241edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|241}}
{{ED intro}}


== Theory ==
== Theory ==
241edo is [[consistency|distinctly consistent]] in the [[15-odd-limit]]. It has a sharp tendency, with [[prime harmonic]]s 3 through 13 all tuned sharp. The equal temperament [[tempering out|tempers out]] [[78732/78125]] in the 5-limit, [[19683/19600]] and [[3136/3125]] in the 7-limit, [[540/539]], 43923/43904, [[65536/65219]], and 151263/151250 in the 11-limit, and [[351/350]], [[676/675]], [[729/728]], [[1001/1000]] and [[2080/2079]] in the 13-limit. It provides the [[optimal patent val]] for [[subpental]].
241edo is [[consistency|distinctly consistent]] in the [[15-odd-limit]]. It has a sharp tendency, with [[prime harmonic]]s 3 through 13 all tuned sharp.  
 
As an equal temperament, it [[tempering out|tempers out]] [[78732/78125]] in the [[5-limit]], [[19683/19600]] and [[3136/3125]] in the [[7-limit]], [[540/539]], [[43923/43904]], [[65536/65219]], and [[151263/151250]] in the [[11-limit]], and [[351/350]], [[676/675]], [[729/728]], [[1001/1000]] and [[2080/2079]] in the [[13-limit]]. It provides the [[optimal patent val]] for [[subpental]].


=== Prime harmonics ===
=== Prime harmonics ===
Line 9: Line 11:


=== Subsets and supersets ===
=== Subsets and supersets ===
241edo is the 53rd [[prime edo]].
241edo is the 53rd [[prime edo]]. As such, it does not contain any nontrivial subset edos. [[1205edo]], which slices its step in five, is a notable higher-limit system.


== Regular temperament properties ==
== Regular temperament properties ==
{{comma basis begin}}
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
|-
| 2.3
| 2.3
| {{monzo| 382 -241 }}
| {{Monzo| 382 -241 }}
| {{mapping| 241 382 }}
| {{Mapping| 241 382 }}
| -0.038
| −0.038
| 0.038
| 0.038
| 0.76
| 0.76
Line 23: Line 34:
| 2.3.5
| 2.3.5
| 78732/78125, {{monzo| 56 -28 -5 }}
| 78732/78125, {{monzo| 56 -28 -5 }}
| {{mapping| 241 382 560 }}
| {{Mapping| 241 382 560 }}
| -0.322
| −0.322
| 0.403
| 0.403
| 8.10
| 8.10
Line 30: Line 41:
| 2.3.5.7
| 2.3.5.7
| 3136/3125, 19683/19600, 829940/823543
| 3136/3125, 19683/19600, 829940/823543
| {{mapping| 241 382 560 677 }}
| {{Mapping| 241 382 560 677 }}
| -0.431
| −0.431
| 0.397
| 0.397
| 7.97
| 7.97
Line 37: Line 48:
| 2.3.5.7.11
| 2.3.5.7.11
| 540/539, 3136/3125, 8019/8000, 15488/15435
| 540/539, 3136/3125, 8019/8000, 15488/15435
| {{mapping| 241 382 560 677 834 }}
| {{Mapping| 241 382 560 677 834 }}
| -0.425
| −0.425
| 0.355
| 0.355
| 7.14
| 7.14
Line 44: Line 55:
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 351/350, 540/539, 676/675, 3136/3125, 10648/10647
| 351/350, 540/539, 676/675, 3136/3125, 10648/10647
| {{mapping| 241 382 560 677 834 892 }}
| {{Mapping| 241 382 560 677 834 892 }}
| -0.397
| −0.397
| 0.330
| 0.330
| 6.63
| 6.63
{{comma basis end}}
|}


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{{rank-2 begin}}
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>ratio*
! Temperaments
|-
|-
| 1
| 1
Line 62: Line 80:
| 50\241
| 50\241
| 248.96
| 248.96
| {{monzo| -26 18 -1 }}
| {{Monzo| -26 18 -1 }}
| [[Monzismic]]
| [[Monzismic]] (5-limit)
|-
|-
| 1
| 1
Line 82: Line 100:
| 4/3
| 4/3
| [[Gary]]
| [[Gary]]
{{rank-2 end}}
|}
{{orf}}
<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]]


[[Category:Subpental]]
[[Category:Subpental]]

Latest revision as of 03:40, 20 September 2026

← 240edo 241edo 242edo →
Prime factorization 241 (prime)
Step size 4.97925 ¢ 
Fifth 141\241 (702.075 ¢)
Semitones (A1:m2) 23:18 (114.5 ¢ : 89.63 ¢)
Consistency limit 15
Distinct consistency limit 15

241 equal divisions of the octave (abbreviated 241edo or 241ed2), also called 241-tone equal temperament (241tet) or 241 equal temperament (241et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 241 equal parts of about 4.98 ¢ each. Each step represents a frequency ratio of 21/241, or the 241st root of 2.

Theory

241edo is distinctly consistent in the 15-odd-limit. It has a sharp tendency, with prime harmonics 3 through 13 all tuned sharp.

As an equal temperament, it tempers out 78732/78125 in the 5-limit, 19683/19600 and 3136/3125 in the 7-limit, 540/539, 43923/43904, 65536/65219, and 151263/151250 in the 11-limit, and 351/350, 676/675, 729/728, 1001/1000 and 2080/2079 in the 13-limit. It provides the optimal patent val for subpental.

Prime harmonics

Approximation of prime harmonics in 241edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.12 +2.07 +2.13 +1.38 +0.97 -0.39 +1.24 -0.89 +1.13 +0.19
Relative (%) +0.0 +2.4 +41.5 +42.7 +27.7 +19.4 -7.9 +24.9 -17.8 +22.7 +3.9
Steps
(reduced) 		241
(0) 		382
(141) 		560
(78) 		677
(195) 		834
(111) 		892
(169) 		985
(21) 		1024
(60) 		1090
(126) 		1171
(207) 		1194
(230)

Subsets and supersets

241edo is the 53rd prime edo. As such, it does not contain any nontrivial subset edos. 1205edo, which slices its step in five, is a notable higher-limit system.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [382 -241⟩ [⟨241 382]] −0.038 0.038 0.76
2.3.5 78732/78125, [56 -28 -5⟩ [⟨241 382 560]] −0.322 0.403 8.10
2.3.5.7 3136/3125, 19683/19600, 829940/823543 [⟨241 382 560 677]] −0.431 0.397 7.97
2.3.5.7.11 540/539, 3136/3125, 8019/8000, 15488/15435 [⟨241 382 560 677 834]] −0.425 0.355 7.14
2.3.5.7.11.13 351/350, 540/539, 676/675, 3136/3125, 10648/10647 [⟨241 382 560 677 834 892]] −0.397 0.330 6.63

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 20\241 99.59 200/189 Quintagar / quinsandric
1 50\241 248.96 [-26 18 -1⟩ Monzismic (5-limit)
1 76\241 378.42 56/45 Subpental
1 89\241 443.15 162/125 Sensipent
1 100\241 497.93 4/3 Gary

* In minimal-generator form

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