232edo: Difference between revisions

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**Imported revision 243978219 - Original comment: **
m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"
 
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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>2011-08-02 16:08:52 UTC</tt>.<br>
: The original revision id was <tt>243978219</tt>.<br>
: The revision comment was: <tt></tt><br>
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>
<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 //232 equal division// divides the octave into 232 equal parts of 5.172 cents each. It provides the [[optimal patent val]] for [[13-limit]] [[Hemifamity temperaments#Mystery|mystery temperament]] and other temperaments tempering out 176/175, for which it gives the optimal patent val for the corresponding rank five temperament. Aside from its patent val, the 232d val is worth considering. Both temper out the wuerschmidt comma, 393216/390625, in the 5-limit. In the 7-limit, the patent val tempers out 5120/5103 and 232d 3125/3097 and 245/243, supporting [[Sensamagic clan#Bohpier|bohpier temperament]]. In the 11-limit, the patent val tempers out 441/440 and 896/891, and 232d 540/539, 1375/1372 and 4000/3993, supporting octoid. In the 13-limit, the patent val tempers out 196/195, 352/351, 364/363, 441/440 and 676/675, which leads to 13-limit mystery, for which it provides the optimal patent val. 232d also tempers out 352/351 and 676/675, which supports variants of octoid.


Considering the 205edo patent val and 13-limit mystery, we note that because it tempers out 441/440 it allows [[werckismic chords]], because it tempers out 196/195 it allows [[mynucumic chords]], because it tempers out 352/351 it allows [[minthmic chords]], and because it tempers out 364/363 it allows [[gentle chords]], and because it tempers out 847/845 it allows the [[cuthbert triad]], making it a very flexible harmonic system.</pre></div>
== Theory ==
<h4>Original HTML content:</h4>
232 = 8 × 29, and 232edo shares its [[3/2|fifth]] with [[29edo]]. The equal temperament [[support]]s and provides the [[optimal patent val]] for the [[13-limit]] [[mystery]] temperament, the rank-3 [[pele]] temperament and the rank-3 [[trimyna]] temperament and other temperaments tempering out [[196/195]], for which it gives the optimal patent val for the corresponding rank-5 temperament.
<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;232edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;em&gt;232 equal division&lt;/em&gt; divides the octave into 232 equal parts of 5.172 cents each. It provides the &lt;a class="wiki_link" href="/optimal%20patent%20val"&gt;optimal patent val&lt;/a&gt; for &lt;a class="wiki_link" href="/13-limit"&gt;13-limit&lt;/a&gt; &lt;a class="wiki_link" href="/Hemifamity%20temperaments#Mystery"&gt;mystery temperament&lt;/a&gt; and other temperaments tempering out 176/175, for which it gives the optimal patent val for the corresponding rank five temperament. Aside from its patent val, the 232d val is worth considering. Both temper out the wuerschmidt comma, 393216/390625, in the 5-limit. In the 7-limit, the patent val tempers out 5120/5103 and 232d 3125/3097 and 245/243, supporting &lt;a class="wiki_link" href="/Sensamagic%20clan#Bohpier"&gt;bohpier temperament&lt;/a&gt;. In the 11-limit, the patent val tempers out 441/440 and 896/891, and 232d 540/539, 1375/1372 and 4000/3993, supporting octoid. In the 13-limit, the patent val tempers out 196/195, 352/351, 364/363, 441/440 and 676/675, which leads to 13-limit mystery, for which it provides the optimal patent val. 232d also tempers out 352/351 and 676/675, which supports variants of octoid. &lt;br /&gt;
 
&lt;br /&gt;
Aside from its [[patent val]], the 232d val {{val| 232 368 539 '''652''' 803 859 }} is worth considering. Both temper out the [[würschmidt comma]], 393216/390625, in the 5-limit. In the 7-limit, the patent val tempers out hemifamity, [[5120/5103]] and the trimyna comma, [[50421/50000]]; and 232d [[4375/4374]] and [[16875/16807]], supporting [[octoid]]. In the 11-limit, the patent val tempers out [[441/440]] and [[896/891]], and 232d [[540/539]], 1375/1372 and [[4000/3993]]. In the 13-limit, the patent val tempers out 196/195, [[352/351]], [[364/363]], [[676/675]], and [[847/845]], which leads to 13-limit mystery, for which it provides the optimal patent val. 232d also tempers out 352/351 and 676/675, which supports a variant of octoid.  
Considering the 205edo patent val and 13-limit mystery, we note that because it tempers out 441/440 it allows &lt;a class="wiki_link" href="/werckismic%20chords"&gt;werckismic chords&lt;/a&gt;, because it tempers out 196/195 it allows &lt;a class="wiki_link" href="/mynucumic%20chords"&gt;mynucumic chords&lt;/a&gt;, because it tempers out 352/351 it allows &lt;a class="wiki_link" href="/minthmic%20chords"&gt;minthmic chords&lt;/a&gt;, and because it tempers out 364/363 it allows &lt;a class="wiki_link" href="/gentle%20chords"&gt;gentle chords&lt;/a&gt;, and because it tempers out 847/845 it allows the &lt;a class="wiki_link" href="/cuthbert%20triad"&gt;cuthbert triad&lt;/a&gt;, making it a very flexible harmonic system.&lt;/body&gt;&lt;/html&gt;</pre></div>
 
Considering the 232edo patent val, 13-limit mystery and 13-limit pele, we note that because it tempers out 441/440 it allows [[werckismic chords]], because it tempers out 196/195 it allows [[mynucumic chords]], because it tempers out 352/351 it allows [[major minthmic chords]], and because it tempers out 364/363 it allows [[minor minthmic chords]], and because it tempers out 847/845 it allows the [[cuthbert chords]], making it a very flexible harmonic system. 232edo is also the first edo that approximates [[6/5]] more accurately than [[19edo]].
 
