525edo: Difference between revisions

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== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{{comma basis begin}}
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal 8ve <br>Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
|-
| 2.3
| 2.3
Line 67: Line 59:
| 729/728, 1089/1088, 1275/1274, 1716/1715, 2025/2023, 2200/2197
| 729/728, 1089/1088, 1275/1274, 1716/1715, 2025/2023, 2200/2197
| {{mapping| 525 832 1219 1474 1816 1943 2146 }}
| {{mapping| 525 832 1219 1474 1816 1943 2146 }}
| -0.0040
| &minus;0.0040
| 0.1091
| 0.1091
| 4.77
| 4.77
Line 81: Line 73:
| 729/728, 1089/1088, 1197/1196, 1275/1274, 1496/1495, 1716/1715, 2024/2023, 2025/2023
| 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 }}
| {{mapping| 525 832 1219 1474 1816 1943 2146 2230 2375 }}
| -0.0007
| &minus;0.0007
| 0.1029
| 0.1029
| 4.50
| 4.50
|}
{{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 [[460edo|460]] and is bettered by [[566edo|566g]]. In the 23-limit it beats [[422edo|422]] and is bettered by [[581edo|581]].  
* 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"
{{rank-2 begin}}
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>Ratio*
! Temperaments
|-
|-
| 1
| 1
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|-
|-
| 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
| 21
| 122\525<br>(22\525)
| 122\525<br />(22\525)
| 278.85<br>(50.29)
| 278.85<br />(50.29)
| 168/143<br>(?)
| 168/143<br />(?)
| [[Vasca]]
| [[Vasca]]
|}
{{rank-2 end}}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
{{orf}}


[[Category:Akjayland]]
[[Category:Akjayland]]

Revision as of 04:48, 16 November 2024

← 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

Template:EDO intro

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

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

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

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