1106edo: Difference between revisions

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-The '''1106 division''' is a [[The_Riemann_Zeta_Function_and_Tuning#Zeta EDO lists|zeta peak edo]] which divides the octave into 1106 parts of 1.805 cents each. 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 171, 270, 342, 441 and 612. It is even stronger in the 11-limit; the only ones being it out now being 270, 342 and 612. It is less strong in the 13 and 17 limits, but even so is distinctly consistent through the 17 limit.
+{{Infobox ET}}
+{{ED intro}}
-[[Category:Equal divisions of the octave|####]] <!-- 4-digit number -->
+== 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|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
+|}
+=== 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
+| 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