# 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

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

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

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

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

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

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

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