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{{Infobox ET}}
{{Infobox ET}}
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.
{{EDO intro|208}}
 
==Theory==
208 = 16 * 13, and has divisors 2, 4, 8, 16, 13, 26, 52, 104.
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.
 
===Odd harmonics===
=13-limit transversal=
{{Harmonics in equal|208}}
[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]
===Subsets and supersets===
208 factors into 2<sup>4</sup> × 13, with subset edos {{EDOs|2, 4, 8, 16, 13, 26, 52, and 104}}.
==Regular temperament properties==
{| 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
|{{monzo|165 -104}}
|{{val|208 330}}
| -0.5966
| 0.5963
| 10.34
|-
|2.3.5
|15625/15552, {{monzo|57 -33 -2}}
|{{val|208 330 483}}
| -0.4301
| 0.5409
| 9.38
|-
|2.3.5.7
|2401/2400, 15625/15552, 179200/177147
|{{val|208 330 483 584}}
| -0.3586
| 0.4845
| 8.40
|-
|2.3.5.7.11
|896/891, 2200/2187, 2401/2400, 3025/3024
|{{val|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
|{{val|208 330 483 584 720 770}}
| -0.4410
| 0.4187
| 7.26
|}
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Generator<br>(reduced)
! Cents<br>(reduced)
! Associated<br>ratio
! Temperaments
|-
|1
|47\208
|251.15
|1024/875
|[[Quasiorwell]]
|-
|1
|55\208
|317.31
|6/5
|[[Hanson]] / [[metakleismic]]
|-
|4
|55\208<br>(3\208)
|317.31<br>(17.31)
|6/5<br>(81/80)
|[[Quadritikleismic]]
|}


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->

Revision as of 19:42, 18 October 2023

← 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

Template:EDO intro

Theory

204edo tempers out 15625/15552, the kleisma, and is the optimal patent val for the kleismic temperament metakleismic, and 7, 11 and 13 limit rank three tolerant temperament. 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.

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

208 factors into 24 × 13, with 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 [165 -104⟩ ⟨208 330] -0.5966 0.5963 10.34
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
(reduced) 		Cents
(reduced) 		Associated
ratio 		Temperaments
1 47\208 251.15 1024/875 Quasiorwell
1 55\208 317.31 6/5 Hanson / metakleismic
4 55\208
(3\208) 		317.31
(17.31) 		6/5
(81/80) 		Quadritikleismic
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