The Biosphere: Difference between revisions
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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. | ||
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). | 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. | 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. | ||
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= Parent Temperaments = | = Parent Temperaments = | ||
== Biome == | == Biome == | ||
Subgroup: 2.3.7.13/5 | Subgroup: 2.3.7.13/5 | ||
| Line 16: | Line 15: | ||
Mapping: | Mapping: | ||
{{val| 1 0 0 1 }} | {{val| 1 0 0 1 }}<br> | ||
{{val| 0 1 0 2 }}<br> | |||
{{val| 0 1 0 2 }} | |||
{{val| 0 0 1 -1 }} | {{val| 0 0 1 -1 }} | ||
| Line 31: | Line 28: | ||
Mapping: | Mapping: | ||
{{val| 1 0 0 0 0 1 }} | {{val| 1 0 0 0 0 1 }}<br> | ||
{{val| 0 1 0 0 0 2 }}<br> | |||
{{val| 0 1 0 0 0 2 }} | {{val| 0 0 1 0 0 1 }}<br> | ||
{{val| 0 0 0 1 0 -1 }}<br> | |||
{{val| 0 0 1 0 0 1 }} | |||
{{val| 0 0 0 1 0 -1 }} | |||
{{val| 0 0 0 0 1 0 }} | {{val| 0 0 0 0 1 0 }} | ||
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= 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. | ||
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Subgroup: 2.3.7.13/5 | Subgroup: 2.3.7.13/5 | ||
Comma list: 91/90 | Comma list: 64/63, 91/90 | ||
[[POTE generator]]: ~4/3 = 486.090 | [[POTE generator]]: ~4/3 = 486.090 | ||
Mapping: [{{val| 1 2 2 3 }}, {{val| 0 -1 2 -4 }}] | [[Mapping]]: [{{val| 1 2 2 3 }}, {{val| 0 -1 2 -4 }}] | ||
{{Val list|legend=1| 27, 32 }} | {{Val list|legend=1| 27, 32 }} | ||
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Subgroup: full 13-limit | Subgroup: full 13-limit | ||
Comma list: | [[Comma list]]: 64/63, 78/77, 91/90, 100/99 | ||
[[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 }}] | [[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 }} | ||
Badness: 0. | [[Badness]]: 0.024673 | ||
=== Ultrapyth === | |||
{{see also| Archytas clan #Ultrapyth }} | |||
Subgroup: full 13-limit | |||
[[Comma list]]: 55/54, 64/63, 91/90, 1573/1568 | |||
[[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 }} | |||
[[Badness]]: 0.049172 | |||
=== Porcupinefish === | === Porcupinefish === | ||
| Line 85: | Line 92: | ||
Subgroup: full 13-limit | Subgroup: full 13-limit | ||
Comma list: | [[Comma list]]: 55/54, 64/63, 91/90, 100/99 | ||
[[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 }}] | [[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 }} | ||
Badness: 0. | [[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. | ||
Subgroup: 2.3.7.13/5 | Subgroup: 2.3.7.13/5 | ||
Comma list: 91/90 | [[Comma list]]: 49/48, 91/90 | ||
[[POTE generator]]: ~7/6 = 251.507 | [[POTE generator]]: ~7/6 = 251.507 | ||
Mapping: [{{val| 1 2 3 2 }}, {{val| 0 -2 -1 -3 }}] | [[Mapping]]: [{{val| 1 2 3 2 }}, {{val| 0 -2 -1 -3 }}] | ||
{{Val list|legend=1| 19, 24 }} | {{Val list|legend=1| 19, 24 }} | ||