525edo
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
Template:Comma basis begin |- | 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 Template:Comma basis end
- 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
Template:Rank-2 begin
|-
| 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
Template:Rank-2 end
Template:Orf