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{{EDO intro|525}}
{{Infobox ET}}
{{ED intro}}


== Theory ==
== Theory ==
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.
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 supports a 140 & 525 temperament with period 35 which sets 7/5 and 10/7 to two "legs" of 35edo (17\35 and 18\35) opposing the tonic and tempers out {{Monzo|34 0 70 -70}}, setting a circle of 35 50/49s equal with the octave. In addition, it suppors 21st-octave period called [[akjayland]].
It allows [[essentially tempered chord]]s of [[squbemic chords]] and [[petrmic chords]] in the 13-odd-limit.  


525's divisors are {{EDOs|1, 3, 5, 7, 15, 21, 25, 35, 75, 105, 175.}}
=== 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}}
{{Harmonics in equal|525}}
[[Category:Equal divisions of the octave|###]]
 
=== 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]]
[[Category:Akjayland]]
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