1665edo: Difference between revisions
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1665edo provides the optimal patent val for the [[rhodium]] temperament in the 11-limit and also in the 13-limit. | == Theory == | ||
1665edo 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. | |||
1665edo 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. | |||
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 | |||