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The ''208 equal division'' divides the [[Octave|octave]] into 208 equal parts of size 5.769 [[cent|cent]]s each. It 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.
{{Infobox ET}}
{{ED intro}}


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}}
[[Category:11-limit]]
 
[[Category:13-limit]]
=== Subsets and supersets ===
[[Category:7-limit]]
Since 208 factors into 2<sup>4</sup> × 13, 208edo has subset edos {{EDOs| 2, 4, 8, 16, 13, 26, 52, and 104 }}.
[[Category:edo]]
 
[[Category:theory]]
== 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
| 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

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