13ed5/2
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Prime factorization
13 (prime)
Step size
122.024¢
Octave
10\13ed5/2 (1220.24¢)
(semiconvergent)
Twelfth
16\13ed5/2 (1952.39¢)
Consistency limit
5
Distinct consistency limit
2
| ← 12ed5/2 | 13ed5/2 | 14ed5/2 → |
(semiconvergent)
13ed5/2 is the equal division of the 5/2 interval into 13 parts of 122.024 cents each. It roughly corresponds to 10edo, and their patent vals match up until the 7-limit.
Theory
Like 10edo, 13ed5/2 tempers out 50/49 in the no-threes 7-limit, supporting 5/2-equivalent jubilic temperament with a generator of ~7/5. In this regard, it could be considered a "no-threes cousin" of 12edo and 13edt, having the basic tuning for the octatonic scale of 5/2-equivalent jubilic (5L 3s⟨5/2⟩). It also tempers out 56/55 in the 11-limit and 26/25, 52/49 and 65/64 in the 13-limit.
Harmonics
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +20.2 | +50.4 | +40.5 | +20.2 | -51.4 | +47.8 | +60.7 | -21.2 | +40.5 | -2.5 | -31.1 |
| Relative (%) | +16.6 | +41.3 | +33.2 | +16.6 | -42.1 | +39.2 | +49.8 | -17.3 | +33.2 | -2.0 | -25.5 | |
| Steps (reduced) |
10 (10) |
16 (3) |
20 (7) |
23 (10) |
25 (12) |
28 (2) |
30 (4) |
31 (5) |
33 (7) |
34 (8) |
35 (9) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -47.7 | -53.9 | -51.4 | -41.1 | -24.0 | -0.9 | +27.5 | +60.7 | -23.7 | +17.7 | -59.2 |
| Relative (%) | -39.1 | -44.2 | -42.1 | -33.6 | -19.7 | -0.8 | +22.5 | +49.8 | -19.5 | +14.5 | -48.5 | |
| Steps (reduced) |
36 (10) |
37 (11) |
38 (12) |
39 (0) |
40 (1) |
41 (2) |
42 (3) |
43 (4) |
43 (4) |
44 (5) |
44 (5) | |
Intervals
| # | Cents | Approximate ratios* | Jubilic[8] notation |
|---|---|---|---|
| 0 | 0.000 | 1/1 | J |
| 1 | 122.024 | 14/13, 35/32 | J&, K@ |
| 2 | 244.048 | 8/7, 28/25 | K |
| 3 | 366.072 | 5/4, 16/13, 49/40 | L |
| 4 | 488.096 | 32/25, 64/49 | L&, M@ |
| 5 | 610.120 | 7/5, 10/7 | M |
| 6 | 732.144 | 20/13, 25/16, 49/32 | M&, N@ |
| 7 | 854.168 | 8/5, 13/8 | N |
| 8 | 976.192 | 7/4, 25/14 | O |
| 9 | 1098.216 | 13/7, 64/35 | O&, P@ |
| 10 | 1220.240 | 2/1, 49/25, 52/25 | P |
| 11 | 1342.264 | 35/16 | Q |
| 12 | 1464.288 | 16/7 | Q&, J@ |
| 13 | 1586.312 | 5/2 | J |
* Based on treating 13ed5/2 as a 5/2.5.7.13 subgroup temperament