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'''525edo''' is the [[EDO|equal division of the octave]] into 525 parts of 2.28571 [[cent]]s each. It is consistent and uniquely consistent through the 25-limit. It tempers out the schisma, 32805/32768 and |8 77 -56> in the 5-limit; 250047/250000, 703125/702464 and 283115520/282475249 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.
{{Infobox ET}}
{{ED intro}}


[[Category:Equal divisions of the octave]]
== Theory ==
525edo is [[distinctly consistent]] through the [[25-odd-limit]]. As an equal temperament, it [[tempering out|tempers out]] the [[schisma]], 32805/32768, and {{monzo| 8 77 -56 }} in the 5-limit; [[250047/250000]], [[703125/702464]] and {{monzo| 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 chord]]s of [[squbemic chords]] and [[petrmic chords]] in the 13-odd-limit.
 
=== Fractional-octave temperaments ===
It supports the 35th-octave temperament [[35th-octave temperaments#Tritonopodismic|tritonopodismic]].
 
525edo supports 21st-octave temperament called [[akjayland]], and the 23-limit extension of akjayland called [[21st-octave temperaments|vasca]], described as {{nowrap|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 ===
{{Harmonics in equal|525}}
 
=== Subsets and supersets ===
Since 525 factors into 3 × 5<sup>2</sup> × 7, 525edo has subset edos {{EDOs| 3, 5, 7, 15, 21, 25, 35, 75, 105, 175 }}.
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 512 -323 }}
| {{mapping| 525 832 }}
| +0.0759
| 0.0759
| 3.32
|-
| 2.3.5
| 32805/32768, {{monzo| 8 77 -56 }}
| {{mapping| 525 832 1219 }}
| +0.0546
| 0.0689
| 3.02
|-
| 2.3.5.7
| 32805/32768, 250047/250000, {{monzo| 21 3 1 -10 }}
| {{mapping| 525 832 1219 1474 }}
| +0.0128
| 0.0940
| 4.11
|-
| 2.3.5.7.11
| 3025/3024, 24057/24010, 32805/32768, 102487/102400
| {{mapping| 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
| {{mapping| 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
| {{mapping| 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
| {{mapping| 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
| {{mapping| 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 [[460edo|460]] and is bettered by [[566edo|566g]]. In the 23-limit it beats [[422edo|422]] and is bettered by [[581edo|581]].
 
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br />per 8ve
! Generator*
! Cents*
! Associated<br />ratio*
! Temperaments
|-
| 1
| 218\525
| 498.29
| 4/3
| [[Helmholtz (temperament)|Helmholtz]]
|-
| 3
| 218\525<br />(43\525)
| 498.29<br />(98.29)
| 4/3<br />(18/17)
| [[Term]]
|-
| 3
| 109\525<br />(66\525)
| 249.14<br />(150.86)
| 15/13<br />(12/11)
| [[Hemiterm]] (525f)
|-
| 7
| 218\525<br />(7\525)
| 498.29<br />(16.00)
| 4/3<br />(99/98)
| [[Septant]]
|-
| 21
| 256\525<br />(6\525)
| 585.14<br />(13.71)
| 91875/65536<br />(126/125)
| [[Akjayland]]
|-
| 21
| 122\525<br />(22\525)
| 278.85<br />(50.29)
| 168/143<br />(?)
| [[Vasca]]
|}
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
 
[[Category:Akjayland]]

Latest revision as of 13:32, 13 March 2026

← 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. As an equal temperament, it 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

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
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

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 distinct

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