208edo: Difference between revisions
Jump to navigation
Jump to search
No edit summary |
Review |
||
| Line 1: | Line 1: | ||
{{Infobox ET}} | {{Infobox ET}} | ||
{{EDO intro|208}} | {{EDO intro|208}} | ||
==Theory== | |||
== Theory == | |||
===Odd harmonics=== | 208et [[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, | === 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" | [[Subgroup]] | ||
! rowspan="2" |[[Comma list|Comma List]] | ! rowspan="2" | [[Comma list|Comma List]] | ||
! rowspan="2" |[[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" |Optimal<br>8ve Stretch (¢) | ! rowspan="2" | Optimal<br>8ve Stretch (¢) | ||
! colspan="2" |Tuning Error | ! colspan="2" | Tuning Error | ||
|- | |- | ||
![[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
![[TE simple badness|Relative]] (%) | ! [[TE simple badness|Relative]] (%) | ||
|- | |- | ||
| 2.3.5 | |||
| 15625/15552, {{monzo| 57 -33 -2 }} | |||
| {{mapping| 208 330 483 }} | |||
|2.3.5 | |||
|15625/15552, {{monzo|57 -33 -2}} | |||
|{{ | |||
| -0.4301 | | -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 | ||
|{{ | | {{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 | ||
|{{ | | {{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 | ||
|{{ | | {{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 | |+Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | ! Periods<br>per 8ve | ||
! Generator | ! Generator* | ||
! Cents | ! Cents* | ||
! Associated<br> | ! Associated<br>Ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
|1 | | 1 | ||
|47\208 | | 47\208 | ||
|251.15 | | 251.15 | ||
|1024/875 | | 1024/875 | ||
|[[Quasiorwell]] | | [[Quasiorwell]] | ||
|- | |- | ||
|1 | | 1 | ||
|55\208 | | 55\208 | ||
|317.31 | | 317.31 | ||
|6/5 | | 6/5 | ||
|[[ | | [[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>(81/80) | ||
|[[Quadritikleismic]] | | [[Quadritikleismic]] (7-limit) | ||
|} | |} | ||
Revision as of 10:19, 13 April 2024
| ← 207edo | 208edo | 209edo → |
Theory
208et 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
| 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
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 47\208 | 251.15 | 1024/875 | Quasiorwell |
| 1 | 55\208 | 317.31 | 6/5 | Metakleismic |
| 4 | 55\208 (3\208) |
317.31 (17.31) |
6/5 (81/80) |
Quadritikleismic (7-limit) |