525edo: Difference between revisions

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{{Infobox ET
{{Infobox ET}}
| Prime factorization = 3 × 5<sup>2</sup> × 7
{{ED intro}}
| Step size = 2.28571¢
| Fifth = 307\525 (701.71¢)
| Semitones = 49:40 (112.00¢ : 91.43¢)
| Consistency = 25
}}
{{EDO intro|525}}


== Theory ==
== Theory ==
525edo is distinctly [[consistent]] through the [[25-odd-limit]]. It 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.
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 the 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 thirty-five [[50/49]]'s equal with the octave. In addition, it supports 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 &amp; 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 ===
=== Prime harmonics ===
{{Harmonics in equal|525|columns=11}}
{{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 ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve <br>stretch (¢)
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning error
|-
|-
Line 31: Line 32:
| 2.3
| 2.3
| {{monzo| 512 -323 }}
| {{monzo| 512 -323 }}
| [{{val| 525 832 }}]
| {{mapping| 525 832 }}
| +0.0759
| +0.0759
| 0.0759
| 0.0759
Line 38: Line 39:
| 2.3.5
| 2.3.5
| 32805/32768, {{monzo| 8 77 -56 }}
| 32805/32768, {{monzo| 8 77 -56 }}
| [{{val| 525 832 1219 }}]
| {{mapping| 525 832 1219 }}
| +0.0546
| +0.0546
| 0.0689
| 0.0689
Line 45: Line 46:
| 2.3.5.7
| 2.3.5.7
| 32805/32768, 250047/250000, {{monzo| 21 3 1 -10 }}
| 32805/32768, 250047/250000, {{monzo| 21 3 1 -10 }}
| [{{val| 525 832 1219 1474 }}]
| {{mapping| 525 832 1219 1474 }}
| +0.0128
| +0.0128
| 0.0940
| 0.0940
Line 52: Line 53:
| 2.3.5.7.11
| 2.3.5.7.11
| 3025/3024, 24057/24010, 32805/32768, 102487/102400
| 3025/3024, 24057/24010, 32805/32768, 102487/102400
| [{{val| 525 832 1219 1474 1816 }}]
| {{mapping| 525 832 1219 1474 1816 }}
| +0.0368
| +0.0368
| 0.0969
| 0.0969
Line 59: Line 60:
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 729/728, 1716/1715, 2200/2197, 3025/3024, 14641/14625
| 729/728, 1716/1715, 2200/2197, 3025/3024, 14641/14625
| [{{val| 525 832 1219 1474 1816 1943 }}]
| {{mapping| 525 832 1219 1474 1816 1943 }}
| +0.0030
| +0.0030
| 0.1164
| 0.1164
| 5.09
| 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 ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per Octave
|-
! Generator<br>(Reduced)
! Periods<br />per 8ve
! Cents<br>(Reduced)
! Generator*
! Associated<br>Ratio
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
Line 78: Line 102:
| 498.29
| 498.29
| 4/3
| 4/3
| [[Helmholtz]]
| [[Helmholtz (temperament)|Helmholtz]]
|-
|-
| 3
| 3
| 218\525<br>(43\525)
| 218\525<br />(43\525)
| 498.29<br>(98.29)
| 498.29<br />(98.29)
| 4/3<br>(18/17)
| 4/3<br />(18/17)
| [[Term]]
| [[Term]]
|-
|-
| 3
| 3
| 109\525<br>(66\525)
| 109\525<br />(66\525)
| 249.14<br>(150.86)
| 249.14<br />(150.86)
| 15/13<br>(12/11)
| 15/13<br />(12/11)
| [[Hemiterm]] (525f)
| [[Hemiterm]] (525f)
|-
|-
| 7
| 7
| 218\525<br>(7\525)
| 218\525<br />(7\525)
| 498.29<br>(16.00)
| 498.29<br />(16.00)
| 4/3<br>(99/98)
| 4/3<br />(99/98)
| [[Septant]]
| [[Septant]]
|-
|-
| 21
| 21
| 256\525<br>(6\525)
| 256\525<br />(6\525)
| 585.14<br>(13.71)
| 585.14<br />(13.71)
| 91875/65536<br>(126/125)
| 91875/65536<br />(126/125)
| [[Akjayland]]
| [[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:Equal divisions of the octave|###]]
[[Category:Akjayland]]
[[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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