Octaphore: Difference between revisions

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{{Infobox Interval|Name=~~Octaphore~~ comma~~, Enneagari~~|Ratio=94450499584/94143178827|Monzo=14 -23 0 8}}

{{Infobox Interval

| 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~~''' 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 generators: ~2, ~28/27, ~5

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

== ~~Temperaments~~ ==

==== Undecimal octaphore ====

~~Tempering~~ out the ~~Octaphore~~ comma ~~in the full~~ 7-~~limit leads to rank~~-~~3 Octophore temperament~~ (~~note~~ the ~~different spelling)~~, ~~and~~ tempering it out in the 2.3.7 ~~subgroup leads to the rank-2 Actinide temperament~~.

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

~~=== '''Octophore''' ===~~

Comma list: 729/728, 3584/3575, 660275/657072

~~Subgroup~~: ~~2.3.5.7~~

~~Comma List~~: ~~94450499584/94143178827~~

Mapping: {{mapping| 1 2 2 4 4 5 | 0 -8 0 -23 2 -25 | 0 0 1 0 -2 0 }}

~~Mapping~~: ~~[⟨1~~ 2 ~~2 4]~~, ~~⟨0 -8 0 -23]~~, ~~⟨0 0 1 0]]~~

Optimal tuning (POTE): ~2 = 1200.000, ~27/26 = 62.281, ~5/4 = 386.512

~~POTE tuning: ~28/27~~ = 62.~~233~~, ~5/~~4 = 386~~.~~314~~

=== 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]].

~~Optimal ET sequence:~~ [[~~19edo|19]], [[39edo|39d]], [[58edo|58]], [[77edo|77]], [[96edo|96d]], [[135edo|135~~]]

[[Subgroup]]: 2.3.7

~~=== Actinide ===~~

[[Comma list]]: 94450499584/94143178827

~~Subgroup~~: ~~2.3.7~~

~~Comma List: 94450499584/94143178827~~

~~Mapping~~: ~~[⟨1~~ 2 4], ~~⟨0 -8 -23]]~~

: mapping generators: ~2, ~28/27

POTE ~~tuning~~: ~28/27 = 62.233

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

Optimal ET sequence~~: [[19edo~~|19]], ~~[[20edo|~~20d]], ~~[[39edo|~~39d]], ~~[[58edo|~~58]], ~~[[77edo|~~77]], ~~[[96edo|~~96d]], ~~[[135edo|~~135]]

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

== See also ==

[[~~Small~~ comma]]

* [[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|Ratio=94450499584/94143178827|Monzo=14 -23 0 8}}
+{{Infobox Interval
+| 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''' 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|legend=1| 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|legend=1| 19, 39d, 58, 77, 96d, 135 }}
-== Temperaments ==
+==== Undecimal octaphore ====
-Tempering out the Octaphore comma in the full 7-limit leads to rank-3 Octophore temperament (note the different spelling), and tempering it out in the 2.3.7 subgroup leads to the rank-2 Actinide temperament.
+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
-=== '''Octophore''' ===
+Comma list: 729/728, 3584/3575, 660275/657072
-Subgroup: 2.3.5.7
-Comma List: 94450499584/94143178827
+Mapping: {{mapping| 1 2 2 4 4 5 | 0 -8 0 -23 2 -25 | 0 0 1 0 -2 0 }}
-Mapping: [⟨1 2 2 4], ⟨0 -8 0 -23], ⟨0 0 1 0]]
+Optimal tuning (POTE): ~2 = 1200.000, ~27/26 = 62.281, ~5/4 = 386.512
-POTE tuning: ~28/27 = 62.233, ~5/4 = 386.314
+=== 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]].
-Optimal ET sequence: [[19edo|19]], [[39edo|39d]], [[58edo|58]], [[77edo|77]], [[96edo|96d]], [[135edo|135]]
+[[Subgroup]]: 2.3.7
-=== Actinide ===
+[[Comma list]]: 94450499584/94143178827
-Subgroup: 2.3.7
-Comma List: 94450499584/94143178827
+{{Mapping|legend=1| 1 2 4 | 0 -8 -23 }}
-Mapping: [⟨1 2 4], ⟨0 -8 -23]]
+: mapping generators: ~2, ~28/27
-POTE tuning: ~28/27 = 62.233
+[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~28/27 = 62.233
-Optimal ET sequence:  [[19edo|19]], [[20edo|20d]], [[39edo|39d]], [[58edo|58]], [[77edo|77]], [[96edo|96d]], [[135edo|135]]
+{{Optimal ET sequence|legend=1| 19, 20d, 39d, 58, 77, 96d, 135 }}
 == See also ==
-[[Small comma]]
+* [[Unicorn family]]
+* [[Unicorn comma]]
+[[Category:Commas named for how they divide the fourth]]
+[[Category:Commas named for the intervals they stack]]