Map of rank-2 temperaments
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 | 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 | Srutal/pajara/injera | |||||||||||
| 4\46 | 104.348 | Srutal/pajara/diaschismic | |||||||||||
| 3\34 | 105.882 | Srutal/pajara/diaschismic | |||||||||||
| 2\22 | 109.091 | Srutal/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 | ||||||||||||
| 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 | ||||||||||
| 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 | 84.2105 | 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 | 145.4545 | ||||||||||
| 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 | 15.78947 | ||||||||||||||||||
| 1\72 | 16.6 | Quadritikleismic | |||||||||||||||||
| 2\140 | 17.14286 | Quadritikleismic | |||||||||||||||||
| 1\68 | 17.64706 | Quadritikleismic | |||||||||||||||||
| 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 | ||||||||||||||||||
| 1\12 | 100 | Diminished | |||||||||||||||||
| 8\92 | 104.34783 | Diminished | |||||||||||||||||
| 7\80 | 105 | Diminished/Bidia | |||||||||||||||||
| 6\68 | 105.88235 | Diminished/Bidia | |||||||||||||||||
| 5\56 | 107.14286 | Diminished/Bidia | |||||||||||||||||
| 4\44 | 109.09 | Diminished | |||||||||||||||||
| 3\32 | 112.5 | Diminished | |||||||||||||||||
| 2\20 | 120 | Diminished | |||||||||||||||||
| 1\8 | 150 | Diminished | |||||||||||||||||
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
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.