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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 '''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 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-odd-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.
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.
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{{val| 0 0 1 -1 }}
{{val| 0 0 1 -1 }}


{{Val list|legend=1| 5, 9, 14, 17, 22, 27, 32, 46 }}
{{Optimal ET sequence|legend=1| 5, 9, 14, 17, 22, 27, 32, 46 }}


=== Biosphere ===
=== Biosphere ===
Subgroup: full 13-limit
Subgroup: 2.3.5.7.11.13


Comma list: 91/90
Comma list: 91/90
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{{val| 0 0 0 0 1 0 }}
{{val| 0 0 0 0 1 0 }}


{{Val list|legend=1| 8d, 9, 10, 14cf, 15, 17c, 19, 22, 27e, 29, 31f, 37, 38df, 46 }}
{{Optimal ET sequence|legend=1| 8d, 9, 10, 14cf, 15, 17c, 19, 22, 27e, 29, 31f, 37, 38df, 46 }}


== Rank two temperaments ==
== Rank two temperaments ==
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[[POTE generator]]: ~4/3 = 486.090
[[POTE generator]]: ~4/3 = 486.090


{{Val list|legend=1| 27, 32 }}
{{Optimal ET sequence|legend=1| 27, 32 }}
 
Scales: [[Oceanfront scales]]


==== Superpyth ====
==== Superpyth ====
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Extends 11-limit superpyth as 22&49.
Extends 11-limit superpyth as 22&49.


Subgroup: full 13-limit
Subgroup: 2.3.5.7.11.13


[[Comma list]]: 64/63, 78/77, 91/90, 100/99
[[Comma list]]: 64/63, 78/77, 91/90, 100/99
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[[POTE generator]]: ~4/3 = 489.521
[[POTE generator]]: ~4/3 = 489.521


{{Val list|legend=1| 22, 27e, 49, 76bcde }}
{{Optimal ET sequence|legend=1| 22, 27e, 49, 76bcde }}


[[Badness]]: 0.024673
[[Badness]]: 0.024673
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{{see also| Archytas clan #Quasisuper }}
{{see also| Archytas clan #Quasisuper }}


Subgroup: full 13-limit
Subgroup: 2.3.5.7.11.13


[[Comma list]]: 64/63, 78/77, 91/90, 121/120
[[Comma list]]: 64/63, 78/77, 91/90, 121/120
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[[POTE generator]]: ~4/3 = 491.996
[[POTE generator]]: ~4/3 = 491.996


{{Val list|legend=1| 17c, 22, 39d, 61df, 100bcdf }}
{{Optimal ET sequence|legend=1| 17c, 22, 39d, 61df, 100bcdf }}


[[Badness]]: 0.030219
[[Badness]]: 0.030219
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[[POTE generator]]: ~4/3 = 486.255
[[POTE generator]]: ~4/3 = 486.255


{{Val list|legend=1| 5, 32, 37 }}
{{Optimal ET sequence|legend=1| 5, 32, 37 }}


===== Full 13-limit ultrapyth =====
===== Full 13-limit ultrapyth =====
Subgroup: full 13-limit
Subgroup: 2.3.5.7.11.13


[[Comma list]]: 55/54, 64/63, 91/90, 1573/1568
[[Comma list]]: 55/54, 64/63, 91/90, 1573/1568
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[[POTE generator]]: ~4/3 = 486.500
[[POTE generator]]: ~4/3 = 486.500


{{Val list|legend=1| 5, 32, 37 }}
{{Optimal ET sequence|legend=1| 5, 32, 37 }}


[[Badness]]: 0.049172
[[Badness]]: 0.049172


===== Counterultrapyth =====
===== Ultramarine =====
Subgroup: full 13-limit
Subgroup: 2.3.5.7.11.13


[[Comma list]]: 64/63, 91/90, 100/99, 847/845
[[Comma list]]: 64/63, 91/90, 100/99, 847/845
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[[POTE generator]]: ~4/3 = 486.189
[[POTE generator]]: ~4/3 = 486.189


{{Val list|legend=1| 5e, 32e, 37, 79bcef, 116bbcef }}
{{Optimal ET sequence|legend=1| 5e, 32e, 37, 79bcef, 116bbcef }}


[[Badness]]: 0.045653
[[Badness]]: 0.045653
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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.


