525edo: Difference between revisions
ArrowHead294 (talk | contribs) mNo edit summary |
m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" |
||
| (6 intermediate revisions by 4 users not shown) | |||
| Line 1: | Line 1: | ||
{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
525edo is [[ | 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. | It allows [[essentially tempered chord]]s of [[squbemic chords]] and [[petrmic chords]] in the 13-odd-limit. | ||
| Line 10: | Line 10: | ||
It supports the 35th-octave temperament [[35th-octave temperaments#Tritonopodismic|tritonopodismic]]. | 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 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]]. | 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 === | === Prime harmonics === | ||
{{Harmonics in equal|525 | {{Harmonics in equal|525}} | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
Since 525 factors into | 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" | ||
|- | |||
! 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 | | 2.3 | ||
| Line 59: | Line 68: | ||
| 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 | ||
| 0.1091 | | 0.1091 | ||
| 4.77 | | 4.77 | ||
| Line 73: | Line 82: | ||
| 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 | ||
| 0.1029 | | 0.1029 | ||
| 4.50 | | 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]]. | * 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" | ||
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |||
|- | |||
! Periods<br />per 8ve | |||
! Generator* | |||
! Cents* | |||
! Associated<br />ratio* | |||
! Temperaments | |||
|- | |- | ||
| 1 | | 1 | ||
| Line 86: | Line 102: | ||
| 498.29 | | 498.29 | ||
| 4/3 | | 4/3 | ||
| [[Helmholtz]] | | [[Helmholtz (temperament)|Helmholtz]] | ||
|- | |- | ||
| 3 | | 3 | ||
| Line 117: | Line 133: | ||
| 168/143<br />(?) | | 168/143<br />(?) | ||
| [[Vasca]] | | [[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]] | ||