The Biosphere
The biosphere is the name given to the collection of temperaments that are children of or related to biome temperament, the rank-3 2.3.7.13/5 subgroup temperament eliminating the biome comma 91/90, and biosphere temperament, its rank-5 full 13-limit extension. The term "biome" loosely means "ecosystem" or "climate."
The next low-numbered triad after 4:5:6 with a 3/2 on the outside is 6:7:9, but its inversion, 14:18:21, can sound extremely dissonant to those not used to 9-limit harmony. On the other hand, you also have 10:13:15, which is another standout triad of low complexity with a fifth on the outside, but its inversion, 26:30:39, is also relatively complex. Tempering out 91/90 makes both of these problems disappear by connecting the two together, such that the utonal inverse of 6:7:9 becomes 10:13:15.
The rank-3 biome temperament is of particular theoretical interest because it generates a rank-3 lattice that is analogous to the 5-limit JI lattice. As 5-limit JI is the basis for which all 5-limit linear temperaments are derived, the rank-3 biome temperament can serve as a basis to derive useful 2.3.7.13/5 linear temperaments. Instead of our base triads being 4:5:6 and its utonal inversion 10:12:15, we instead treat 6:7:9 and its utonal inversion 10:13:15 as fundamental to the system. The three dimensions of the system can be thought of as 2/1, 3/2, and 7/6 (or 9/7, or 13/10). 46EDO is a great tuning for biome, giving nearly-pure harmonies all around, somewhat analogous to the accuracy of 34EDO or 53EDO in approximating 5-limit JI.
This lattice can also be extended to deal with "higher primes", as can 5-limit JI. However, we instead expand the subgroup outward from the center, so that the "higher primes" we look at are things like like 5, 11, and 13. However, it may prove more useful at first to think purely within the 2.3.7.13/5 subgroup, so as to first come to understand the xenharmonic possibilities of the system.
Parent Temperaments
Biome
Subgroup: 2.3.7.13/5
Comma list: 91/90
Mapping:
⟨1 0 0 1]
⟨0 1 0 2]
⟨0 0 1 -1]
Optimal ET sequence: 5, 9, 14, 17, 22, 27, 32, 46
Biosphere
Subgroup: full 13-limit
Comma list: 91/90
Mapping:
⟨1 0 0 0 0 1]
⟨0 1 0 0 0 2]
⟨0 0 1 0 0 1]
⟨0 0 0 1 0 -1]
⟨0 0 0 0 1 0]
Optimal ET sequence: 8d, 9, 10, 14cf, 15, 17c, 19, 22, 27e, 29, 31f, 37, 38df, 46
Rank two temperaments
Oceanfront
Oceanfront is very similar to the familiar 7-limit superpyth temperament, in which 16/9 is equated with 7/4, 32/27 equated with 7/6, and 81/64 with 9/7. Oceanfront aims to equate 81/64 with 13/10 instead, however, so the fifths are even sharper than those of superpyth - 713.910 cents is the optimal POTE generator. The general structure of this scale is similar to that of meantone[7], except that the "major" triads in this scale are 10:13:15, and the minor triads are 6:7:9.
The sharp fifths of this scale can be a little more dissonant than meantone ears are used to, as can the flat fifths of something like mavila. This scale is very much like a brighter cousin of mavila in that regard.
Subgroup: 2.3.7.13/5
Comma list: 64/63, 91/90
Mapping: [⟨1 2 2 3], ⟨0 -1 2 -4]]
POTE generator: ~4/3 = 486.090
Superpyth
- See also: Archytas clan #Superpyth
Extends 11-limit superpyth as 22&49.
