525edo: Difference between revisions

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== 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]]. 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; [[1089/1088]], 1275/1274, and [[2025/2023]] in the 17-limit; 2376/2375 in the 19-limit.

~~525'~~s ~~divisors are {{EDOs| 1, 3, 5, 7, 15, 21, 25, 35, 75, 105, 175}}~~.

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 ~~period~~ called [[akjayland]], and the 23-limit extension of akjayland called [[21st-octave temperaments|vasca]], ~~defined~~ 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-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 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 ===

=== Subsets and supersets ===

Since 525 factors into 3 × 52 × 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" | [[Comma list|Comma List]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve ~~stretch~~ (¢)

! rowspan="2" | Optimal 8ve Stretch (¢)

! colspan="2" | Tuning ~~error~~

! colspan="2" | Tuning Error

|-

! [[TE error|Absolute]] (¢)

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| 4.66

|}

* 525edo has lower absolute errors than any previous equal temperaments in the 19- and 23-limit. In the 19-limit it beats [[460edo|460]] and in the 23-limit it beats [[422edo|422]]. It is bettered by [[581edo|581]] in either subgroup.

=== Rank-2 temperaments ===

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| [[Akjayland]]

|-

|21

| 21

|122\525 (22\525)

| 122\525 (22\525)

|278.85 (50.29)

| 278.85 (50.29)

|168/143 (?)

| 168/143 (?)

|[[Vasca]]

| [[Vasca]]

|}

~~[[Category:Equal divisions of the octave|###]]~~

[[Category:Akjayland]]

@@ Line 3: / Line 3: @@
 == Theory ==
-edo 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.
+edo 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; [[1089/1088]], 1275/1274, and [[2025/2023]] in the 17-limit; 2376/2375 in the 19-limit.
-'s divisors are {{EDOs| 1, 3, 5, 7, 15, 21, 25, 35, 75, 105, 175}}.
+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]].
-edo supports 21st-octave period called [[akjayland]], and the 23-limit extension of akjayland called [[21st-octave temperaments|vasca]], defined 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-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]].
+edo 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]].
 === Prime harmonics ===
 {{Harmonics in equal|525|columns=11}}
+=== 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" | [[Comma list|Comma List]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal 8ve <br>stretch (¢)
+! rowspan="2" | Optimal 8ve <br>Stretch (¢)
-! colspan="2" | Tuning error
+! colspan="2" | Tuning Error
 |-
 ! [[TE error|Absolute]] (¢)
@@ Line 75: / Line 78: @@
 | 4.66
 |}
+* 525edo has lower absolute errors than any previous equal temperaments in the 19- and 23-limit. In the 19-limit it beats [[460edo|460]] and in the 23-limit it beats [[422edo|422]]. It is bettered by [[581edo|581]] in either subgroup.
 === Rank-2 temperaments ===
@@ Line 115: / Line 119: @@
 | [[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]]
 |}
-[[Category:Equal divisions of the octave|###]]
 [[Category:Akjayland]]