299edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
In the 5-limit, 299et [[tempering out|tempers out]] the [[kleisma]], 15625/15552, in the 7-limit [[10976/10935]], in the 11-limit [[385/384]]; and in the 13-limit [[325/324]], [[625/624]] and [[676/675]]. It provides the [[optimal patent val]] for the 13-limit rank-3 [[enlil]] temperament, and the rank-4 temperament tempering out 325/324 and 385/384. | In the 5-limit, 299et [[tempering out|tempers out]] the [[kleisma]], 15625/15552, in the [[7-limit]] [[10976/10935]], in the [[11-limit]] [[385/384]]; and in the [[13-limit]] [[325/324]], [[625/624]] and [[676/675]]. It provides the [[optimal patent val]] for the 13-limit rank-3 [[enlil]] temperament, and the rank-4 temperament tempering out 325/324 and 385/384. | ||
=== Prime harmonics === | === Prime harmonics === | ||
| Line 9: | Line 9: | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
Since 299 factors into | Since 299 factors into 13 × 23, 299edo contains [[13edo]] and [[23edo]] as subsets. | ||
== Regular temperament properties == | == 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 | | 2.3 | ||
| {{ | | {{Monzo| 474 -299 }} | ||
| {{ | | {{Mapping| 299 474 }} | ||
| | | −0.1218 | ||
| 0.1218 | | 0.1218 | ||
| 3.04 | | 3.04 | ||
| Line 23: | Line 32: | ||
| 2.3.5 | | 2.3.5 | ||
| 15625/15552, {{monzo| 80 -49 -1 }} | | 15625/15552, {{monzo| 80 -49 -1 }} | ||
| {{ | | {{Mapping| 299 474 694 }} | ||
| +0.0665 | | +0.0665 | ||
| 0.2844 | | 0.2844 | ||
| Line 30: | Line 39: | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 10976/10935, 15625/15552, 823543/819200 | | 10976/10935, 15625/15552, 823543/819200 | ||
| {{ | | {{Mapping| 299 474 694 839 }} | ||
| +0.1925 | | +0.1925 | ||
| 0.3291 | | 0.3291 | ||
| Line 37: | Line 46: | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 385/384, 6250/6237, 10976/10935, 12005/11979 | | 385/384, 6250/6237, 10976/10935, 12005/11979 | ||
| {{ | | {{Mapping| 299 474 694 839 1034 }} | ||
| +0.2399 | | +0.2399 | ||
| 0.3092 | | 0.3092 | ||
| Line 44: | Line 53: | ||
| 2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
| 325/324, 385/384, 625/624, 10648/10647, 10976/10935 | | 325/324, 385/384, 625/624, 10648/10647, 10976/10935 | ||
| {{ | | {{Mapping| 299 474 694 839 1034 1106 }} | ||
| +0.2779 | | +0.2779 | ||
| 0.2948 | | 0.2948 | ||
| Line 51: | Line 60: | ||
| 2.3.5.7.11.13.17 | | 2.3.5.7.11.13.17 | ||
| 325/324, 385/384, 595/594, 625/624, 2058/2057, 8624/8619 | | 325/324, 385/384, 595/594, 625/624, 2058/2057, 8624/8619 | ||
| {{ | | {{Mapping| 299 474 694 839 1034 1106 1222 }} | ||
| +0.2595 | | +0.2595 | ||
| 0.2767 | | 0.2767 | ||
| Line 58: | Line 67: | ||
| 2.3.5.7.11.13.17.19 | | 2.3.5.7.11.13.17.19 | ||
| 325/324, 343/342, 385/384, 595/594, 625/624, 1216/1215, 1445/1444 | | 325/324, 343/342, 385/384, 595/594, 625/624, 1216/1215, 1445/1444 | ||
| {{ | | {{Mapping| 299 474 694 839 1034 1106 1222 1270 }} | ||
| +0.2424 | | +0.2424 | ||
| 0.2627 | | 0.2627 | ||
| 6.54 | | 6.54 | ||
|} | |||
=== Rank-2 temperaments === | === 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 | |||
| 25\299 | |||
| 100.33 | |||
| 1323/1250 | |||
| [[Heptacot]] (7-limit) | |||
|- | |- | ||
| 1 | | 1 | ||
| Line 84: | Line 106: | ||
| 75/56 | | 75/56 | ||
| [[Marfifths]] | | [[Marfifths]] | ||
|} | |||
<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:Enlil]] | [[Category:Enlil]] | ||
[[Category:Keenanismic]] | [[Category:Keenanismic]] | ||