472edo: Difference between revisions

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-'''472edo''' is the [[EDO|equal division of the octave]] into 472 parts of 2.54237 cents each. It is consistent to the 11-limit, tempering out 32805/32768 (schisma) and 1224440064/1220703125 (parakleisma) in the 5-limit; 2401/2400, 2460375/2458624, and 30623756184/30517578125 in the 7-limit; 9801/9800, 46656/46585, 117649/117612, and 234375/234256 in the 11-limit, [[support|supporting]] the [[Breedsmic temperaments|maviloid temperament]], the [[Schismatic family|bisesqui temperament]], and the [[Schismatic family|octant temperament]]. Using the patent val, it tempers out 729/728, 1575/1573, 2200/2197, 4096/4095, and 21168/21125 in the 13-limit, so it also supports the 13-limit octant.
+{{Infobox ET}}
+{{ED intro}}
-It is a [[zeta peak integer edo]].
+== Theory ==
+edo is [[enfactoring|enfactored]] in the 5-limit, with the same tuning as [[118edo]], defined by [[tempering out]] the [[schisma]] and the [[parakleisma]], but the approximation to higher harmonics are much improved. It is a [[zeta peak integer edo]], [[consistent]] to the [[11-odd-limit]] or the no-13 [[29-odd-limit]].
-{{Primes in edo|472|prec=3}}
+In the 7-limit, the equal temperament tempers out [[2401/2400]], 2460375/2458624, and 30623756184/30517578125; in the 11-limit, [[9801/9800]], 46656/46585, 117649/117612, and 234375/234256, [[support]]ing the [[maviloid]] temperament, the [[Schismatic family #Bisesqui|bisesqui]] temperament, and the [[octant]] temperament. Using the [[patent val]], it tempers out [[729/728]], [[1575/1573]], [[2200/2197]], [[4096/4095]], and 21168/21125 in the 13-limit, so it also supports the 13-limit octant.
-[[Category:Equal divisions of the octave]]
-[[Category:Zeta]]
+=== Prime harmonics ===
+{{Harmonics in equal|472}}
+=== Subsets and supersets ===
+Since 472 factors into {{factorization|472}}, 472edo has subset edos {{EDOs| 2, 4, 8, 59, 118, and 236 }}.
+== 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]] (%)
+|-
+| 2.3.5.7
+| 2401/2400, 32805/32768, {{monzo| 8 14 -13 }}
+| [{{val| 472 748 1096 1325 }}]
+| +0.0435
+| 0.0814
+| 3.20
+|-
+| 2.3.5.7.11
+| 2401/2400, 9801/9800, 32805/32768, 46656/46585
+| [{{val| 472 748 1096 1325 1633 }}]
+| +0.0130
+| 0.0950
+| 3.74
+|-
+| 2.3.5.7.11.13
+| 729/728, 1575/1573, 2200/2197, 2401/2400, 4096/4095
+| [{{val| 472 748 1096 1325 1633 1747 }}]
+| −0.0341
+| 0.1365
+| 5.37
+|}
+=== Rank-2 temperaments ===
+Note: 5-limit temperaments supported by [[118edo|118et]] are not included.
+{| 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
+| 69\472
+| 175.42
+| 448/405
+| [[Sesquiquartififths]]
+|-
+| 1
+| 137\472
+| 348.31
+| 57344/46875
+| [[Subneutral]]
+|-
+| 1
+| 205\472
+| 521.19
+| 875/648
+| [[Maviloid]]
+|-
+| 2
+| 69\472
+| 175.42
+| 448/405
+| [[Bisesqui]]
+|-
+| 8
+| 196\472<br />(19\472)
+| 498.31<br />(48.31)
+| 4/3<br />(36/35)
+| [[Octant]]
+|}
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct