21ed5/2: Difference between revisions
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| Line 96: | Line 96: | ||
|- | |- | ||
|16 | |16 | ||
| | |1208.620 | ||
| | |P | ||
|[[2/1]] | |[[2/1]] | ||
|- | |- | ||
|17 | |17 | ||
| | |1284.159 | ||
| | |P&, Q@ | ||
|[[52/25]] | |[[52/25]] | ||
|- | |- | ||
|18 | |18 | ||
| | |1359.698 | ||
| | |Q | ||
|[[11/5]] | |[[11/5]] | ||
|- | |- | ||
|19 | |19 | ||
| | |1435.236 | ||
| | |Q& | ||
|[[16/7]] | |[[16/7]] | ||
|- | |- | ||
|20 | |20 | ||
| | |1510.775 | ||
| | |J@ | ||
|[[26/11]] | |[[26/11]] | ||
|- | |- | ||
|21 | |21 | ||
| | |1586.314 | ||
| | |J | ||
|[[5/2]] | |[[5/2]] | ||
|} | |} | ||
<nowiki>*</nowiki> Based on treating 21ed5/2 as a no-threes 13-limit temperament | <nowiki>*</nowiki> Based on treating 21ed5/2 as a no-threes 13-limit temperament | ||
Revision as of 03:17, 13 July 2023
| ← 20ed5/2 | 21ed5/2 | 22ed5/2 → |
(semiconvergent)
(semiconvergent)
21ed5/2 is the equal division of the 5/2 interval into 21 parts of approximately 75.539 cents each. It roughly corresponds to 16edo.
Theory
From a no-threes point of view, 21ed5/2 tempers out 50/49 in the 7-limit (being a jubilic system similar to 13ed5/2), 625/616 and 176/175 in the 11-limit, and 143/140, 715/686 and 847/845 in the 13-limit. It is not particularly excellent as a no-threes system with the 5/4 and 7/4 being noticeably off, but can work for 5/2-equivalent jubilic.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +8.6 | -13.5 | +17.2 | +8.6 | -4.9 | +30.4 | +25.9 | -27.0 | +17.2 | +3.3 | +3.8 |
| Relative (%) | +11.4 | -17.9 | +22.8 | +11.4 | -6.4 | +40.3 | +34.2 | -35.7 | +22.8 | +4.4 | +5.0 | |
| Steps (reduced) |
16 (16) |
25 (4) |
32 (11) |
37 (16) |
41 (20) |
45 (3) |
48 (6) |
50 (8) |
53 (11) |
55 (13) |
57 (15) | |
Interval table
| Steps | Cents | Jubilic[8] notation | Approximate ratios* |
|---|---|---|---|
| 0 | 0.000 | J | 1/1 |
| 1 | 75.539 | J& | 26/25 |
| 2 | 151.078 | K@ | 35/32 |
| 3 | 226.616 | K | 8/7, 28/25 |
| 4 | 302.155 | K&, L@ | 13/11, 77/64 |
| 5 | 377.694 | L | 5/4, 11/9, 16/13, 49/40 |
| 6 | 453.233 | L& | 13/10, 32/25 |
| 7 | 528.771 | M@ | 11/8, 35/26 |
| 8 | 604.310 | M | 7/5, 10/7 |
| 9 | 679.849 | M& | 16/11, 52/35 |
| 10 | 755.388 | N@ | 11/7, 20/13, 25/16, 49/32 |
| 11 | 830.926 | N | 8/5, 13/8 |
| 12 | 906.465 | N&, O@ | 22/13, 55/32 |
| 13 | 982.004 | O | 7/4, 25/14 |
| 14 | 1057.543 | O& | 13/7, 20/11 |
| 15 | 1133.081 | P@ | 25/13 |
| 16 | 1208.620 | P | 2/1 |
| 17 | 1284.159 | P&, Q@ | 52/25 |
| 18 | 1359.698 | Q | 11/5 |
| 19 | 1435.236 | Q& | 16/7 |
| 20 | 1510.775 | J@ | 26/11 |
| 21 | 1586.314 | J | 5/2 |
* Based on treating 21ed5/2 as a no-threes 13-limit temperament