Subgroup: full 13-limit
Subgroup: 2.3.5.7.11.13


[[Comma list]]: 55/54, 64/63, 91/90, 100/99
[[Comma list]]: 55/54, 64/63, 91/90, 100/99
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[[POTE generator]]: ~10/9 = 162.277
[[POTE generator]]: ~10/9 = 162.277


{{Val list|legend=1| 15, 22, 37, 59 }}
{{Optimal ET sequence|legend=1| 15, 22, 37, 59 }}


[[Badness]]: 0.025314
[[Badness]]: 0.025314
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[[POTE generator]]: ~7/6 = 251.507
[[POTE generator]]: ~7/6 = 251.507


{{Val list|legend=1| 19, 24 }}
{{Optimal ET sequence|legend=1| 19, 24 }}
 
==== Godzilla ====
{{see also| Meantone family #Godzilla }}
 
Subgroup: 2.3.5.7.13
 
[[Comma list]]: 49/48, 81/80, 91/90
 
[[Mapping]]: [{{val|1 0 -4 2 -5}}, {{val|0 2 8 1 11}}]
 
[[POTE generator]]: ~7/6 = 252.429
 
{{Optimal ET sequence|legend=1| 5, 14cf, 19 }}
 
===== Full 13-limit godzilla =====
Subgroup: 2.3.5.7.11.13
 
[[Comma list]]: 45/44, 49/48, 78/77, 81/80
 
[[Mapping]]: [{{val|1 0 -4 2 -6 -5}}, {{val|0 2 8 1 12 11}}]
 
[[POTE generator]]: ~7/6 = 253.603
 
{{Optimal ET sequence|legend=1| 5e, 14cf, 19, 33cdff, 52cdff }}
 
[[Badness]]: 0.022503
 
===== Varan =====
Subgroup: 2.3.5.7.11.13
 
[[Comma list]]: 49/48, 66/65, 77/75, 81/80
 
[[Mapping]]: [{{val|1 0 -4 2 -10 -5}}, {{val|0 2 8 1 17 11}}]
 
[[POTE generator]]: ~7/6 = 251.165
 
{{Optimal ET sequence|legend=1| 19e, 24, 43de }}
 
[[Badness]]: 0.025676
 
===== Baragon =====
Subgroup: 2.3.5.7.11.13
 
[[Comma list]]: 49/48, 56/55, 81/80, 91/90
 
[[Mapping]]: [{{val|1 0 -4 2 9 -5}}, {{val|0 2 8 1 -7 11}}]
 
[[POTE generator]]: ~7/6 = 251.198
 
{{Optimal ET sequence|legend=1| 5, 14cef, 19, 24, 43d }}
 
[[Badness]]: 0.026703


==== Anguirus ====
==== Anguirus ====
{{see also| Diaschismic family #Anguirus }}
{{see also| Diaschismic family #Anguirus }}


Subgroup: full 13-limit
Subgroup: 2.3.5.7.11.13


[[Comma list]]: 49/48, 56/55, 91/90, 352/351
[[Comma list]]: 49/48, 56/55, 91/90, 352/351
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[[POTE generator]]: ~8/7 = 247.691
[[POTE generator]]: ~8/7 = 247.691


{{Val list|legend=1| 10, 24, 34, 58d, 92def }}
{{Optimal ET sequence|legend=1| 10, 24, 34, 58d, 92def }}


[[Badness]]: 0.030829
[[Badness]]: 0.030829
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13-limit echidnic temperament, the 10&46 temperament, is about as accurate as a biosphere temperament can get.
13-limit echidnic temperament, the 10&46 temperament, is about as accurate as a biosphere temperament can get.


Subgroup: full 13-limit
Subgroup: 2.3.5.7.11.13


[[Comma list]]: 91/90, 169/168, 385/384, 441/440
[[Comma list]]: 91/90, 169/168, 385/384, 441/440
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[[POTE generator]]: ~8/7 = 235.088
[[POTE generator]]: ~8/7 = 235.088


{{Val list|legend=1| 10, 46, 102, 148f, 194bcdf }}
{{Optimal ET sequence|legend=1| 10, 46, 102, 148f, 194bcdf }}


[[Badness]]: 0.028874
[[Badness]]: 0.028874


[[Category:Theory]]
[[Category:Regular temperament theory]]
[[Category:Temperament collection]]
[[Category:Commatic realms]]
[[Category:Biome]]
[[Category:Biome]]
[[Category:Biosphere]]
[[Category:Biosphere]]
Retrieved from "https://en.xen.wiki/w/The_Biosphere"