1106edo: Difference between revisions

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 {{Infobox ET}}
-{{EDO intro|1106}}
+{{ED intro}}
-edo is a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]]. It is strong as a 7-limit system; the only edos lower than it with a lower 7-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] being {{EDOs| 171, 270, 342, 441 and 612 }}. It is even stronger in the 11-limit; the only ones beating it out now being {{EDOs| 270, 342 and 612 }}. It is less strong in the 13 and 17 limits, but even so is distinctly [[consistent]] through the [[17-odd-limit]].
+== Theory ==
+edo is a [[zeta peak edo]]. It is strong as a 7-limit system; the only edos lower than it with a lower 7-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] being {{EDOs| 171, 270, 342, 441, and 612 }}. It is even stronger in the 11-limit; the only ones beating it out now being {{EDOs| 270, 342, and 612 }}. It is less strong in the 13- and 17-limit, but even so is [[consistency|distinctly consistent]] through the [[17-odd-limit]].
+The equal temperament [[tempering out|tempers out]] {{monzo| -53 10 16 }} (kwazy comma) and {{monzo| -13 -46 37 }} (supermajor comma) in the 5-limit; [[4375/4374]] and 52734375/52706752 in the 7-limit; [[3025/3024]] and [[9801/9800]] in the 11-limit; [[4096/4095]], 78125/78078, and 105644/105625 in the 13-limit; [[2500/2499]], [[4914/4913]], and 8624/8619 in the 17-limit. It notably supports [[supermajor]], [[brahmagupta]], and [[orga]] in the 7-limit, and [[semisupermajor]] in the 11-limit. In the higher limits, it supports the 79th-octave temperament [[gold]].
 === Prime harmonics ===
-{{Harmonics in equal|1106}}
+{{Harmonics in equal|1106|columns=12}}
+=== Subsets and supersets ===
+Since 1106 factors into {{factorization|1106}}, it has subset edos {{EDOs| 2, 7, 14, 79, 158, and 553 }}.
+== 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| 1753 -1106 }}
+| {{Mapping| 1106 1753 }}
+| −0.010
+| 0.010
+| 0.99
+|-
+| 2.3.5
+| {{Monzo| -53 10 16 }}, {{monzo| -13 -46 37 }}
+| {{Mapping| 1106 1753 2568 }}
+| +0.001
+| 0.019
+| 1.73
+|-
+| 2.3.5.7
+| 4375/4374, 52734375/52706752, {{monzo| 46 -14 -3 -6 }}
+| {{Mapping| 1106 1753 2568 3105 }}
+| −0.006
+| 0.020
+| 1.83
+|-
+| 2.3.5.7.11
+| 3025/3024, 4375/4374, 5767168/5764801, 35156250/35153041
+| {{Mapping| 1106 1753 2568 3105 3826 }}
+| +0.004
+| 0.026
+| 2.38
+|-
+| 2.3.5.7.11.13
+| 3025/3024, 4096/4095, 4375/4374, 78125/78078, 105644/105625
+| {{Mapping| 1106 1753 2568 3105 3826 4093 }}
+| −0.012
+| 0.043
+| 3.94
+|-
+| 2.3.5.7.11.13.17
+| 2500/2499, 3025/3024, 4096/4095, 4375/4374, 4914/4913, 8624/8619
+| {{Mapping| 1106 1753 2568 3105 3826 4093 4521 }}
+| −0.021
+| 0.045
+| 4.11
+|}
-=== Divisors ===
+=== Rank-2 temperaments ===
-Since 1106 factors into 2 × 7 × 79, it has subset edos {{EDOs| 2, 7, 14, 79, 158, and 553 }}.
+{| 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
+| 213\1106
+| 231.103
+| 8/7
+| [[Orga]] (11-limit)
+|-
+| 1
+| 401\1106
+| 435.081
+| 9/7
+| [[Supermajor]]
+|-
+| 2
+| 150\1106
+| 162.749
+| 1125/1024
+| [[Crazy]]
+|-
+| 2
+| 401\1106<br>(152\1106)
+| 435.081<br>(164.919)
+| 9/7<br>(11/10)
+| [[Semisupermajor]]
+|-
+| 7
+| 479\1106<br>(5\1106)
+| 519.711<br>(5.424)
+| 27/20<br>(5120/5103)
+| [[Brahmagupta]] (7-limit)
+|-
+| 79
+| 459\1106<br>(11\1106)
+| 498.011<br>(11.935)
+| 4/3<br>(?)
+| [[Gold]]
+|}
+<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|minimal form]] in parentheses if distinct