Xenharmonic Wiki

Map of rank-2 temperaments

Revision as of 03:25, 3 March 2024 by BudjarnLambeth (talk | contribs) (→Five periods per octave: Added cents)

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 0.000
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 Vishnu
3\50 72.000
1\16 75.000
2\30 80.000
3\44 81.818
4\58 82.759 Harry
9\130 83.077 Harry
5\72 83.333 Harry
1\14 85.714
2\26 92.308 Injera
3\38 94.737 Injera
1\12 100.000 Injera/diaschismic
5\58 103.448 Diaschismic
4\46 104.348 Diaschismic/srutal
7\80 105.000 Srutal
3\34 105.882 Srutal/pajara
5\56 107.143 Pajara
2\22 109.091 Pajara
3\32 112.500
1\10 120.000 Octokaidecal/Nimona
3\28 128.571 Octokaidecal/Nimona
2\18 133.333 Octokaidecal/Nimona
3\26 138.462
6\60 140.000
11\94 140.126 Fifive
15\128 140.625 Fifive
4\34 141.176 Fifive
5\42 142.857
1\8 150.000
4\30 160.000
3\22 163.636 Hedgehog/echidna
11\80 165.000 Hedgehog/echidna
8\58 165.517 Hedgehog/echidna
5\36 166.667
2\14 171.429
3\20 180.000
7\46 182.609 Unidec/hendec
11\72 183.333 Unidec/hendec
4\26 184.615 Unidec/hendec
5\32 187.500
1\6 200.000
6\34 211.765
5\28 214.286 Antikythera
9\50 216.000 Antikythera/Wizard
13\72 216.667 Antikythera/Wizard
4\22 218.182 Antikythera/Wizard/Astrology
15\82 219.512 Astrology
11\60 220.000 Astrology
7\38 221.053
3\16 225.000 Lemba
5\26 230.769 Lemba
12\62 232.258 Lemba
7\36 233.333
9\46 234.783 Echidnic
29\148 235.135 Echidnic
20\102 235.294 Echidnic
11\56 235.714
2\10 240.000 Decimal
5\24 250.000 Decimal
8\38 252.632 Decimal
3\14 257.143 Decimal
4\18 266.667
5\22 272.727 Doublewide
16\70 274.286 Doublewide
11\48 275.000 Doublewide
6\26 276.923
1\4 300.000

Three periods per octave

Generator Cents Comments
0\3 0.000
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
11\159 83.019 Tritikleismic
5\72 83.333 Tritikleismic
4\57 84.211 Tritikleismic
3\42 85.714
2\27 88.889 Augmented/augene
3\39 92.308 Augmented/augene
1\12 100.000 Augmented/augene/august
3\33 109.091 August
2\21 114.286 August
3\30 120.000
1\9 133.333
4\33 145.455
3\24 150.000 Triforce
5\39 153.846 Triforce
2\15 160.000 Triforce
3/21 171.429
1\6 200.000

Four periods per octave

Generator Cents Comments
0\4 0
1\76 15.789
1\72 16.667 Quadritikleismic
2\140 17.143 Quadritikleismic
1\68 17.647 Quadritikleismic
1\64 18.75
1\60 20
1\56 21.429
1\52 23.068
1\48 25
1\44 27.273
1\40 30
1\36 33.333
1\32 37.5
1\28 42.857
1\24 50 Darian calendar
1\20 60
1\16 75
1\12 100 Diminished
7\80 105 Diminished/Bidia
6\68 105.882 Diminished/Bidia
5\56 107.143 Diminished/Bidia
4\44 109.091 Diminished
3\32 112.5 Diminished
2\20 120 Diminished
1\8 150 Diminished

Five periods per octave

Generator Cents Comments
0\5 0
1\30 40
1\25 48
2\45 53.3
1\20 60
3\55 65.455
2\35 68.571
3\50 72
1\15 80 Blackwood/Blacksmith
4\55 87.273 Blackwood/Blacksmith
3\40 90 Blackwood/Blacksmith
5\65 92.308 Blackwood
2\25 96 Blackwood
5\60 100 Blackwood
3\35 102.857 Blackwood
4\45 106.667 Blackwood
5\55 109.091 Blackwood
6\65 110.769 Blackwood
7\75 112 Blackwood
8\85 112.941 Blackwood/Elderthing
9\95 113.684 Blackwood
10\105 114.286 Blackwood
1\10 120 Blackwood

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.

See also

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