=== Odd harmonics ===
{{Harmonics in equal|232}}
 
=== Subsets and supersets ===
Since 232 factors into 2<sup>3</sup> × 29, 232edo has subset edos {{EDOs| 2, 4, 8, 29, 58, and 116 }}.
 
== 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]] (%)
|-
| 2.3.5
| 393216/390625, {{monzo| 46 -29 0 }}
| {{mapping| 232 368 539 }}
| −0.5461
| 0.3989
| 7.71
|}
 
=== 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
| 61\232
| 315.52
| 6/5
| [[Acrokleismic]] (7-limit, 232d)
|-
| 1
| 75\232
| 387.93
| 5/4
| [[Würschmidt]] (5-limit)
|-
| 8
| 113\232<br>(3\232)
| 584.48<br>(15.52)
| 7/5<br>(100/99)
| [[Octoid]] (232d)
|-
| 29
| 96\232<br>(3\232)
| 496.55<br>(15.52)
| 4/3<br>(105/104)
| [[Mystery]] (232)
|}
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
== Scales ==
* [[Mystery58]]
 
[[Category:Mystery]]
[[Category:Pele]]
[[Category:Trimyna]]
[[Category:Mynucumic]]

Latest revision as of 13:31, 13 March 2026

← 231edo 232edo 233edo →
Prime factorization 23 × 29
Step size 5.17241 ¢ 
Fifth 136\232 (703.448 ¢) (→ 17\29)
Semitones (A1:m2) 24:16 (124.1 ¢ : 82.76 ¢)
Consistency limit 5
Distinct consistency limit 5

232 equal divisions of the octave (abbreviated 232edo or 232ed2), also called 232-tone equal temperament (232tet) or 232 equal temperament (232et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 232 equal parts of about 5.17 ¢ each. Each step represents a frequency ratio of 21/232, or the 232nd root of 2.

Theory

232 = 8 × 29, and 232edo shares its fifth with 29edo. The equal temperament supports and provides the optimal patent val for the 13-limit mystery temperament, the rank-3 pele temperament and the rank-3 trimyna temperament and other temperaments tempering out 196/195, for which it gives the optimal patent val for the corresponding rank-5 temperament.

Aside from its patent val, the 232d val ⟨232 368 539 652 803 859] is worth considering. Both temper out the würschmidt comma, 393216/390625, in the 5-limit. In the 7-limit, the patent val tempers out hemifamity, 5120/5103 and the trimyna comma, 50421/50000; and 232d 4375/4374 and 16875/16807, supporting octoid. In the 11-limit, the patent val tempers out 441/440 and 896/891, and 232d 540/539, 1375/1372 and 4000/3993. In the 13-limit, the patent val tempers out 196/195, 352/351, 364/363, 676/675, and 847/845, which leads to 13-limit mystery, for which it provides the optimal patent val. 232d also tempers out 352/351 and 676/675, which supports a variant of octoid.

Considering the 232edo patent val, 13-limit mystery and 13-limit pele, we note that because it tempers out 441/440 it allows werckismic chords, because it tempers out 196/195 it allows mynucumic chords, because it tempers out 352/351 it allows major minthmic chords, and because it tempers out 364/363 it allows minor minthmic chords, and because it tempers out 847/845 it allows the cuthbert chords, making it a very flexible harmonic system. 232edo is also the first edo that approximates 6/5 more accurately than 19edo.

Odd harmonics

Approximation of odd harmonics in 232edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +1.49 +1.62 -1.58 -2.19 +2.13 +2.58 -2.06 -1.51 +2.49 -0.09 -2.41
Relative (%) +28.9 +31.3 -30.6 -42.3 +41.2 +49.8 -39.9 -29.1 +48.1 -1.8 -46.6
Steps
(reduced) 		368
(136) 		539
(75) 		651
(187) 		735
(39) 		803
(107) 		859
(163) 		906
(210) 		948
(20) 		986
(58) 		1019
(91) 		1049
(121)

Subsets and supersets

Since 232 factors into 23 × 29, 232edo has subset edos 2, 4, 8, 29, 58, and 116.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3.5 393216/390625, [46 -29 0⟩ [⟨232 368 539]] −0.5461 0.3989 7.71

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 61\232 315.52 6/5 Acrokleismic (7-limit, 232d)
1 75\232 387.93 5/4 Würschmidt (5-limit)
8 113\232
(3\232) 		584.48
(15.52) 		7/5
(100/99) 		Octoid (232d)
29 96\232
(3\232) 		496.55
(15.52) 		4/3
(105/104) 		Mystery (232)

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

Scales

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