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 === | ||
| Line 15: | Line 14: | ||
[[Comma list]]: 94450499584/94143178827 | [[Comma list]]: 94450499584/94143178827 | ||
{{Mapping|legend=1| 1 2 | {{Mapping|legend=1| 1 2 2 4 | 0 -8 0 -23 | 0 0 1 0 }} | ||
: 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 }} | ||
=== 2.3.7 Unicorn === | ==== Undecimal octaphore ==== | ||
{{ See also | Unicorn }} | 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]]. | ||
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]]. | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 200704/200475, 94450499584/94143178827 | |||
Mapping: {{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 [[729/728|squbema]], or equivalently by tempering out the [[28812/28561|tesseract comma]]. | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 729/728, 3584/3575, 660275/657072 | |||
Mapping: {{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) === | |||
{{See also | Unicorn }} | |||
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 | ||
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: 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 }} | ||