525edo
| ← 524edo | 525edo | 526edo → |
525 equal divisions of the octave (abbreviated 525edo or 525ed2), also called 525-tone equal temperament (525tet) or 525 equal temperament (525et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 525 equal parts of about 2.29 ¢ each. Each step represents a frequency ratio of 21/525, or the 525th root of 2.
Theory
525edo is distinctly consistent through the 25-odd-limit. The equal temperament tempers out the schisma, 32805/32768, and [8 77 -56⟩ in the 5-limit; 250047/250000, 703125/702464 and [21 3 1 -10⟩ in the 7-limit; 3025/3024, 24057/24010, 102487/102400 and 180224/180075 in the 11-limit; 729/728, 1716/1715, 2200/2197, 4096/4095 and 14641/14625 in the 13-limit; 1089/1088, 1275/1274, and 2025/2023 in the 17-limit; 2376/2375 in the 19-limit; and 1197/1196, 1496/1495, 2024/2023, and 2025/2024 in the 23-limit.
It allows essentially tempered chords of squbemic chords and petrmic chords in the 13-odd-limit.
Fractional-octave temperaments
It supports the 35th-octave temperament tritonopodismic.
525edo supports 21st-octave temperament called akjayland, and the 23-limit extension of akjayland called vasca, described as 357 & 525. It is more suitable to view this temperament as vasca in 525edo as opposed to simply akjayland, since 525edo is consistent in the 23-odd-limit, while other edos which support akjayland are not. As a corollary of supporting vasca, 525edo also supports the relationship that sets 11\21 to 23/16.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.24 | -0.03 | +0.32 | -0.46 | +0.62 | +0.19 | -0.37 | +0.30 | -1.01 | +0.11 |
| Relative (%) | +0.0 | -10.5 | -1.2 | +13.9 | -20.2 | +26.9 | +8.2 | -16.2 | +13.0 | -44.0 | +4.7 | |
| Steps (reduced) |
525 (0) |
832 (307) |
1219 (169) |
1474 (424) |
1816 (241) |
1943 (368) |
2146 (46) |
2230 (130) |
2375 (275) |
2550 (450) |
2601 (501) | |
Subsets and supersets
Since 525 factors into 3 × 52 × 7, 525edo has subset edos 3, 5, 7, 15, 21, 25, 35, 75, 105, 175.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [512 -323⟩ | [⟨525 832]] | +0.0759 | 0.0759 | 3.32 |
| 2.3.5 | 32805/32768, [8 77 -56⟩ | [⟨525 832 1219]] | +0.0546 | 0.0689 | 3.02 |
| 2.3.5.7 | 32805/32768, 250047/250000, [21 3 1 -10⟩ | [⟨525 832 1219 1474]] | +0.0128 | 0.0940 | 4.11 |
| 2.3.5.7.11 | 3025/3024, 24057/24010, 32805/32768, 102487/102400 | [⟨525 832 1219 1474 1816]] | +0.0368 | 0.0969 | 4.24 |
| 2.3.5.7.11.13 | 729/728, 1716/1715, 2200/2197, 3025/3024, 14641/14625 | [⟨525 832 1219 1474 1816 1943]] | +0.0030 | 0.1164 | 5.09 |
| 2.3.5.7.11.13.17 | 729/728, 1089/1088, 1275/1274, 1716/1715, 2025/2023, 2200/2197 | [⟨525 832 1219 1474 1816 1943 2146]] | -0.0040 | 0.1091 | 4.77 |
| 2.3.5.7.11.13.17.19 | 729/728, 1089/1088, 1275/1274, 1716/1715, 2025/2023, 2200/2197, 2376/2375 | [⟨525 832 1219 1474 1816 1943 2146 2230]] | +0.0074 | 0.1064 | 4.66 |
| 2.3.5.7.11.13.17.19.23 | 729/728, 1089/1088, 1197/1196, 1275/1274, 1496/1495, 1716/1715, 2024/2023, 2025/2023 | [⟨525 832 1219 1474 1816 1943 2146 2230 2375]] | -0.0007 | 0.1029 | 4.50 |
- 525et has lower absolute errors than any previous equal temperaments in the 19- and 23-limit. In the 19-limit it beats 460 and is bettered by 566g. In the 23-limit it beats 422 and is bettered by 581.
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 218\525 | 498.29 | 4/3 | Helmholtz |
| 3 | 218\525 (43\525) |
498.29 (98.29) |
4/3 (18/17) |
Term |
| 3 | 109\525 (66\525) |
249.14 (150.86) |
15/13 (12/11) |
Hemiterm (525f) |
| 7 | 218\525 (7\525) |
498.29 (16.00) |
4/3 (99/98) |
Septant |
| 21 | 256\525 (6\525) |
585.14 (13.71) |
91875/65536 (126/125) |
Akjayland |
| 21 | 122\525 (22\525) |
278.85 (50.29) |
168/143 (?) |
Vasca |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct