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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|208}}
{{ED intro}}
==Theory==
 
204edo tempers out 15625/15552, the kleisma, and is the [[Optimal_patent_val|optimal patent val]] for the kleismic temperament [[Kleismic_family|metakleismic]], and 7, 11 and 13 limit rank three [[Tolermic_family|tolerant]] temperament. It is also the optimal patent val for the rank four [[11-limit|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.
== Theory ==
===Odd harmonics===
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.
 
=== Odd harmonics ===
{{Harmonics in equal|208}}
{{Harmonics in equal|208}}
===Subsets and supersets===
 
208 factors into 2<sup>4</sup> × 13, with subset edos {{EDOs|2, 4, 8, 16, 13, 26, 52, and 104}}.
=== Subsets and supersets ===
==Regular temperament properties==
Since 208 factors into 2<sup>4</sup> × 13, 208edo has subset edos {{EDOs| 2, 4, 8, 16, 13, 26, 52, and 104 }}.
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" |[[Subgroup]]
! rowspan="2" |[[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! colspan="2" |Tuning Error
|-
![[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
|-
|-
|2.3
! rowspan="2" | [[Subgroup]]
|{{monzo|165 -104}}
! rowspan="2" | [[Comma list]]
|{{val|208 330}}
! rowspan="2" | [[Mapping]]
| -0.5966
! rowspan="2" | Optimal<br />8ve stretch (¢)
| 0.5963
! colspan="2" | Tuning error
| 10.34
|-
|-
|2.3.5
! [[TE error|Absolute]] (¢)
|15625/15552, {{monzo|57 -33 -2}}
! [[TE simple badness|Relative]] (%)
|{{val|208 330 483}}
|-
| -0.4301
| 2.3.5
| 15625/15552, {{monzo| 57 -33 -2 }}
| {{mapping| 208 330 483 }}
| −0.4301
| 0.5409
| 0.5409
| 9.38
| 9.38
|-
|-
|2.3.5.7
| 2.3.5.7
|2401/2400, 15625/15552, 179200/177147
| 2401/2400, 15625/15552, 179200/177147
|{{val|208 330 483 584}}
| {{mapping| 208 330 483 584 }}
| -0.3586
| −0.3586
| 0.4845
| 0.4845
| 8.40
| 8.40
|-
|-
|2.3.5.7.11
| 2.3.5.7.11
|896/891, 2200/2187, 2401/2400, 3025/3024
| 896/891, 2200/2187, 2401/2400, 3025/3024
|{{val|208 330 483 584 720}}
| {{mapping| 208 330 483 584 720 }}
| -0.4330
| −0.4330
| 0.4582
| 0.4582
| 7.94
| 7.94
|-
|-
|2.3.5.7.11.13
| 2.3.5.7.11.13
|325/324, 352/351, 364/363, 676/675, 2401/2400
| 325/324, 352/351, 364/363, 676/675, 2401/2400
|{{val|208 330 483 584 720 770}}
| {{mapping| 208 330 483 584 720 770 }}
| -0.4410
| −0.4410
| 0.4187
| 0.4187
| 7.26
| 7.26
|}
|}
=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per 8ve
|-
! Generator<br>(reduced)
! Periods<br />per 8ve
! Cents<br>(reduced)
! Generator*
! Associated<br>ratio
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
|1
| 1
|47\208
| 47\208
|251.15
| 271.15
|1024/875
| 1024/875
|[[Quasiorwell]]
| [[Quasiorwell]]
|-
|-
|1
| 1
|55\208
| 55\208
|317.31
| 317.31
|6/5
| 6/5
|[[Hanson]] / [[metakleismic]]
| [[Metakleismic]]
|-
|-
|4
| 4
|55\208<br>(3\208)
| 55\208<br>(3\208)
|317.31<br>(17.31)
| 317.31<br>(17.31)
|6/5<br>(81/80)
| 6/5<br>(126/125)
|[[Quadritikleismic]]
| [[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:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Metakleismic]]
[[Category:11-limit]]
[[Category:Tolerant]]
[[Category:13-limit]]
[[Category:Pentacircle]]
[[Category:7-limit]]
Retrieved from "https://en.xen.wiki/w/208edo"