Xenharmonic Wiki

181edo: Difference between revisions

← Older editNewer edit →
VisualWikitext
Contribution (talk | contribs)
3,308 edits
No edit summary
Cleanup
Line 1: Line 1:
{{Infobox ET}}
{{Infobox ET}}
The '''181 equal divisions of the octave''' ('''181edo'''), or the '''181(-tone) equal temperament''' ('''181tet''', '''181et''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 181 parts of about 6.63 [[cent]]s each.
{{EDO intro}}


== Theory ==
== Theory ==
181et tempers out 2109375/2097152 ([[semicomma]]) and {{monzo| 14 -22 9 }} in the 5-limit; [[2401/2400]], [[5120/5103]], and 390625/387072 in the 7-limit (supporting the [[hemififths]] and the [[cotritone]]). Using the patent val, it tempers out [[385/384]], 1375/1372, 2200/2187, and [[4000/3993]] in the 11-limit; [[325/324]], [[352/351]], [[847/845]], and [[1575/1573]] in the 13-limit.
181et [[tempering out|tempers out]] 2109375/2097152 ([[semicomma]]) and {{monzo| 14 -22 9 }} in the 5-limit; [[2401/2400]], [[5120/5103]], and 390625/387072 in the 7-limit ([[support]]ing the [[hemififths]] and the [[cotritone]]). Using the patent val, it tempers out [[385/384]], 1375/1372, [[2200/2187]], and [[4000/3993]] in the 11-limit; [[325/324]], [[352/351]], [[847/845]], and [[1575/1573]] in the 13-limit.
 
181edo is the 42nd [[prime edo]].


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|181}}
{{Harmonics in equal|181}}
=== Subsets and supersets ===
181edo is the 42nd [[prime edo]].


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning Error
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
Line 23: Line 24:
| 2.3
| 2.3
| {{monzo| 287 -181 }}
| {{monzo| 287 -181 }}
| [{{val| 181 287 }}]
| {{mapping| 181 287 }}
| -0.255
| -0.255
| 0.255
| 0.255
Line 30: Line 31:
| 2.3.5
| 2.3.5
| 2109375/2097152, {{monzo| 14 -22 9 }}
| 2109375/2097152, {{monzo| 14 -22 9 }}
| [{{val| 181 287 420 }}]
| {{mapping| 181 287 420 }}
| +0.086
| +0.086
| 0.525
| 0.525
Line 37: Line 38:
| 2.3.5.7
| 2.3.5.7
| 2401/2400, 5120/5103, 390625/387072
| 2401/2400, 5120/5103, 390625/387072
| [{{val| 181 287 420 508 }}]
| {{mapping| 181 287 420 508 }}
| +0.142
| +0.142
| 0.465
| 0.465
Line 44: Line 45:
| 2.3.5.7.11
| 2.3.5.7.11
| 385/384, 1375/1372, 2200/2187, 4000/3993
| 385/384, 1375/1372, 2200/2187, 4000/3993
| [{{val| 181 287 420 508 626 }}]
| {{mapping| 181 287 420 508 626 }}
| +0.174
| +0.174
| 0.421
| 0.421
Line 51: Line 52:
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 325/324, 352/351, 385/384, 1375/1372, 1575/1573
| 325/324, 352/351, 385/384, 1375/1372, 1575/1573
| [{{val| 181 287 420 508 626 670 }}]
| {{mapping| 181 287 420 508 626 670 }}
| +0.079
| +0.079
| 0.439
| 0.439
Line 58: Line 59:
| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 325/324, 352/351, 375/374, 385/384, 595/594, 1275/1274
| 325/324, 352/351, 375/374, 385/384, 595/594, 1275/1274
| [{{val| 181 287 420 508 626 670 740 }}]
| {{mapping| 181 287 420 508 626 670 740 }}
| +0.028
| +0.028
| 0.425
| 0.425
Line 65: Line 66:
| 2.3.5.7.11.13.17.19
| 2.3.5.7.11.13.17.19
| 325/324, 352/351, 375/374, 385/384, 400/399, 595/594, 1275/1274
| 325/324, 352/351, 375/374, 385/384, 400/399, 595/594, 1275/1274
| [{{val| 181 287 420 508 626 670 740 769 }}]
| {{mapping| 181 287 420 508 626 670 740 769 }}
| +0.000
| +0.000
| 0.404
| 0.404
Line 75: Line 76:
|+Table of rank-2 temperaments by generator
|+Table of rank-2 temperaments by generator
! Periods<br>per octave
! Periods<br>per octave
! Generator<br>(reduced)
! Generator*
! Cents<br>(reduced)
! Cents*
! Associated<br>ratio
! Associated<br>Ratio*
! Temperaments
! Temperaments
|-
|-
Line 116: Line 117:
| [[Cotritone]] (181f)
| [[Cotritone]] (181f)
|}
|}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
== See also ==
[[Category:Prime EDO]]
 
== See Also ==
 
* [[181edo and stretched hemififths]]
* [[181edo and stretched hemififths]]
Retrieved from "https://en.xen.wiki/w/181edo"