Octaphore: Difference between revisions
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{{Infobox Interval | {{Infobox Interval | ||
| Name = | | Name = octaphore, enneagari comma | ||
| Ratio = 94450499584/94143178827 | | Ratio = 94450499584/94143178827 | ||
| Monzo = 14 -23 0 8 | | Monzo = 14 -23 0 8 | ||
}} | }} | ||
The '''octaphore''', also known as the '''enneagari comma''', is a [[small comma|small]] [[7-limit]] (also 2.3.7-[[subgroup]]) [[comma]] measuring about 5.64 [[cent]]s. 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|garischisma (33554432/33480783)]] and the [[septimal ennealimma|septimal ennealimma (40353607/40310784)]]. | |||
The '''octaphore''', also known as the '''enneagari comma''', is a [[small comma|small]] [[7-limit]] (also 2.3.7-[[subgroup]]) [[comma]] measuring about 5.64 [[cent]]s. It is so named because it is the amount by which eight [[28/27]] third-tones exceed the [[4/3]] perfect fourth. | |||
== Temperaments == | == 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 | [[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 === | === Octaphore === | ||
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: mapping generators: ~2, ~28/27, ~5 | : mapping generators: ~2, ~28/27, ~5 | ||
[[Optimal tuning]] ([[POTE]]): ~2 = | [[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~28/27 = 62.233, ~5/4 = 386.314 | ||
{{Optimal ET sequence|legend=1| 19, 39d, 58, 77, 96d, 135 }} | {{Optimal ET sequence|legend=1| 19, 39d, 58, 77, 96d, 135 }} | ||
==== Undecimal | ==== Undecimal octaphore ==== | ||
By noticing that the interval at | By noticing that the interval at {{monzo| 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 | Subgroup: 2.3.5.7.11 | ||
Comma list: 94450499584/94143178827 | Comma list: 200704/200475, 94450499584/94143178827 | ||
Mapping: | Mapping: {{mapping| 1 2 2 4 4 | 0 -8 0 -23 2 | 0 0 1 0 -2 }} | ||
Optimal tuning (POTE): ~2 = | Optimal tuning (POTE): ~2 = 1200.000, ~28/27 = 62.233, ~5/4 = 386.481 | ||
==== Tridecimal | ==== 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 [[729/728| | By noticing that two generators is extremely close to 14/13, we can add prime 13 to the mapping by tempering out the [[729/728|squbema]], or equivalently by tempering out the [[28812/28561|tesseract comma]]. | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
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Comma list: 729/728, 3584/3575, 660275/657072 | Comma list: 729/728, 3584/3575, 660275/657072 | ||
Mapping: | Mapping: {{mapping| 1 2 2 4 4 5 | 0 -8 0 -23 2 -25 | 0 0 1 0 -2 0 }} | ||
Optimal tuning (POTE): ~2 = | Optimal tuning (POTE): ~2 = 1200.000, ~27/26 = 62.281, ~5/4 = 386.512 | ||
===2.3.7 | === Unicorn (2.3.7 subgroup) === | ||
{{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]], where it finds prime 5 by interpreting five gens as a flat [[~]][[6/5]] by tempering [[126/125]]. | If we temper out the octaphore in its minimal prime subgroup of 2.3.7, we get the 2.3.7-subgroup version of [[unicorn]], where it finds prime 5 by interpreting five gens as a flat [[~]][[6/5]] by tempering out [[126/125]]. | ||
[[Subgroup]]: 2.3.7 | [[Subgroup]]: 2.3.7 | ||
| Line 57: | Line 56: | ||
: mapping generators: ~2, ~28/27 | : mapping generators: ~2, ~28/27 | ||
[[Optimal tuning]] ([[POTE]]): ~2 = | [[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~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 }} | ||
==See also== | == See also == | ||
*[[Unicorn family]] | * [[Unicorn family]] | ||
*[[Unicorn comma]] | * [[Unicorn comma]] | ||
[[Category:Commas named for how they divide the fourth]] | [[Category:Commas named for how they divide the fourth]] | ||
[[Category:Commas named for the intervals they stack]] | [[Category:Commas named for the intervals they stack]] | ||
Latest revision as of 10:41, 3 April 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]]
- mapping generators: ~2, ~28/27, ~5
Optimal tuning (POTE): ~2 = 1200.000, ~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: 200704/200475, 94450499584/94143178827
Mapping: [⟨1 2 2 4 4], ⟨0 -8 0 -23 2], ⟨0 0 1 0 -2]]
Optimal tuning (POTE): ~2 = 1200.000, ~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 = 1200.000, ~27/26 = 62.281, ~5/4 = 386.512
Unicorn (2.3.7 subgroup)
If we temper out the octaphore in its minimal prime subgroup of 2.3.7, we get the 2.3.7-subgroup version of unicorn, where it finds prime 5 by interpreting five gens as a flat ~6/5 by tempering out 126/125.
Subgroup: 2.3.7
Comma list: 94450499584/94143178827
Mapping: [⟨1 2 4], ⟨0 -8 -23]]
- mapping generators: ~2, ~28/27
Optimal tuning (POTE): ~2 = 1200.000, ~28/27 = 62.233
Optimal ET sequence: 19, 20d, 39d, 58, 77, 96d, 135