Octaphore: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Unque (talk | contribs)
217 edits
No edit summary
Cleanup
 
(19 intermediate revisions by 5 users not shown)
Line 1: Line 1:
{{Infobox Interval|Name=Octopus Comma, Enneagari|Ratio=94450499584/94143178827|Monzo=14 -23 0 8}}The '''Octophore 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).
{{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)]].  


== Temperaments ==
== Temperaments ==
Tempering out the Octaphore comma in the full 7-limit leads to rank-3 Octophore 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.


=== '''Octophore''' ===
=== Octaphore ===
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Comma List: 94450499584/94143178827
[[Comma list]]: 94450499584/94143178827


Mapping: [⟨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 }}


POTE tuning: ~28/27 = 62.233, ~5/4 = 386.314
: mapping generators: ~2, ~28/27, ~5


Optimal ET sequence: [[19edo|19]], [[39edo|39d]], [[58edo|58]], [[77edo|77]], [[96edo|96d]], [[135edo|135]]
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~28/27 = 62.233, ~5/4 = 386.314


=== Actinide ===
{{Optimal ET sequence|legend=1| 19, 39d, 58, 77, 96d, 135 }}
Subgroup: 2.3.7


Comma List: 94450499584/94143178827
==== 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]].


Mapping: [⟨1 2 4], ⟨0 -8 -23]]
Subgroup: 2.3.5.7.11


POTE tuning: ~28/27 = 62.233
Comma list: 200704/200475, 94450499584/94143178827


Optimal ET sequence: [[19edo|19]], [[20edo|20d]], [[39edo|39d]], [[58edo|58]], [[77edo|77]], [[96edo|96d]], [[135edo|135]]
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
 
[[Comma list]]: 94450499584/94143178827
 
{{Mapping|legend=1| 1 2 4 | 0 -8 -23 }}
 
: mapping generators: ~2, ~28/27
 
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~28/27 = 62.233
 
{{Optimal ET sequence|legend=1| 19, 20d, 39d, 58, 77, 96d, 135 }}


== See also ==
== 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]]

Latest revision as of 10:41, 3 April 2025

Interval information
Ratio 94450499584/94143178827
Factorization 214 × 3-23 × 78
Monzo [14 -23 0 8⟩
Size in cents 5.642232¢
Names octaphore,
enneagari comma
FJS name [math]\displaystyle{ \text{5d6}^{7,7,7,7,7,7,7,7} }[/math]
Special properties reduced
Tenney height (log2 nd) 72.913
Weil height (log2 max(n, d)) 72.9177
Wilson height (sopfr(nd)) 153
Open this interval in xen-calc

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)

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

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

See also

Retrieved from "https://en.xen.wiki/index.php?title=Octaphore&oldid=189697"