Subgroup: full 13-limit
Comma list: 64/63, 78/77, 91/90, 100/99
Mapping: [⟨1 2 6 2 10 9], ⟨0 -1 -9 2 -16 -13]]
POTE generator: ~4/3 = 489.521
Optimal ET sequence: 22, 27e, 49, 76bcde
Badness: 0.024673
Quasisupra
- See also: Archytas clan #Quasisuper
Subgroup: full 13-limit
Comma list: 64/63, 78/77, 91/90, 121/120
Mapping: [⟨1 2 -3 2 1 0], ⟨0 -1 13 2 6 9]]
POTE generator: ~4/3 = 491.996
Optimal ET sequence: 17c, 22, 39d, 61df, 100bcdf
Badness: 0.030219
Ultrapyth
- See also: Archytas clan #Ultrapyth
Subgroup: 2.3.5.7.13
Comma list: 64/63, 91/90, 4394/4375
Mapping: [⟨1 2 8 2 11], ⟨0 -1 -14 2 -18]]
POTE generator: ~4/3 = 486.255
Optimal ET sequence: 5, 32, 37
Full 13-limit ultrapyth
Subgroup: full 13-limit
Comma list: 55/54, 64/63, 91/90, 1573/1568
Mapping: [⟨1 2 8 2 -1 11], ⟨0 -1 -14 2 11 -18]]
POTE generator: ~4/3 = 486.500
Optimal ET sequence: 5, 32, 37
Badness: 0.049172
Ultramarine
Subgroup: full 13-limit
Comma list: 64/63, 91/90, 100/99, 847/845
Mapping: [⟨1 2 8 2 14 11], ⟨0 -1 -14 2 -26 -18]]
POTE generator: ~4/3 = 486.189
Optimal ET sequence: 5e, 32e, 37, 79bcef, 116bbcef
Badness: 0.045653
Porcupinefish
- See also: Porcupine family #Porcupinefish
Porcupinefish is the 13-limit extension of porcupine that you get by adding 91/90 to the usual mix of porcupine temperaments. Its name is derived from that it is a combination of the porcupine and oceanfront temperaments.
Subgroup: full 13-limit
Comma list: 55/54, 64/63, 91/90, 100/99
Mapping: [⟨1 2 3 2 4 6], ⟨0 -3 -5 6 -4 -17]]
POTE generator: ~10/9 = 162.277
Optimal ET sequence: 15, 22, 37, 59
Badness: 0.025314
Tropic
Tropic is the merger of the biosphere and the archipelago. It is also a subgroup relative of semaphore temperament, since 49/48 vanishes. Of note is that 676/675 vanishes, so that two 7/6's (or 15/13)'s is equated with 4/3. While this temperament doesn't take advantage of the nearly pure harmonies that biome tempering can offer, particularly where 7/4 is involved, it still has some use, particularly for those who don't mind a bit more error in their tunings.
Subgroup: 2.3.7.13/5
Comma list: 49/48, 91/90
Mapping: [⟨1 2 3 2], ⟨0 -2 -1 -3]]
POTE generator: ~7/6 = 251.507
Godzilla
- See also: Meantone family #Godzilla
Subgroup: 2.3.5.7.13
Comma list: 49/48, 81/80, 91/90
Mapping: [⟨1 0 -4 2 -5], ⟨0 2 8 1 11]]
POTE generator: ~7/6 = 252.429
Optimal ET sequence: 5, 14cf, 19
Full 13-limit godzilla
Subgroup: full 13-limit
Comma list: 45/44, 49/48, 78/77, 81/80
Mapping: [⟨1 0 -4 2 -6 -5], ⟨0 2 8 1 12 11]]
POTE generator: ~7/6 = 253.603
Optimal ET sequence: 5e, 14cf, 19, 33cdff, 52cdff
Badness: 0.022503
Varan
Subgroup: full 13-limit
Comma list: 49/48, 66/65, 77/75, 81/80
Mapping: [⟨1 0 -4 2 -10 -5], ⟨0 2 8 1 17 11]]
POTE generator: ~7/6 = 251.165
Optimal ET sequence: 19e, 24, 43de
Badness: 0.025676
Baragon
Subgroup: full 13-limit
Comma list: 49/48, 56/55, 81/80, 91/90
Mapping: [⟨1 0 -4 2 9 -5], ⟨0 2 8 1 -7 11]]
POTE generator: ~7/6 = 251.198
Optimal ET sequence: 5, 14cef, 19, 24, 43d
Badness: 0.026703
Anguirus
- See also: Diaschismic family #Anguirus
Subgroup: full 13-limit
Comma list: 49/48, 56/55, 91/90, 352/351
Mapping: [⟨2 4 3 6 9 7], ⟨0 -2 4 -1 -5 1]]
POTE generator: ~8/7 = 247.691
Optimal ET sequence: 10, 24, 34, 58d, 92def
Badness: 0.030829
Echidnic
- See also: Diaschismic family #Echidnic
13-limit echidnic temperament, the 10&46 temperament, is about as accurate as a biosphere temperament can get.
Subgroup: full 13-limit
Comma list: 91/90, 169/168, 385/384, 441/440
Mapping: [⟨2 2 7 6 3 7], ⟨0 3 -6 -1 10 1]]
POTE generator: ~8/7 = 235.088
Optimal ET sequence: 10, 46, 102, 148f, 194bcdf
Badness: 0.028874