Octaphore: Difference between revisions

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{{Infobox Interval

| Name = octaphore ~~comma~~, enneagari comma

| Name = octaphore, enneagari comma

| Ratio = 94450499584/94143178827

| 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 ~~comma~~''', 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)]].

== Temperaments ==

Tempering out the octaphore comma in the full 7-limit leads to rank-3 octaphore temperament, and ~~tempering it out in~~ the 2.3.7 subgroup ~~leads to the~~ rank-2 ~~actinide~~ temperament.

[[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 ===

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[[Comma list]]: 94450499584/94143178827

: mapping generators: ~2, ~28/27, ~5

[[Optimal tuning]] ([[POTE]]): ~2 = ~~1\1~~, ~28/27 = 62.233, ~5/4 = 386.314

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~28/27 = 62.233, ~5/4 = 386.314

=== ~~Actinide~~ ===

==== Undecimal octaphore ====

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

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) ===

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

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: mapping generators: ~2, ~28/27

[[Optimal tuning]] ([[POTE]]): ~2 = ~~1\1~~, ~28/27 = 62.233

[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~28/27 = 62.233

{{Optimal ET sequence|legend=1| 19, 20d, 39d, 58, 77, 96d, 135 }}

== See also ==

* [[Unicorn family]]

* [[Unicorn comma]]

[[Category:Commas named for how they divide the fourth]]

[[Category:Commas named for the intervals they stack]]

@@ Line 1: / Line 1: @@
 {{Infobox Interval
-| Name = octaphore comma, enneagari comma
+| Name = octaphore, enneagari comma
 | Ratio = 94450499584/94143178827
 | 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 comma''', 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)]].
 == Temperaments ==
-Tempering out the octaphore comma in the full 7-limit leads to rank-3 octaphore temperament, and tempering it out in the 2.3.7 subgroup leads to the rank-2 actinide temperament.
+[[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 ===
@@ Line 15: / Line 14: @@
 [[Comma list]]: 94450499584/94143178827
-{{Mapping|legend=1| 1 2 0 4 | 0 -8 0 -23 | 0 0 1 0 }}
+{{Mapping|legend=1| 1 2 2 4 | 0 -8 0 -23 | 0 0 1 0 }}
 : mapping generators: ~2, ~28/27, ~5
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/27 = 62.233, ~5/4 = 386.314
+[[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 }}
-=== Actinide ===
+==== Undecimal octaphore ====
+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
+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
@@ Line 32: / Line 56: @@
 : mapping generators: ~2, ~28/27
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/27 = 62.233
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~28/27 = 62.233
 {{Optimal ET sequence|legend=1| 19, 20d, 39d, 58, 77, 96d, 135 }}
+== See also ==
+* [[Unicorn family]]
+* [[Unicorn comma]]
+[[Category:Commas named for how they divide the fourth]]
+[[Category:Commas named for the intervals they stack]]