263edo: Difference between revisions

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 {{Infobox ET}}
-'''263EDO''' is the [[EDO|equal division of the octave]] into 263 parts of 4.5627 [[cent]]s each. It tempers out 393216/390625 (Würschmidt comma) and |50 -33 1&gt; in the 5-limit. Using the patent val, it tempers out 4375/4374, 50421/50000, and 458752/455625 in the 7-limit; 441/440, 3388/3375, 16384/16335, and 26411/26244 in the 11-limit; 364/363, 2080/2079, 2197/2187, and 3584/3575 in the 13-limit; 595/594, 833/832, 936/935, and 1156/1155 in the 17-limit. 263EDO is acculate for the 17th harmonic, as the denominator of a convergent to log<sub>2</sub>17, after [[80edo|80]] and before [[343edo|343]]. Using the 263d val, it tempers out 5120/5103, 16875/16807, and 1959552/1953125 in the 7-limit; 540/539, 1375/1372, 16384/16335, and 43923/43750 in the 11-limit; 351/350, 1001/1000, 1573/1568, 2197/2187, and 4225/4224 in the 13-limit. Using the 263df val, it tempers out 352/351, 640/637, 729/728, and 3584/3575 in the 13-limit.
+{{ED intro}}
-EDO is the 56th [[prime EDO]].
+== Theory ==
+et [[tempering out|tempers out]] 393216/390625 ([[würschmidt comma]]) and {{monzo| 50 -33 1 }} in the 5-limit. Using the [[patent val]], it tempers out [[4375/4374]], [[50421/50000]], and 458752/455625 in the 7-limit; [[441/440]], [[3388/3375]], [[16384/16335]], and 26411/26244 in the 11-limit; [[364/363]], [[2080/2079]], [[2197/2187]], and 3584/3575 in the 13-limit; [[595/594]], [[833/832]], [[936/935]], and [[1156/1155]] in the 17-limit.
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
+Using the 263d val, it tempers out [[5120/5103]], [[16875/16807]], and 1959552/1953125 in the 7-limit; [[540/539]], 1375/1372, 16384/16335, and 43923/43750 in the 11-limit; [[351/350]], [[1001/1000]], [[1573/1568]], 2197/2187, and [[4225/4224]] in the 13-limit.
-[[Category:Prime EDO]]
+Using the 263df val, it tempers out [[352/351]], [[640/637]], [[729/728]], and 3584/3575 in the 13-limit.
+Finally, it is accurate for the 17th harmonic, as the denominator of a convergent to log<sub>2</sub>17, after [[80edo|80]] and before [[343edo|343]].
+=== Prime harmonics ===
+{{Harmonics in equal|263}}
+=== Subsets and supersets ===
+edo is the 56th [[prime edo]].
+Notable supersets include [[789edo]], which triples it to achieve extreme accuracy in the [[2.5.7 subgroup]], and [[1578edo]], which sextuples it to be extremely strong in the [[11-limit]] add-17 and in higher limits.
+== 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| 417 -263 }}
+| {{val| 263 417 }}
+| −0.2229
+| 0.2229
+| 4.89
+|-
+| 2.3.5
+| 393216/390625, {{monzo| 50 -33 1 }}
+| {{val| 263 417 611 }}
+| −0.3666
+| 0.2728
+| 5.98
+|}
+=== 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
+| 40\263
+| 182.51
+| 10/9
+| [[Minortone]]
+|-
+| 1
+| 85\263
+| 387.83
+| 5/4
+| [[Würschmidt]]
+|}
+<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