# 525edo

 ← 524edo 525edo 526edo →
Prime factorization 3 × 52 × 7
Step size 2.28571¢
Fifth 307\525 (701.714¢)
Semitones (A1:m2) 49:40 (112¢ : 91.43¢)
Consistency limit 25
Distinct consistency limit 25

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.

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

Approximation of prime harmonics in 525edo
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
• 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

Table of rank-2 temperaments by generator
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