13ed5/2: Difference between revisions
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| Line 18: | Line 18: | ||
|0.000 | |0.000 | ||
|[[1/1]] | |[[1/1]] | ||
| | |J | ||
|- | |- | ||
|1 | |1 | ||
|122.024 | |122.024 | ||
|[[14/13]], [[35/32]] | |[[14/13]], [[35/32]] | ||
| | |J&, K@ | ||
|- | |- | ||
|2 | |2 | ||
|244.048 | |244.048 | ||
|[[8/7]], [[28/25]] | |[[8/7]], [[28/25]] | ||
| | |K | ||
|- | |- | ||
|3 | |3 | ||
|366.072 | |366.072 | ||
|[[5/4]], [[16/13]], [[49/40]] | |[[5/4]], [[16/13]], [[49/40]] | ||
| | |L | ||
|- | |- | ||
|4 | |4 | ||
|488.096 | |488.096 | ||
|[[32/25]], [[64/49]] | |[[32/25]], [[64/49]] | ||
| | |L&, M@ | ||
|- | |- | ||
|5 | |5 | ||
|610.120 | |610.120 | ||
|[[7/5]], [[10/7]] | |[[7/5]], [[10/7]] | ||
| | |M | ||
|- | |- | ||
|6 | |6 | ||
|732.144 | |732.144 | ||
|[[20/13]], [[25/16]], [[49/32]] | |[[20/13]], [[25/16]], [[49/32]] | ||
| | |M&, N@ | ||
|- | |- | ||
|7 | |7 | ||
|854.168 | |854.168 | ||
|[[8/5]], [[13/8]] | |[[8/5]], [[13/8]] | ||
| | |N | ||
|- | |- | ||
|8 | |8 | ||
|976.192 | |976.192 | ||
|[[7/4]], [[25/14]] | |[[7/4]], [[25/14]] | ||
| | |O | ||
|- | |- | ||
|9 | |9 | ||
|1098.216 | |1098.216 | ||
|[[13/7]], [[64/35]] | |[[13/7]], [[64/35]] | ||
| | |O&, P@ | ||
|- | |- | ||
|10 | |10 | ||
|1220.240 | |1220.240 | ||
|[[2/1]], [[49/25]], 52/25 | |[[2/1]], [[49/25]], 52/25 | ||
| | |P | ||
|- | |- | ||
|11 | |11 | ||
|1342.264 | |1342.264 | ||
|35/16 | |35/16 | ||
| | |Q | ||
|- | |- | ||
|12 | |12 | ||
|1464.288 | |1464.288 | ||
|[[16/7]] | |[[16/7]] | ||
| | |Q&, J@ | ||
|- | |- | ||
|13 | |13 | ||
|1586.312 | |1586.312 | ||
|[[5/2]] | |[[5/2]] | ||
| | |J | ||
|} | |} | ||
<nowiki>*</nowiki> Based on treating 13ed5/2 as a 5/2.5.7.13 subgroup temperament | <nowiki>*</nowiki> Based on treating 13ed5/2 as a 5/2.5.7.13 subgroup temperament | ||
Revision as of 00:52, 12 July 2023
| ← 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.
| 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) | |
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