1665edo: Difference between revisions
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1665edo | == 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. | |||
==Regular temperament properties== | 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 === | === Rank-2 temperaments === | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
|+Table of rank-2 temperaments by generator | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | |- | ||
! Generator | ! Periods<br />per 8ve | ||
! Cents | ! Generator* | ||
! Associated<br> | ! Cents* | ||
! Associated<br />ratio* | |||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| Line 23: | Line 29: | ||
| 127\1665 | | 127\1665 | ||
| 91.531 | | 91.531 | ||
| {{monzo|9 -32 18}} | | {{monzo| 9 -32 18 }} | ||
| [[Gross]] | | [[Gross]] | ||
|- | |- | ||
| 37 | | 37 | ||
| 377\1665<br>(17\1665) | | 377\1665<br />(17\1665) | ||
| 271.711<br>(12.252) | | 271.711<br />(12.252) | ||
| 117/100<br>(?) | | 117/100<br />(?) | ||
| [[Dzelic]] | | [[Dzelic]] | ||
|- | |- | ||
| 45 | | 45 | ||
| 1301\1665<br>(6\1665) | | 1301\1665<br />(6\1665) | ||
| 937.657<br>(4.324) | | 937.657<br />(4.324) | ||
| 55/32<br>(?) | | 55/32<br />(?) | ||
| [[Rhodium]] | | [[Rhodium]] | ||
|} | |} | ||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | |||
Latest revision as of 06:13, 21 February 2025
| ← 1664edo | 1665edo | 1666edo → |
1665 equal divisions of the octave (abbreviated 1665edo or 1665ed2), also called 1665-tone equal temperament (1665tet) or 1665 equal temperament (1665et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1665 equal parts of about 0.721 ¢ each. Each step represents a frequency ratio of 21/1665, or the 1665th root of 2.
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
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.027 | -0.007 | -0.177 | +0.033 | -0.167 | +0.270 | +0.145 | +0.194 | +0.333 | +0.190 |
| Relative (%) | +0.0 | +3.7 | -1.0 | -24.6 | +4.6 | -23.2 | +37.4 | +20.1 | +26.9 | +46.2 | +26.3 | |
| Steps (reduced) |
1665 (0) |
2639 (974) |
3866 (536) |
4674 (1344) |
5760 (765) |
6161 (1166) |
6806 (146) |
7073 (413) |
7532 (872) |
8089 (1429) |
8249 (1589) | |
Subsets and supersets
Since 1665 factors into 32 × 5 × 37, 1665edo has subset edos 3, 5, 9, 15, 37, 45, 111, 185, 333, and 555.
Regular temperament properties
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 127\1665 | 91.531 | [9 -32 18⟩ | Gross |
| 37 | 377\1665 (17\1665) |
271.711 (12.252) |
117/100 (?) |
Dzelic |
| 45 | 1301\1665 (6\1665) |
937.657 (4.324) |
55/32 (?) |
Rhodium |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct