Xenharmonic Wiki

The Biosphere: Difference between revisions

← Older editNewer edit →
VisualWikitext
Xenllium (talk | contribs)
autopatrolled, emailconfirmed
5,030 edits
No edit summary
Xenllium (talk | contribs)
autopatrolled, emailconfirmed
5,030 edits
No edit summary
Line 7: Line 7:
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.
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 ==
=== Biome ===
Subgroup: 2.3.7.13/5
Subgroup: 2.3.7.13/5


Line 21: Line 21:
{{Val list|legend=1| 5, 9, 14, 17, 22, 27, 32, 46 }}
{{Val list|legend=1| 5, 9, 14, 17, 22, 27, 32, 46 }}


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


Line 36: Line 36:
{{Val list|legend=1| 8d, 9, 10, 14cf, 15, 17c, 19, 22, 27e, 29, 31f, 37, 38df, 46 }}
{{Val list|legend=1| 8d, 9, 10, 14cf, 15, 17c, 19, 22, 27e, 29, 31f, 37, 38df, 46 }}


= Rank two temperaments =
== Rank two temperaments ==
== Oceanfront ==
=== 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.
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.


Line 44: Line 44:
Subgroup: 2.3.7.13/5
Subgroup: 2.3.7.13/5


Comma list: 64/63, 91/90
[[Comma list]]: 64/63, 91/90
 
[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -1 2 -4 }}]


[[POTE generator]]: ~4/3 = 486.090
[[POTE generator]]: ~4/3 = 486.090
[[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -1 2 -4 }}]


{{Val list|legend=1| 27, 32 }}
{{Val list|legend=1| 27, 32 }}


== Oceanfront extensions ==
==== Superpyth ====
=== Superpyth ===
{{see also| Archytas clan #Superpyth }}
{{see also| Archytas clan #Superpyth }}


Line 61: Line 60:


[[Comma list]]: 64/63, 78/77, 91/90, 100/99
[[Comma list]]: 64/63, 78/77, 91/90, 100/99
[[Mapping]]: [{{val| 1 2 6 2 10 9 }}, {{val| 0 -1 -9 2 -16 -13 }}]


[[POTE generator]]: ~4/3 = 489.521
[[POTE generator]]: ~4/3 = 489.521
[[Mapping]]: [{{val| 1 2 6 2 10 9 }}, {{val| 0 -1 -9 2 -16 -13 }}]


{{Val list|legend=1| 22, 27e, 49, 76bcde }}
{{Val list|legend=1| 22, 27e, 49, 76bcde }}
Line 70: Line 69:
[[Badness]]: 0.024673
[[Badness]]: 0.024673


=== Ultrapyth ===
==== Quasisupra ====
{{see also| Archytas clan #Quasisuper }}
 
Subgroup: full 13-limit
 
[[Comma list]]: 64/63, 78/77, 91/90, 121/120
 
[[Mapping]]: [{{val| 1 2 -3 2 1 0 }}, {{val| 0 -1 13 2 6 9 }}]
 
[[POTE generator]]: ~4/3 = 491.996
 
{{Val list|legend=1| 17c, 22, 39d, 61df, 100bcdf }}
 
[[Badness]]: 0.030219
 
==== Ultrapyth ====
{{see also| Archytas clan #Ultrapyth }}
{{see also| Archytas clan #Ultrapyth }}


Subgroup: 2.3.5.7.13
[[Comma list]]: 64/63, 91/90, 4394/4375
[[Mapping]]: [{{val|1 2 8 2 11}}, {{val|0 -1 -14 2 -18}}]
[[POTE generator]]: ~4/3 = 486.255
{{Val list|legend=1| 5, 32, 37 }}
===== Full 13-limit ultrapyth =====
Subgroup: full 13-limit
Subgroup: full 13-limit


[[Comma list]]: 55/54, 64/63, 91/90, 1573/1568
[[Comma list]]: 55/54, 64/63, 91/90, 1573/1568
[[Mapping]]: [{{val| 1 2 8 2 -1 11 }}, {{val| 0 -1 -14 2 11 -18 }}]


[[POTE generator]]: ~4/3 = 486.500
[[POTE generator]]: ~4/3 = 486.500
[[Mapping]]: [{{val| 1 2 8 2 -1 11 }}, {{val| 0 -1 -14 2 11 -18 }}]


{{Val list|legend=1| 5, 32, 37 }}
{{Val list|legend=1| 5, 32, 37 }}
Line 85: Line 110:
[[Badness]]: 0.049172
[[Badness]]: 0.049172


=== Porcupinefish ===
===== Counterultrapyth =====
Subgroup: full 13-limit
 
[[Comma list]]: 64/63, 91/90, 100/99, 847/845
 
[[Mapping]]: [{{val| 1 2 8 2 14 11 }}, {{val| 0 -1 -14 2 -26 -18 }}]
 
[[POTE generator]]: ~4/3 = 486.189
 
{{Val list|legend=1| 5e, 32e, 37, 79bcef, 116bbcef }}
 
[[Badness]]: 0.045653
 
==== Porcupinefish ====
{{see also| Porcupine family #Porcupinefish }}
{{see also| Porcupine family #Porcupinefish }}


Line 93: Line 131:


[[Comma list]]: 55/54, 64/63, 91/90, 100/99
[[Comma list]]: 55/54, 64/63, 91/90, 100/99
[[Mapping]]: [{{val| 1 2 3 2 4 6 }}, {{val| 0 -3 -5 6 -4 -17 }}]


[[POTE generator]]: ~10/9 = 162.277
[[POTE generator]]: ~10/9 = 162.277
[[Mapping]]: [{{val| 1 2 3 2 4 6 }}, {{val| 0 -3 -5 6 -4 -17 }}]


{{Val list|legend=1| 15, 22, 37, 59 }}
{{Val list|legend=1| 15, 22, 37, 59 }}
Line 102: Line 140:
[[Badness]]: 0.025314
[[Badness]]: 0.025314


== Tropic ==
=== Tropic ===
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.
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.


Line 108: Line 146:


[[Comma list]]: 49/48, 91/90
[[Comma list]]: 49/48, 91/90
[[Mapping]]: [{{val| 1 2 3 2 }}, {{val| 0 -2 -1 -3 }}]


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


[[Mapping]]: [{{val| 1 2 3 2 }}, {{val| 0 -2 -1 -3 }}]
{{Val list|legend=1| 19, 24 }}
 
==== Anguirus ====
{{see also| Diaschismic family #Anguirus }}
 
Subgroup: full 13-limit
 
[[Comma list]]: 49/48, 56/55, 91/90, 352/351
 
[[Mapping]]: [{{val| 2 4 3 6 9 7 }}, {{val| 0 -2 4 -1 -5 1 }}]
 
[[POTE generator]]: ~8/7 = 247.691
 
{{Val list|legend=1| 10, 24, 34, 58d, 92def }}


{{Val list|legend=1| 19, 24 }}
[[Badness]]: 0.030829


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


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
[[Comma list]]: 91/90, 169/168, 385/384, 441/440
[[Mapping]]: [{{val| 2 2 7 6 3 7 }}, {{val| 0 3 -6 -1 10 1 }}]
[[POTE generator]]: ~8/7 = 235.088
{{Val list|legend=1| 10, 46, 102, 148f, 194bcdf }}
[[Badness]]: 0.028874


[[Category:Theory]]
[[Category:Theory]]
Retrieved from "https://en.xen.wiki/w/The_Biosphere"