The Biosphere: Difference between revisions
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The biosphere is the name given to the collection of temperaments that are children of or related to | 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 [[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 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. | ||
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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). 46-EDO is a great tuning for biome, giving nearly-pure harmonies all around, somewhat analogous to the accuracy of 34-EDO or 53-EDO in approximating 5-limit JI. | 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). 46-EDO is a great tuning for biome, giving nearly-pure harmonies all around, somewhat analogous to the accuracy of 34-EDO or 53-EDO in approximating 5-limit JI. | ||
This lattice can also be extended to deal with "higher primes, | 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= | = Parent Temperaments = | ||
== Biome == | |||
= | |||
Subgroup: 2.3.7.13/5 | Subgroup: 2.3.7.13/5 | ||
| Line 24: | Line 23: | ||
EDOs: 14, 17, 22, 27, 32, 46 | EDOs: 14, 17, 22, 27, 32, 46 | ||
= | == Biosphere == | ||
Subgroup: Full 13-limit | Subgroup: Full 13-limit | ||
| Line 43: | Line 42: | ||
EDOs: 46 | EDOs: 46 | ||
=Rank two temperaments= | = Rank two temperaments = | ||
== Oceanfront == | |||
= | |||
Subgroup: 2.3.7.13/5 | Subgroup: 2.3.7.13/5 | ||
Commas: 91/90, 64/63 | Commas: 91/90, 64/63 | ||
[[ | [[POTE generator]]: ~4/3 = 486.090 | ||
Map: [<1 2 2 3|, < 0 -1 2 -4|] | Map: [<1 2 2 3|, < 0 -1 2 -4|] | ||
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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. | 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. | ||
== | == Oceanfront Children == | ||
=== Superpyth === | |||
===Superpyth=== | |||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Commas: 91/90, 64/63, 78/77, 245/243 | Commas: 91/90, 64/63, 78/77, 245/243 | ||
[[ | [[POTE generator]]: ~4/3 = 489.521 | ||
Map: [< 1 2 6 2 10 9|, <0 -1 -9 2 -16 -13|] | Map: [< 1 2 6 2 10 9|, <0 -1 -9 2 -16 -13|] | ||
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Extends 11-limit superpyth as 22&49. | Extends 11-limit superpyth as 22&49. | ||
===Porcupinefish=== | === Porcupinefish === | ||
Subgroup: 13-limit | Subgroup: 13-limit | ||
Commas: 91/90, 64/63, 250/243, 121/120 | Commas: 91/90, 64/63, 250/243, 121/120 | ||
[[ | [[POTE generator]]: ~10/9 = 162.277 | ||
Map: [<1 2 3 2 4 6|, <0 -3 -5 6 -4 -17|] | Map: [<1 2 3 2 4 6|, <0 -3 -5 6 -4 -17|] | ||
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Porcupinefish is the 13-limit extension of [[Porcupine|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. | Porcupinefish is the 13-limit extension of [[Porcupine|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. | ||
=Tropic= | == Tropic == | ||
Subgroup: 2.3.7.13/5 | Subgroup: 2.3.7.13/5 | ||
Commas: 91/90, 49/48 | Commas: 91/90, 49/48 | ||
[[ | [[POTE generator]]: ~7/6 = 251.507 | ||
Map: [<1 2 3 2|, <0 -2 -1 -3|] | Map: [<1 2 3 2|, <0 -2 -1 -3|] | ||
| Line 103: | Line 100: | ||
EDOs: 19, 24 | EDOs: 19, 24 | ||
Tropic is the merger of the biosphere and the [[ | Tropic is the merger of the biosphere and the [[The Archipelago|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. | ||
=Avian= | == Avian == | ||
Subgroup: 2.3.5.7.13 | Subgroup: 2.3.5.7.13 | ||
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EDOs: 19, 27, 46 | EDOs: 19, 27, 46 | ||
=Echidnic= | == Echidnic == | ||
13-limit [[ | 13-limit [[Diaschismic family #Echidnic|echidnic]] temperament, the 10&46 temperament, is about as accurate as a biosphere temperament can get. | ||
[[Category:Theory]] | |||
[[Category: | [[Category:Temperament collection]] | ||
[[Category: | [[Category:Biome]] | ||
[[Category: | [[Category:Biosphere]] | ||
[[Category: | |||