Octaphore: Difference between revisions
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{{Mapping|legend=1| 1 2 2 4 | 0 -8 0 -23 | 0 0 1 0 }} | {{Mapping|legend=1| 1 2 2 4 | 0 -8 0 -23 | 0 0 1 0 }} | ||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/27 = 62.233, ~5/4 = 386.314 | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/27 = 62.233, ~5/4 = 386.314 | ||
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===2.3.7 Unicorn=== | ===2.3.7 Unicorn=== | ||
{{See also | Unicorn }} | {{See also | Unicorn }} | ||
If we temper the octaphore in its minimal prime subgroup of 2.3.7, we get the 2.3.7-subgroup version of [[unicorn]] | If we temper the octaphore in its minimal prime subgroup of 2.3.7, we get the 2.3.7-subgroup version of [[unicorn]]: | ||
[[Subgroup]]: 2.3.7 | [[Subgroup]]: 2.3.7 | ||
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{{Mapping|legend=1| 1 2 4 | 0 -8 -23 }} | {{Mapping|legend=1| 1 2 4 | 0 -8 -23 }} | ||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/27 = 62.233 | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/27 = 62.233 | ||
{{Optimal ET sequence|legend=1| 19, 20d, 39d, 58, 77, 96d, 135 }} | {{Optimal ET sequence|legend=1| 19, 20d, 39d, 58, 77, 96d, 135 }} | ||
==== Septimal Unicorn ==== | |||
By noticing that the interval at five generators is nearly [[6/5]], we can introduce prime 5 to the mapping by tempering out [[126/125]]. | |||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: 10976/10935, 126/125 | |||
{{Mapping|legend=1| 1 2 3 4| 0 -8 -13 -23 }} | |||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/27 = 62.324 | |||
{{Optimal ET sequence|legend=1| 19, 39d, 58, 77, 135c, 212c}} | |||
==See also== | ==See also== | ||
Revision as of 14:17, 29 March 2025
| Interval information |
enneagari comma
The octaphore, also known as the enneagari comma, is a small 7-limit (also 2.3.7-subgroup) comma measuring about 5.64 cents. It is so named because it is the amount by which eight 28/27 third-tones exceed the 4/3 perfect fourth. It can also be found as the amount by which seven 28/27 third-tones exceed the 9/7 supermajor third, or as the sum of the garischisma (33554432/33480783) and the septimal ennealimma (40353607/40310784).
Temperaments
Tempering out the octaphore comma in the full 7-limit leads to rank-3 octaphore temperament, and excluding prime 5 from the subgroup leads to the 2.3.7 subgroup rank-2 Unicorn temperament.
Octaphore
Subgroup: 2.3.5.7
Comma list: 94450499584/94143178827
Mapping: [⟨1 2 2 4], ⟨0 -8 0 -23], ⟨0 0 1 0]]
Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.233, ~5/4 = 386.314
Optimal ET sequence: 19, 39d, 58, 77, 96d, 135
Undecimal Octaphore
By noticing that the interval at ⟨4 2 -2] is quite close to 11/8, we can add prime 11 to the mapping by tempering out the Reef comma.
Subgroup: 2.3.5.7.11
Comma list: 94450499584/94143178827, 200704/200475
Mapping: [⟨1 2 2 4 4], ⟨0 -8 0 -23 2], ⟨0 0 1 0 -2]]
Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.233, ~5/4 = 386.481
Tridecimal Octaphore
By noticing that two generators is extremely close to 14/13, we can add prime 13 to the mapping by tempering out the Squbema, or equivalently by tempering out the Tesseract Comma.
Subgroup: 2.3.5.7.11.13
Comma list: 729/728, 3584/3575, 660275/657072
Mapping: [⟨1 2 2 4 4 5], ⟨0 -8 0 -23 2 -25], ⟨0 0 1 0 -2 0]]
Optimal tuning (POTE): ~2 = 1\1, ~27/26 = 62.281, ~5/4 = 386.512
2.3.7 Unicorn
If we temper the octaphore in its minimal prime subgroup of 2.3.7, we get the 2.3.7-subgroup version of unicorn:
Subgroup: 2.3.7
Comma list: 94450499584/94143178827
Mapping: [⟨1 2 4], ⟨0 -8 -23]]
Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.233
Optimal ET sequence: 19, 20d, 39d, 58, 77, 96d, 135
Septimal Unicorn
By noticing that the interval at five generators is nearly 6/5, we can introduce prime 5 to the mapping by tempering out 126/125.
Subgroup: 2.3.5.7
Comma list: 10976/10935, 126/125
Mapping: [⟨1 2 3 4], ⟨0 -8 -13 -23]]
Optimal tuning (POTE): ~2 = 1\1, ~28/27 = 62.324
Optimal ET sequence: 19, 39d, 58, 77, 135c, 212c