1665edo: Difference between revisions

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 {{Infobox ET}}
-{{EDO intro|1665}}
+{{ED intro}}
-edo is a very strong 5-limit (as well as 2.3.5.11 subgroup) tuning and it is consistent in the 15-odd-limit. In the 5-limit, 1665edo is a tuning for the [[gross]] temperament.
+== Theory ==
+edo is a very strong 5-limit (as well as 2.3.5.11 [[subgroup]]) tuning and it is [[consistent]] in the [[15-odd-limit]]. In the 5-limit, 1665edo is a tuning for the [[gross]] temperament.
-edo provides the optimal patent val for the [[rhodium]] temperament in the 11-limit and also in the 13-limit.
+edo provides the [[optimal patent val]] for the [[rhodium]] temperament in the 11-limit and also in the 13-limit. In addition, it provides the optimal patent val for [[dzelic]] temperament in the 13-limit.
-===Prime harmonics===
-{{harmonics in equal|1665}}
+The 1665cc val is a tuning for the [[roentgenium]] temperament, and the patent val tunes the unnamed 111 & 1665 temperament in the 13-limit which has a comma basis {6656/6655, 123201/123200, 250047/250000, 91182091/91125000}.
+=== Prime harmonics ===
+{{Harmonics in equal|1665}}
+=== Subsets and supersets ===
+Since 1665 factors into {{factorization|1665}}, 1665edo has subset edos {{EDOs| 3, 5, 9, 15, 37, 45, 111, 185, 333, and 555 }}.
+== Regular temperament properties ==
+=== 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
+| 127\1665
+| 91.531
+| {{monzo| 9 -32 18 }}
+| [[Gross]]
+|-
+| 37
+| 377\1665<br />(17\1665)
+| 271.711<br />(12.252)
+| 117/100<br />(?)
+| [[Dzelic]]
+|-
+| 45
+| 1301\1665<br />(6\1665)
+| 937.657<br />(4.324)
+| 55/32<br />(?)
+| [[Rhodium]]
+|}
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct