21ed5/2: Difference between revisions
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<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 | ||
{{Interval table}} | {{Interval table}} | ||
Revision as of 03:07, 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 | 8/7, 28/25 | |||
| 4 | 13/11, 77/64 | |||
| 5 | 5/4, 11/9, 16/13, 49/40 | |||
| 6 | 13/10, 32/25 | |||
| 7 | 11/8, 35/26 | |||
| 8 | 7/5, 10/7 | |||
| 9 | 16/11, 52/35 | |||
| 10 | 11/7, 20/13, 25/16, 49/32 | |||
| 11 | 8/5, 13/8 | |||
| 12 | 22/13, 55/32 | |||
| 13 |
* Based on treating 21ed5/2 as a no-threes 13-limit temperament
| Steps | Cents | Approximate ratios |
|---|---|---|
| 0 | 0 | 1/1 |
| 1 | 75.5 | 21/20, 22/21, 23/22, 24/23, 25/24 |
| 2 | 151.1 | 11/10, 12/11, 13/12, 23/21, 25/23 |
| 3 | 226.6 | 8/7, 17/15, 25/22 |
| 4 | 302.2 | 6/5, 13/11, 25/21 |
| 5 | 377.7 | 5/4, 21/17 |
| 6 | 453.2 | 13/10, 17/13, 22/17 |
| 7 | 528.8 | 15/11, 23/17 |
| 8 | 604.3 | 10/7, 17/12, 24/17 |
| 9 | 679.8 | 25/17 |
| 10 | 755.4 | 17/11, 20/13 |
| 11 | 830.9 | 13/8, 21/13 |
| 12 | 906.5 | 17/10, 22/13 |
| 13 | 982 | 7/4, 23/13 |
| 14 | 1057.5 | 11/6, 13/7, 24/13 |
| 15 | 1133.1 | 21/11, 23/12, 25/13 |
| 16 | 1208.6 | 2/1 |
| 17 | 1284.2 | 19/9, 21/10, 23/11, 25/12 |
| 18 | 1359.7 | 11/5, 24/11 |
| 19 | 1435.2 | 16/7, 23/10, 25/11 |
| 20 | 1510.8 | 12/5 |
| 21 | 1586.3 | 5/2 |