Map of rank-2 temperaments

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This is intended to be a map of all interesting rank-2 temperaments that are compatible with octave equivalence. The only rank-2 temperaments not appearing here should be ones like Bohlen-Pierce that completely lack octaves.

Please make sure each fraction of an octave is always the mediant of the ones directly above and below.

One period per octave

Since this is the largest subset, it has its own page: Map of linear temperaments.

Two periods per octave

Generator Cents Comments
0\2 0
1\26 46.154
1\24 50.000 Shrutar
2\46 52.174 Shrutar
1\22 54.545 Shrutar
1\20 60.000
1\18 66.667
2\34 70.588 Vishnu
7\118 71.186 Vishnu
5\84 71.429
3\50 72.000
1\16 75.000
2\30 80.000
3\44 81.818
4\58 82.759 Harry
5\72 83.333 Harry
1\14 85.714
2\26 92.308 Injera
3\38 94.737 Injera
1\12 100.000 Srutal/pajara/injera
4\46 104.348 Srutal/pajara/diaschismic
3\34 105.882 Srutal/pajara/diaschismic
2\22 109.091 Srutal/pajara
1\10 120.000
2\18 133.333 Octokaidecal
3\26 138.462
4\34 141.176 Fifive
1\8 150.000
4\30 160.000
3\22 163.636 Hedgehog/echidna
8\58 165.517 Hedgehog/echidna
5\36 166.667 Hedgehog/echidna
2\14 171.429
3\20 180.000
7\46 182.609 Unidec/hendec
11\72 183.333 Unidec/hendec
4\26 184.615
5\32 187.500
6\38 189.474
7\44 190.909
8\50 192.000
9\56 192.857
10\62 193.544
1\6 200.000
11\64 206.250
10\58 206.897
9\52 207.692
8\46 208.696
7\40 210.000
6\34 211.765
5\28 214.286
9\50 216.000
13\72 216.667 Antikythera/astrology/wizard
4\22 218.182 Antikythera/astrology
3\16 225.000
8\42 228.571
5\26 230.769 Lemba
7\36 233.333 Lemba
9\46 234.783 Echidnic
2\10 240.000 Decimal
7\34 247.059 Decimal
5\24 250.000 Decimal
3\14 257.143
4\18 266.667
5\22 272.727 Doublewide
11\48 275.000 Doublewide
6\26 276.923 Doublewide
7\30 280.000
8/34 282.353
9/38 284.2105
10/42 285.714
11\46 286.9565
1\4 300.000

Three periods per octave

Generator Cents Comments
0\3 0
1\30 40.000
1\27 44.444 Semiaug
2\51 47.059 Semiaug
1\24 50.000 Semiaug
1\21 57.143
1\18 66.667
1\15 80.000
6\87 82.759 Tritikleismic
5\72 83.333 Tritikleismic
4\57 84.2105
3\42 85.714
2\27 88.889 Augmented/augene
1\12 100.000 Augmented/augene/august
3\33 109.091 Augmented/august
2\21 114.286
3\30 120.000
4\39 123.077
5\48 125.000
6\57 126.316
7\66 127.273
1\9 133.333
8\69 139.134
7\60 140.000
6\51 141.176
5\42 142.857
4\33 145.4545
3\24 150.000 Triforce
2\15 160.000
3/21 171.429
4/27 177.778
5/33 181.818
6\39 184.615
7\45 186.667
8\51 118.235
1\6 200.000

Four periods per octave

Generator Cents Comments
0\4 0
1\76 15.78947
1\72 16.6 Quadritikleismic
2\140 17.14286
1\68 17.64706
1\64 18.75
1\60 20
1\56 21.42857
1\52 23.06792
1\48 25
1\44 27.27
1\40 30
1\36 33.3
1\32 37.5
1\28 42.85714
1\24 50
1\20 60
1\16 75 Diminished
2\28 85.714285
3\40 90
4\52 92.30769
5\64 93.75
6\76 94.73684
7\88 95.45
8\100 96
9\112 96.42857
10\124 96.77419
11\136 97.08852
12\148 97.297
13\160 97.5
14\172 97.67442
15\184 97.82609
16\196 97.95918
1\12 100 Diminished
17\200 102
16\188 102.12766
15\176 102.27
14\164 102.43902
13\152 102.63158
12\140 102.85714
11\128 103.125
10\116 103.448275
9\104 103.84615
8\92 104.34783
7\80 105
6\68 105.88235 Bidia
5\56 107.14286
4\44 109.09
3\32 112.5
2\20 120
3\28 128.57143
4\36 133.3
5\44 136.36
6\52 138.46154
7\60 140
8\68 141.17646
9\76 142.10526
10\84 142.85714
11\92 143.47826
12\100 144
13\108 144.4
14\116 144.82759
15\124 145.16129
16\132 145.45
17\140 145.714285
1\8 150

Five periods per octave

  • Blackwood/blacksmith - The prime 3, and in blacksmith also 7, is represented using 5edo. The generator gets you to all intervals of 5.
  • Elderthing - generator of phi. Two generators up to 3, two down to 7, other primes are more complex. (One generator up or one down are ambiguous 13.)

Six periods per octave

  • Hexe - The 2.5.7 subgroup is represented using 6edo, and the generator gets you to 4/3 and 3/2. Makes little sense not to additionally temper down to 12edo.

Seven periods per octave

  • Whitewood - Analogue of blackwood. The prime 3 is represented using 7edo, the generator is used for 5.
  • Jamesbond/septimal - The 5-limit (and in septimal the prime 11) is represented using 7edo, and the generator is only used for intervals of 7.
  • Sevond - 10/9 is tempered to be exactly 1\7 of an octave. Therefore 3/2 is 1 generator sharp of a 7edo step and 5/4 is 2 generators sharp.
  • Absurdity - A complex temperament (perhaps "absurdly" so).

Eight periods per octave

  • Octoid - 16-cent generator, sub-cent accuracy.

Nine periods per octave

  • Ennealimmal - The generator is 49.02 cents, and don't forget the ".02" because it really is that accurate.

Twelve periods per octave

See also: Pythagorean family

Temperaments in this family are interesting because they can be thought of as 12edo with microtonal alterations.

  • Compton - 3-limit as in 12edo; intervals of 5 are off by one generator. In the 7-limit (sometimes called waage), intervals of 7 are off by two generators. In the 11-limit, intervals of 11 are off by 3 generators. Thinking of 72edo might make this more concrete.
  • Catler - 5-limit as in 12edo; intervals of 7 are off by one generator.
  • Atomic - Does not temper out the schisma, so 3/2 is one schisma sharp of its 12edo value. In atomic, since twelve fifths are sharp of seven octaves by twelve schismas, the Pythagorean comma is twelve schismas, and hence 81/80, the Didymus comma, is eleven schismas. In fact eleven schismas is sharp of 81/80, and twelve schismas of the Pythaorean comma, by the microscopic interval of the atom, which atomic tempers out. Extremely accurate.