3395edo: Difference between revisions
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=== Rank-2 temperaments === | === 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 | | 1 | ||
| Line 34: | Line 41: | ||
| 4/3<br />(?) | | 4/3<br />(?) | ||
| [[Berkelium]] | | [[Berkelium]] | ||
|} | |||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | |||
[[Category:Jacobin]] | [[Category:Jacobin]] | ||
[[Category:Quartismic]] | [[Category:Quartismic]] | ||
Revision as of 13:15, 16 November 2024
| ← 3394edo | 3395edo | 3396edo → |
Theory
3395edo is an extremely strong 17- and 19-limit system, and a zeta peak, integral and gap edo. It has a lower 17-limit TE relative error than any edo until 7033, and a lower 19-limit relative error than any edo until 8269. Besides, it provides the optimal patent val for the 13-limit rank-5 temperament tempering out 6656/6655, the jacobin comma, and for quartismic, which also tempers out 123201/123200. A basis for the 17-limit commas is {6656/6655, 12376/12375, 28561/28560, 31213/31212, 37180/37179, 937125/937024}, and for the 19-limit, {6656/6655, 12376/12375, 12636/12635, 13377/13376, 14365/14364, 23409/23408, 28561/28560}.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.018 | +0.019 | +0.011 | +0.081 | +0.003 | +0.022 | +0.101 | -0.174 | +0.055 | -0.175 |
| Relative (%) | +0.0 | +5.2 | +5.4 | +3.0 | +23.0 | +0.7 | +6.4 | +28.6 | -49.3 | +15.5 | -49.6 | |
| Steps (reduced) |
3395 (0) |
5381 (1986) |
7883 (1093) |
9531 (2741) |
11745 (1560) |
12563 (2378) |
13877 (297) |
14422 (842) |
15357 (1777) |
16493 (2913) |
16819 (3239) | |
Subsets and supersets
Since 3395 factors into 5 × 7 × 97, 3395edo has subset edos 5, 7, 35, 97, 485, and 679.
Regular temperament properties
3395edo has a lower 17-limit TE relative error than any edo until 7033, and a lower 19-limit relative error than any edo until 8269.
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 2319\3395 | 819.676 | 55115776/34328125 | Genojacobin |
| 35 | 1409\3395 (51\3395) |
498.027 (18.026) |
4/3 (?) |
Bromine |
| 97 | 1409\3395 (9\3395) |
498.027 (3.181) |
4/3 (?) |
Berkelium |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct