar to an equalizer subgroup.
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.
The sections of this article will showcase temperaments and scales from different equalizer subgroups. Feel free to add more.
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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]]
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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 ====


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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
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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
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==== Scales ====
==== Scales ====


== 2.3.n subgroups =====  
== 2.3.n subgroups ==


2.3.13.n subgroups ===
=== 2.3.13.n subgroups ===
Examples of valid equalizer subgroups in this category (not exhaustive):
Examples of valid equalizer subgroups in this category (not exhaustive):
* 2.3.13.14.17
* 2.3.13.14.17
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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== 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]]
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