241edo: Difference between revisions

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== Theory ==
== Theory ==
241et tempers out [[78732/78125]] in the 5-limit, [[19683/19600]] and [[3136/3125]] in the 7-limit, [[65536/65219]], [[540/539]], 43923/43904, 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. 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 the 53rd [[prime edo]].


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|241}}
{{Harmonics in equal|241}}
=== Subsets and supersets ===
241edo is the 53rd [[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 8ve <br>stretch (¢)
! rowspan="2" | Optimal 8ve <br>Stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning Error
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
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| 2.3
| 2.3
| {{monzo| 382 -241 }}
| {{monzo| 382 -241 }}
| [{{val| 241 382 }}]
| {{mapping| 241 382 }}
| -0.038
| -0.038
| 0.038
| 0.038
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| 2.3.5
| 2.3.5
| 78732/78125, {{monzo| 56 -28 -5 }}
| 78732/78125, {{monzo| 56 -28 -5 }}
| [{{val| 241 382 560 }}]
| {{mapping| 241 382 560 }}
| -0.322
| -0.322
| 0.403
| 0.403
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| 2.3.5.7
| 2.3.5.7
| 3136/3125, 19683/19600, 829940/823543
| 3136/3125, 19683/19600, 829940/823543
| [{{val| 241 382 560 677 }}]
| {{mapping| 241 382 560 677 }}
| -0.431
| -0.431
| 0.397
| 0.397
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| 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
| [{{val| 241 382 560 677 834 }}]
| {{mapping| 241 382 560 677 834 }}
| -0.425
| -0.425
| 0.355
| 0.355
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| 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
| [{{val| 241 382 560 677 834 892 }}]
| {{mapping| 241 382 560 677 834 892 }}
| -0.397
| -0.397
| 0.330
| 0.330
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{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+Table of rank-2 temperaments by generator
! Periods<br>per octave
! Periods<br>per 8ve
! Generator<br>(reduced)
! Generator*
! Cents<br>(reduced)
! Cents*
! Associated<br>ratio
! Associated<br>Ratio*
! Temperaments
! Temperaments
|-
|-
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| [[Gary]]
| [[Gary]]
|}
|}
<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 -->
[[Category:Prime EDO]]
[[Category:Subpental]]
[[Category:Subpental]]

Revision as of 12:28, 25 March 2024

← 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

Template:EDO intro

Theory

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

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
1 76\241 378.42 56/45 Subpental
1 89\241 443.15 162/125 Sensipent
1 100\241 497.93 4/3 Gary

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

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