253edo: Difference between revisions

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-'''253edo''' is the [[EDO|equal division of the octave]] into 253 parts of 4.743083 [[cent]]s each. It is [[consistent]] to the [[17-odd-limit]], approximating the fifth by 148\253 (0.021284 cents sharper than the just 3/2), and the prime harmonics from 5 to 17 are all slightly flat. It tempers out [[32805/32768]] in the 5-limit; [[2401/2400]] in the 7-limit; [[385/384]], 1375/1372 and [[4000/3993]] in the 11-limit; [[325/324]], [[1575/1573]] and [[2200/2197]] in the 13-limit; 375/374 and 595/594 in the 17-limit. It provides a good tuning for higher-limit [[sesquiquartififths]] temperament.
+{{Infobox ET
+| Prime factorization = 11 × 23
+| Step size = 4.74308¢
+| Fifth = 148\253 (701.98¢)
+| Semitones = 24:19 (113.83¢ : 90.12¢)
+| Consistency = 17
+}}
+{{EDO intro|253}}
+== Theory ==
+edo is [[consistent]] to the [[17-odd-limit]], approximating the fifth by 148\253 (0.021284 cents sharper than the just 3/2), and the prime harmonics from 5 to 17 are all slightly flat. It tempers out [[32805/32768]] in the 5-limit; [[2401/2400]] in the 7-limit; [[385/384]], 1375/1372 and [[4000/3993]] in the 11-limit; [[325/324]], [[1575/1573]] and [[2200/2197]] in the 13-limit; 375/374 and 595/594 in the 17-limit. It provides a good tuning for higher-limit [[sesquiquartififths]] temperament.
 = 11 × 23, and has subset edos [[11edo]] and [[23edo]].
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 === Prime harmonics ===
 {{Harmonics in equal|253}}
+== Regular temperament properties ==
+{| class="wikitable center-4 center-5 center-6"
+! rowspan="2" | Subgroup
+! rowspan="2" | [[Comma list]]
+! rowspan="2" | [[Mapping]]
+! rowspan="2" | Optimal 8ve <br>stretch (¢)
+! colspan="2" | Tuning error
+|-
+! [[TE error|Absolute]] (¢)
+! [[TE simple badness|Relative]] (%)
+|-
+| 2.3
+| {{monzo| 401 -253 }}
+| [{{val| 253 401 }}]
+| -0.007
+| 0.007
+| 0.14
+|-
+| 2.3.5
+| 32805/32768, {{monzo| -4 -37 27 }}
+| [{{val| 253 401 587 }}]
+| +0.300
+| 0.435
+| 9.16
+|-
+| 2.3.5.7
+| 2401/2400, 32805/32768, 390625/387072
+| [{{val| 253 401 587 710 }}]
+| +0.335
+| 0.381
+| 8.03
+|-
+| 2.3.5.7.11
+| 385/384, 1375/1372, 4000/3993, 19712/19683
+| [{{val| 253 401 587 710 875 }}]
+| +0.333
+| 0.341
+| 7.19
+|-
+| 2.3.5.7.11.13
+| 325/324, 385/384, 1375/1372, 1575/1573, 2200/2197
+| [{{val| 253 401 587 710 875 936 }}]
+| +0.323
+| 0.312
+| 6.58
+|-
+| 2.3.5.7.11.13.17
+| 325/324, 375/374, 385/384, 595/594, 1275/1274, 2200/2197
+| [{{val| 253 401 587 710 875 936 1034 }}]
+| +0.298
+| 0.295
+| 6.22
+|}
+=== Rank-2 temperaments ===
+{| class="wikitable center-all left-5"
+|+Table of rank-2 temperaments by generator
+! Periods<br>per octave
+! Generator<br>(reduced)
+! Cents<br>(reduced)
+! Associated<br>ratio
+! Temperaments
+|-
+| 1
+| 37\253
+| 175.49
+| 448/405
+| [[Sesquiquartififths]]
+|-
+| 1
+| 105\253
+| 498.02
+| 4/3
+| [[Helmholtz]]
+|-
+| 1
+| 123\253
+| 583.40
+| 7/5
+| [[Cotritone]]
+|}
 == Scales ==
-* 63 32 63 63 32: [[3L_2s|Pentatonic]]
+* 63 32 63 63 32: [[3L 2s|Pentic]]
-* 43 43 19 43 43 43 19: [[5L_2s|Pythagorean tuning]]
+* 43 43 19 43 43 43 19: [[Helmholtz]][7]
-* 41 41 24 41 41 41 24: [[Meantone|Meantonic tuning]]
+* 41 41 24 41 41 41 24: [[Meantone]][7]
-* 35 35 35 35 35 35 35 8: [[7L_1s|Porcupine tuning]]
+* 35 35 35 35 35 35 35 8: [[Porcupine]][8]
 * 33 33 33 11 33 33 33 33 11: [[23edo|"The Hendecapliqued superdiatonic of the Icositriphony"]]
-* 31 31 31 18 31 31 31 31 18: [[7L_2s|Superdiatonic tuning]] in the way of Mavila
+* 31 31 31 18 31 31 31 31 18: [[Mavila]][9]
-* 26 26 15 26 26 26 15 26 26 26 15: [[sensi11|Sensi tuning]]
+* 26 26 15 26 26 26 15 26 26 26 15: [[Sensi]][11]
-* 20 20 20 11 20 20 20 20 11 20 20 20 20 11: [[11L_3s|Ketradektriatoh tuning]]
+* 20 20 20 11 20 20 20 20 11 20 20 20 20 11: [[11L 3s|Ketradektriatoh scale]]
 [[Category:Equal divisions of the octave]]
 [[Category:Sesquiquartififths]]