214edo: Difference between revisions

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-'''214edo''' is the equal division of the [[octave]] into 214 parts of 5.6075 [[cent]]s each. It is (uniquely) consistent through the [[7-odd-limit]]. The patent val for 214edo is {{val| 214 339 497 601 740 792 }}, which tempers out the following commas: 78732/78125 ([[sensipent comma]]) and {{monzo| -51 19 9 }} (untriton comma) in the 5-limit; [[6144/6125]] (porwell), 16875/16807 (mirkwai), 321489/320000 (varunisma), and {{monzo| 22 -1 -10 1 }} (quasiorwellisma) in the 7-limit; [[540/539]] and 1375/1372 in the 11-limit; 1188/1183, [[351/350]] and [[847/845]] in the 13-limit. It can be viewed as a 2.13/5 subgroup temperament, as its approximations for lower prime limits are very poor but this makes 214edo an exceptionally xenharmonic tuning.
+{{Infobox ET}}
+{{ED intro}}
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
+== Theory ==
+edo is (uniquely) consistent through the [[7-odd-limit]]. The patent val for 214edo is {{val| 214 339 497 601 740 792 }}, which [[tempering out|tempers out]] the following commas: 78732/78125 ([[sensipent comma]]) and {{monzo| -51 19 9 }} (untriton comma) in the 5-limit; 6144/6125 ([[porwell comma]]), 16875/16807 ([[mirkwai comma]]), 321489/320000 (varunisma), and {{monzo| 22 -1 -10 1 }} (quasiorwellisma) in the 7-limit; [[540/539]], 1375/1372, [[5632/5625]], in the 11-limit; [[351/350]], [[847/845]], [[1001/1000]], [[1188/1183]], [[1573/1568]], and [[4096/4095]] in the 13-limit. It can be viewed as a 2.3.5.13.19.23 [[subgroup]] temperament, as its approximations for lower prime limits are very poor but this makes 214edo an exceptionally xenharmonic tuning.
+=== Prime harmonics ===
+{{Harmonics in equal|214}}
+=== Subsets and supersets ===
+Since 214 factors into {{factorisation|214}}, 214edo contains [[2edo]] and [[107edo]] as its subsets.
+== 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| -339 214 }}
+| {{mapping| 214 339 }}
+| +0.3219
+| 0.3220
+| 5.74
+|-
+| 2.3.5
+| 78732/78125, {{monzo| -49 28 2 }}
+| {{mapping| 214 339 497 }}
+| +0.1281
+| 0.3797
+| 6.77
+|-
+| 2.3.5.7
+| 6144/6125, 16875/16807, 78732/78125
+| {{mapping| 214 339 497 601 }}
+| −0.0169
+| 0.4137
+| 7.38
+|-
+| 2.3.5.7.11
+| 540/539, 1375/1372, 5632/5625, 72171/71680
+| {{mapping| 214 339 497 601 740 }}
+| +0.0897
+| 0.4270
+| 7.61
+|-
+| 2.3.5.7.11.13
+| 351/350, 540/539, 847/845, 1375/1372, 4096/4095
+| {{mapping| 214 339 497 601 740 792 }}
+| +0.0480
+| 0.4008
+| 7.15
+|-
+| 2.3.5.7.11.13.17
+| 351/350, 540/539, 715/714, 847/845, 936/935, 4096/4095
+| {{mapping| 214 339 497 601 740 792 875 }}
+| −0.0144
+| 0.4012
+| 7.15
+|}
+=== 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
+| 27\214
+| 151.40
+| 12/11
+| [[Browser]]
+|-
+| 1
+| 69\214
+| 386.92
+| 5/4
+| [[Grendel]]
+|-
+| 1
+| 79\214
+| 442.99
+| 162/125
+| [[Sensipent]]
+|-
+| 1
+| 105\214
+| 588.79
+| 7/5
+| [[Aufo]]
+|-
+| 2
+| 28\214
+| 157.01
+| 35/32
+| [[Bison]] (214e)
+|-
+| 2
+| 29\214
+| 162.62
+| 1125/1024
+| [[Kwazy]]
+|}
+<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:Browser]]