241edo: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Infobox ET}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
{{ED intro}}
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2012-05-21 22:39:00 UTC</tt>.<br>
 
: The original revision id was <tt>338146166</tt>.<br>
== Theory ==
: The revision comment was: <tt></tt><br>
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]].
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
 
<h4>Original Wikitext content:</h4>
=== Prime harmonics ===
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">The 241 equal temperament divides the octave in 241 parts of 4.979 cents each. It 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 to [[Hemimean clan#Subpental|subpental temperament]]. 241 is a prime number.</pre></div>
{{Harmonics in equal|241}}
<h4>Original HTML content:</h4>
 
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;241edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The 241 equal temperament divides the octave in 241 parts of 4.979 cents each. It 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 to &lt;a class="wiki_link" href="/Hemimean%20clan#Subpental"&gt;subpental temperament&lt;/a&gt;. 241 is a prime number.&lt;/body&gt;&lt;/html&gt;</pre></div>
=== Subsets and supersets ===
241edo is the 53rd [[prime edo]].
 
== Regular temperament properties ==
{| 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
| {{monzo| 382 -241 }}
| {{mapping| 241 382 }}
| −0.038
| 0.038
| 0.76
|-
| 2.3.5
| 78732/78125, {{monzo| 56 -28 -5 }}
| {{mapping| 241 382 560 }}
| −0.322
| 0.403
| 8.10
|-
| 2.3.5.7
| 3136/3125, 19683/19600, 829940/823543
| {{mapping| 241 382 560 677 }}
| −0.431
| 0.397
| 7.97
|-
| 2.3.5.7.11
| 540/539, 3136/3125, 8019/8000, 15488/15435
| {{mapping| 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
| {{mapping| 241 382 560 677 834 892 }}
| −0.397
| 0.330
| 6.63
|}
 
=== Rank-2 temperaments ===
{| 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
| 20\241
| 99.59
| 200/189
| [[Quintagar]] / [[quinsandric]]
|-
| 1
| 50\241
| 248.96
| {{monzo| -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]]
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 
[[Category:Subpental]]

Latest revision as of 14:23, 20 February 2025

← 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.

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 distinct

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