1665edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{EDO intro|1665}} | {{EDO intro|1665}} | ||
==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 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. | ||
| Line 7: | Line 7: | ||
===Prime harmonics=== | ===Prime harmonics=== | ||
{{harmonics in equal|1665}} | {{harmonics in equal|1665}} | ||
==Regular temperament properties== | |||
=== Rank-2 temperaments === | |||
{| class="wikitable center-all left-5" | |||
|+Table of rank-2 temperaments by generator | |||
! Periods<br>per 8ve | |||
! Generator<br>(Reduced) | |||
! Cents<br>(Reduced) | |||
! 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]] | |||
|} | |||
Revision as of 02:30, 18 March 2023
| ← 1664edo | 1665edo | 1666edo → |
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.
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) | |
Regular temperament properties
Rank-2 temperaments
| Periods per 8ve |
Generator (Reduced) |
Cents (Reduced) |
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 |