208edo: Difference between revisions

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**Imported revision 378018578 - 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>2012-10-31 11:50:26 UTC</tt>.<br>
: The original revision id was <tt>378018578</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 //208 equal division// divides the [[octave]] into 208 equal parts of size 5.769 [[cent]]s each. It tempers out 15625/15552, the kleisma, and is the [[optimal patent val]] for the kleismic temperament [[Kleismic family|metakleismic]]. It is also the optimal patent val for the rank four [[11-limit]] temperament tempering out 896/891, the pentacircle temperament. Other commas it tempers out include 2200/2187 in the 11-limit and 325/324, 352/351, 364/363 and 625/624 in the 13-limit.


208 = 16 * 13, and has divisors 2, 4, 8, 16, 13, 26, 52, 104.
== Theory ==
208edo is closely related to [[104edo]], but the mappings for [[harmonic]] [[5/1|5]] differ. As an equal temperament, it [[tempering out|tempers out]] [[15625/15552]], the kleisma, and is the [[optimal patent val]] for the kleismic temperament [[metakleismic]], and 7-, 11- and 13-limit rank-3 [[tolerant]] temperament. It is also the optimal patent val for the rank-4 [[11-limit]] temperament tempering out [[896/891]], the [[pentacircle]] temperament. Other commas it tempers out include [[2200/2187]] in the 11-limit and [[325/324]], [[352/351]], [[364/363]] and [[625/624]] in the 13-limit.


=13-limit transversal=
=== Odd harmonics ===
[196/195, 100/99, 91/90, 64/63, 55/54, 49/48, 40/39, 77/75, 36/35, 28/27, 80/77, 25/24, 245/234, 22/21, 21/20, 81/77, 35/33, 52/49, 16/15, 77/72, 15/14, 14/13, 250/231, 13/12, 49/45, 12/11, 35/32, 100/91, 11/10, 54/49, 10/9, 49/44, 39/35, 28/25, 55/49, 9/8, 147/130, 25/22, 91/80, 8/7, 55/48, 147/128, 15/13, 196/169, 64/55, 7/6, 90/77, 75/64, 147/125, 13/11, 77/65, 25/21, 105/88, 117/98, 6/5, 77/64, 40/33, 63/52, 128/105, 11/9, 49/40, 16/13, 154/125, 26/21, 56/45, 96/77, 5/4, 49/39, 44/35, 63/50, 80/63, 14/11, 125/98, 32/25, 77/60, 9/7, 35/27, 100/77, 13/10, 64/49, 55/42, 21/16, 120/91, 33/25, 65/49, 4/3, 147/110, 75/56, 35/26, 66/49, 27/20, 49/36, 15/11, 175/128, 48/35, 11/8, 135/98, 18/13, 245/176, 39/28, 7/5, 108/77, 45/32, 147/104, 64/45, 77/54, 10/7, 56/39, 351/245, 13/9, 196/135, 16/11, 35/24, 143/98, 22/15, 72/49, 40/27, 49/33, 52/35, 112/75, 220/147, 3/2, 98/65, 50/33, 91/60, 32/21, 55/36, 49/32, 20/13, 77/50, 54/35, 14/9, 120/77, 25/16, 196/125, 11/7, 63/40, 100/63, 35/22, 78/49, 8/5, 77/48, 45/28, 21/13, 125/77, 13/8, 49/30, 18/11, 105/64, 104/63, 33/20, 81/49, 5/3, 147/88, 117/70, 42/25, 130/77, 22/13, 245/144, 75/44, 77/45, 12/7, 55/32, 169/98, 26/15, 256/147, 96/55, 7/4, 135/77, 44/25, 260/147, 16/9, 98/55, 25/14, 70/39, 88/49, 9/5, 49/27, 20/11, 91/50, 64/35, 11/6, 90/49, 24/13, 231/125, 13/7, 28/15, 144/77, 15/8, 49/26, 66/35, 91/48, 40/21, 21/11, 245/128, 25/13, 77/40, 27/14, 35/18, 150/77, 39/20, 49/25, 55/28, 63/32, 125/63, 99/50, 195/98, 2]
{{Harmonics in equal|208}}


</pre></div>
=== Subsets and supersets ===
<h4>Original HTML content:</h4>
Since 208 factors into 2<sup>4</sup> × 13, 208edo has subset edos {{EDOs| 2, 4, 8, 16, 13, 26, 52, and 104 }}.
<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;208edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;em&gt;208 equal division&lt;/em&gt; divides the &lt;a class="wiki_link" href="/octave"&gt;octave&lt;/a&gt; into 208 equal parts of size 5.769 &lt;a class="wiki_link" href="/cent"&gt;cent&lt;/a&gt;s each. It tempers out 15625/15552, the kleisma, and is the &lt;a class="wiki_link" href="/optimal%20patent%20val"&gt;optimal patent val&lt;/a&gt; for the kleismic temperament &lt;a class="wiki_link" href="/Kleismic%20family"&gt;metakleismic&lt;/a&gt;. It is also the optimal patent val for the rank four &lt;a class="wiki_link" href="/11-limit"&gt;11-limit&lt;/a&gt; temperament tempering out 896/891, the pentacircle temperament. Other commas it tempers out include 2200/2187 in the 11-limit and 325/324, 352/351, 364/363 and 625/624 in the 13-limit.&lt;br /&gt;
 
&lt;br /&gt;
== Regular temperament properties ==
208 = 16 * 13, and has divisors 2, 4, 8, 16, 13, 26, 52, 104.&lt;br /&gt;
{| class="wikitable center-4 center-5 center-6"
&lt;br /&gt;
|-
&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="x13-limit transversal"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;13-limit transversal&lt;/h1&gt;
! rowspan="2" | [[Subgroup]]
[196/195, 100/99, 91/90, 64/63, 55/54, 49/48, 40/39, 77/75, 36/35, 28/27, 80/77, 25/24, 245/234, 22/21, 21/20, 81/77, 35/33, 52/49, 16/15, 77/72, 15/14, 14/13, 250/231, 13/12, 49/45, 12/11, 35/32, 100/91, 11/10, 54/49, 10/9, 49/44, 39/35, 28/25, 55/49, 9/8, 147/130, 25/22, 91/80, 8/7, 55/48, 147/128, 15/13, 196/169, 64/55, 7/6, 90/77, 75/64, 147/125, 13/11, 77/65, 25/21, 105/88, 117/98, 6/5, 77/64, 40/33, 63/52, 128/105, 11/9, 49/40, 16/13, 154/125, 26/21, 56/45, 96/77, 5/4, 49/39, 44/35, 63/50, 80/63, 14/11, 125/98, 32/25, 77/60, 9/7, 35/27, 100/77, 13/10, 64/49, 55/42, 21/16, 120/91, 33/25, 65/49, 4/3, 147/110, 75/56, 35/26, 66/49, 27/20, 49/36, 15/11, 175/128, 48/35, 11/8, 135/98, 18/13, 245/176, 39/28, 7/5, 108/77, 45/32, 147/104, 64/45, 77/54, 10/7, 56/39, 351/245, 13/9, 196/135, 16/11, 35/24, 143/98, 22/15, 72/49, 40/27, 49/33, 52/35, 112/75, 220/147, 3/2, 98/65, 50/33, 91/60, 32/21, 55/36, 49/32, 20/13, 77/50, 54/35, 14/9, 120/77, 25/16, 196/125, 11/7, 63/40, 100/63, 35/22, 78/49, 8/5, 77/48, 45/28, 21/13, 125/77, 13/8, 49/30, 18/11, 105/64, 104/63, 33/20, 81/49, 5/3, 147/88, 117/70, 42/25, 130/77, 22/13, 245/144, 75/44, 77/45, 12/7, 55/32, 169/98, 26/15, 256/147, 96/55, 7/4, 135/77, 44/25, 260/147, 16/9, 98/55, 25/14, 70/39, 88/49, 9/5, 49/27, 20/11, 91/50, 64/35, 11/6, 90/49, 24/13, 231/125, 13/7, 28/15, 144/77, 15/8, 49/26, 66/35, 91/48, 40/21, 21/11, 245/128, 25/13, 77/40, 27/14, 35/18, 150/77, 39/20, 49/25, 55/28, 63/32, 125/63, 99/50, 195/98, 2]&lt;/body&gt;&lt;/html&gt;</pre></div>
! 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
| 15625/15552, {{monzo| 57 -33 -2 }}
| {{mapping| 208 330 483 }}
| −0.4301
| 0.5409
| 9.38
|-
| 2.3.5.7
| 2401/2400, 15625/15552, 179200/177147
| {{mapping| 208 330 483 584 }}
| −0.3586
| 0.4845
| 8.40
|-
| 2.3.5.7.11
| 896/891, 2200/2187, 2401/2400, 3025/3024
| {{mapping| 208 330 483 584 720 }}
| −0.4330
| 0.4582
| 7.94
|-
| 2.3.5.7.11.13
| 325/324, 352/351, 364/363, 676/675, 2401/2400
| {{mapping| 208 330 483 584 720 770 }}
| −0.4410
| 0.4187
| 7.26
|}
 
=== 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
| 47\208
| 271.15
| 1024/875
| [[Quasiorwell]]
|-
| 1
| 55\208
| 317.31
| 6/5
| [[Metakleismic]]
|-
| 4
| 55\208<br>(3\208)
| 317.31<br>(17.31)
| 6/5<br>(126/125)
| [[Quadritikleismic]] (7-limit)
|}
<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
 
[[Category:Metakleismic]]
[[Category:Tolerant]]
[[Category:Pentacircle]]

Latest revision as of 13:32, 13 March 2026

← 207edo 208edo 209edo →
Prime factorization 24 × 13
Step size 5.76923 ¢ 
Fifth 122\208 (703.846 ¢) (→ 61\104)
Semitones (A1:m2) 22:14 (126.9 ¢ : 80.77 ¢)
Consistency limit 7
Distinct consistency limit 7

208 equal divisions of the octave (abbreviated 208edo or 208ed2), also called 208-tone equal temperament (208tet) or 208 equal temperament (208et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 208 equal parts of about 5.77 ¢ each. Each step represents a frequency ratio of 21/208, or the 208th root of 2.

Theory

208edo is closely related to 104edo, but the mappings for harmonic 5 differ. As an equal temperament, it tempers out 15625/15552, the kleisma, and is the optimal patent val for the kleismic temperament metakleismic, and 7-, 11- and 13-limit rank-3 tolerant temperament. It is also the optimal patent val for the rank-4 11-limit temperament tempering out 896/891, the pentacircle temperament. Other commas it tempers out include 2200/2187 in the 11-limit and 325/324, 352/351, 364/363 and 625/624 in the 13-limit.

Odd harmonics

Approximation of odd harmonics in 208edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +1.89 +0.22 +0.40 -1.99 +2.53 +1.78 +2.12 -1.11 +2.49 +2.30 +0.57
Relative (%) +32.8 +3.9 +7.0 -34.4 +43.8 +30.9 +36.7 -19.2 +43.1 +39.8 +9.9
Steps
(reduced) 		330
(122) 		483
(67) 		584
(168) 		659
(35) 		720
(96) 		770
(146) 		813
(189) 		850
(18) 		884
(52) 		914
(82) 		941
(109)

Subsets and supersets

Since 208 factors into 24 × 13, 208edo has subset edos 2, 4, 8, 16, 13, 26, 52, and 104.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3.5 15625/15552, [57 -33 -2⟩ [⟨208 330 483]] −0.4301 0.5409 9.38
2.3.5.7 2401/2400, 15625/15552, 179200/177147 [⟨208 330 483 584]] −0.3586 0.4845 8.40
2.3.5.7.11 896/891, 2200/2187, 2401/2400, 3025/3024 [⟨208 330 483 584 720]] −0.4330 0.4582 7.94
2.3.5.7.11.13 325/324, 352/351, 364/363, 676/675, 2401/2400 [⟨208 330 483 584 720 770]] −0.4410 0.4187 7.26

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 47\208 271.15 1024/875 Quasiorwell
1 55\208 317.31 6/5 Metakleismic
4 55\208
(3\208) 		317.31
(17.31) 		6/5
(126/125) 		Quadritikleismic (7-limit)

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

Retrieved from "https://en.xen.wiki/index.php?title=208edo&oldid=225861"