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{{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]]

Revision as of 09:14, 25 April 2024

← 180edo 181edo 182edo →
Prime factorization 181 (prime)
Step size 6.62983 ¢ 
Fifth 106\181 (702.762 ¢)
Semitones (A1:m2) 18:13 (119.3 ¢ : 86.19 ¢)
Consistency limit 7
Distinct consistency limit 7

Template:EDO intro

Theory

181et tempers out 2109375/2097152 (semicomma) and [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.

Prime harmonics

Approximation of prime harmonics in 181edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.81 -1.78 -0.87 -1.04 +1.46 +1.12 +0.83 +1.56 -1.95 +1.93
Relative (%) +0.0 +12.2 -26.9 -13.1 -15.7 +22.0 +16.9 +12.5 +23.5 -29.5 +29.0
Steps
(reduced) 		181
(0) 		287
(106) 		420
(58) 		508
(146) 		626
(83) 		670
(127) 		740
(16) 		769
(45) 		819
(95) 		879
(155) 		897
(173)

Subsets and supersets

181edo is the 42nd prime edo.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3 [287 -181⟩ [⟨181 287]] -0.255 0.255 3.84
2.3.5 2109375/2097152, [14 -22 9⟩ [⟨181 287 420]] +0.086 0.525 7.92
2.3.5.7 2401/2400, 5120/5103, 390625/387072 [⟨181 287 420 508]] +0.142 0.465 7.01
2.3.5.7.11 385/384, 1375/1372, 2200/2187, 4000/3993 [⟨181 287 420 508 626]] +0.174 0.421 6.35
2.3.5.7.11.13 325/324, 352/351, 385/384, 1375/1372, 1575/1573 [⟨181 287 420 508 626 670]] +0.079 0.439 6.62
2.3.5.7.11.13.17 325/324, 352/351, 375/374, 385/384, 595/594, 1275/1274 [⟨181 287 420 508 626 670 740]] +0.028 0.425 6.40
2.3.5.7.11.13.17.19 325/324, 352/351, 375/374, 385/384, 400/399, 595/594, 1275/1274 [⟨181 287 420 508 626 670 740 769]] +0.000 0.404 6.09

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per octave 		Generator* Cents* Associated
Ratio* 		Temperaments
1 18\181 119.34 15/14 Septidiasemi
1 35\181 232.04 8/7 Quadrawell
1 39\181 258.56 [-32 13 5⟩ Lafa
1 41\181 271.82 75/64 Orson
1 53\181 351.38 49/40 Hemififths (7-limit)
1 88\181 583.43 7/5 Cotritone (181f)

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct

See also

Retrieved from "https://en.xen.wiki/index.php?title=181edo&oldid=141865"