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]]. 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.

~~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. ===

=== Fractional-octave temperaments ===

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~~.

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. ~~In addition to splitting octave into 21 parts~~, 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 ===

=== 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" | [[Mapping]]

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

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

! colspan="2" | Tuning error

|-

Line 28:

Line 32:

| 2.3

| {{monzo| 512 -323 }}

| [{{~~val~~| 525 832 }}]

| {{mapping| 525 832 }}

| +0.0759

| 0.0759

Line 35:

Line 39:

| 2.3.5

| 32805/32768, {{monzo| 8 77 -56 }}

| [{{~~val~~| 525 832 1219 }}]

| {{mapping| 525 832 1219 }}

| +0.0546

| 0.0689

Line 42:

Line 46:

| 2.3.5.7

| 32805/32768, 250047/250000, {{monzo| 21 3 1 -10 }}

| [{{~~val~~| 525 832 1219 1474 }}]

| {{mapping| 525 832 1219 1474 }}

| +0.0128

| 0.0940

Line 49:

Line 53:

| 2.3.5.7.11

| 3025/3024, 24057/24010, 32805/32768, 102487/102400

| [{{~~val~~| 525 832 1219 1474 1816 }}]

| {{mapping| 525 832 1219 1474 1816 }}

| +0.0368

| 0.0969

Line 56:

Line 60:

| 2.3.5.7.11.13

| 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

| 0.1164

Line 63:

Line 67:

| 2.3.5.7.11.13.17

| 729/728, 1089/1088, 1275/1274, 1716/1715, 2025/2023, 2200/2197

| [{{~~val~~| 525 832 1219 1474 1816 1943 2146 }}]

| {{mapping| 525 832 1219 1474 1816 1943 2146 }}

| -0.0040

| −0.0040

| 0.1091

| 4.77

Line 70:

Line 74:

| 2.3.5.7.11.13.17.19

| 729/728, 1089/1088, 1275/1274, 1716/1715, 2025/2023, 2200/2197, 2376/2375

| [{{~~val~~| 525 832 1219 1474 1816 1943 2146 2230 }}]

| {{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"

|+Table of rank-2 temperaments by generator

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

! Periods per ~~Octave~~

|-

! Generator~~ (Reduced)~~

! Periods per 8ve

! Cents~~ (Reduced)~~

! Generator*

! Associated ~~Ratio~~

! Cents*

! Associated ratio*

! Temperaments

|-

Line 89:

Line 102:

| 498.29

| 4/3

| [[Helmholtz]]

| [[Helmholtz (temperament)|Helmholtz]]

|-

| 3

| 218\525 (43\525)

| 218\525 (43\525)

| 498.29 (98.29)

| 498.29 (98.29)

| 4/3 (18/17)

| 4/3 (18/17)

| [[Term]]

|-

| 3

| 109\525 (66\525)

| 109\525 (66\525)

| 249.14 (150.86)

| 249.14 (150.86)

| 15/13 (12/11)

| 15/13 (12/11)

| [[Hemiterm]] (525f)

|-

| 7

| 218\525 (7\525)

| 218\525 (7\525)

| 498.29 (16.00)

| 498.29 (16.00)

| 4/3 (99/98)

| 4/3 (99/98)

| [[Septant]]

|-

| 21

| 256\525 (6\525)

| 256\525 (6\525)

| 585.14 (13.71)

| 585.14 (13.71)

| 91875/65536 (126/125)

| 91875/65536 (126/125)

| [[Akjayland]] / [[~~vasca~~]]

| [[Akjayland]]

|-

| 21

| 122\525 (22\525)

| 278.85 (50.29)

| 168/143 (?)

| [[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]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|525}}
+{{ED intro}}
 == 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]]. 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.
-'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. ===
+=== Fractional-octave temperaments ===
-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.
+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. In addition to splitting octave into 21 parts, 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 {{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 ===
-{{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 ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
 ! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal 8ve <br>stretch (¢)
+! rowspan="2" | Optimal<br />8ve stretch (¢)
 ! colspan="2" | Tuning error
 |-
@@ Line 28: / Line 32: @@
 | 2.3
 | {{monzo| 512 -323 }}
-| [{{val| 525 832 }}]
+| {{mapping| 525 832 }}
 | +0.0759
 | 0.0759
@@ Line 35: / Line 39: @@
 | 2.3.5
 | 32805/32768, {{monzo| 8 77 -56 }}
-| [{{val| 525 832 1219 }}]
+| {{mapping| 525 832 1219 }}
 | +0.0546
 | 0.0689
@@ Line 42: / Line 46: @@
 | 2.3.5.7
 | 32805/32768, 250047/250000, {{monzo| 21 3 1 -10 }}
-| [{{val| 525 832 1219 1474 }}]
+| {{mapping| 525 832 1219 1474 }}
 | +0.0128
 | 0.0940
@@ Line 49: / Line 53: @@
 | 2.3.5.7.11
 | 3025/3024, 24057/24010, 32805/32768, 102487/102400
-| [{{val| 525 832 1219 1474 1816 }}]
+| {{mapping| 525 832 1219 1474 1816 }}
 | +0.0368
 | 0.0969
@@ Line 56: / Line 60: @@
 | 2.3.5.7.11.13
 | 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
 | 0.1164
@@ Line 63: / Line 67: @@
 | 2.3.5.7.11.13.17
 | 729/728, 1089/1088, 1275/1274, 1716/1715, 2025/2023, 2200/2197
-| [{{val| 525 832 1219 1474 1816 1943 2146 }}]
+| {{mapping| 525 832 1219 1474 1816 1943 2146 }}
-| -0.0040
+| −0.0040
 | 0.1091
 | 4.77
@@ Line 70: / Line 74: @@
 | 2.3.5.7.11.13.17.19
 | 729/728, 1089/1088, 1275/1274, 1716/1715, 2025/2023, 2200/2197, 2376/2375
-| [{{val| 525 832 1219 1474 1816 1943 2146 2230 }}]
+| {{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"
-|+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
 |-
@@ Line 89: / Line 102: @@
 | 498.29
 | 4/3
-| [[Helmholtz]]
+| [[Helmholtz (temperament)|Helmholtz]]
 |-
 | 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]]
 |-
 | 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)
 |-
 | 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]]
 |-
 | 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]] / [[vasca]]
+| [[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]]