299edo: Difference between revisions

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-The ''299 equal division'' divides the octave into 299 parts of 4.013 cents each. In the 5-limit it 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.
+{{Infobox ET}}
+{{ED intro}}
+== Theory ==
+et [[tempering out|tempers out]] the kleisma, [[15625/15552]], in the [[5-limit]]; [[10976/10935]] in the [[7-limit]]; [[385/384]] in the [[11-limit]]; and [[325/324]], [[625/624]] and [[676/675]] in the [[13-limit]]. 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 ===
+{{Harmonics in equal|299}}
+=== Subsets and supersets ===
+Since 299 factors into primes as {{nowrap| 13 × 23 }}, 299edo contains [[13edo]] and [[23edo]] as subset edos.
+== 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
+| {{Monzo| 474 -299 }}
+| {{Mapping| 299 474 }}
+| −0.1218
+| 0.1218
+| 3.04
+|-
+| 2.3.5
+| 15625/15552, {{monzo| 80 -49 -1 }}
+| {{Mapping| 299 474 694 }}
+| +0.0665
+| 0.2844
+| 7.09
+|-
+| 2.3.5.7
+| 10976/10935, 15625/15552, 823543/819200
+| {{Mapping| 299 474 694 839 }}
+| +0.1925
+| 0.3291
+| 8.20
+|-
+| 2.3.5.7.11
+| 385/384, 6250/6237, 10976/10935, 12005/11979
+| {{Mapping| 299 474 694 839 1034 }}
+| +0.2399
+| 0.3092
+| 7.70
+|-
+| 2.3.5.7.11.13
+| 325/324, 385/384, 625/624, 10648/10647, 10976/10935
+| {{Mapping| 299 474 694 839 1034 1106 }}
+| +0.2779
+| 0.2948
+| 7.34
+|-
+| 2.3.5.7.11.13.17
+| 325/324, 385/384, 595/594, 625/624, 2058/2057, 8624/8619
+| {{Mapping| 299 474 694 839 1034 1106 1222 }}
+| +0.2595
+| 0.2767
+| 6.89
+|-
+| 2.3.5.7.11.13.17.19
+| 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
+| 0.2627
+| 6.54
+|}
+=== 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
+| 79\299
+| 317.06
+| 6/5
+| [[Hanson]]
+|-
+| 1
+| 124\299
+| 497.66
+| 4/3
+| [[Cotoneum]] (7-limit)
+|-
+| 1
+| 124\299
+| 505.69
+| 75/56
+| [[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:Keenanismic]]