524edo: Difference between revisions

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{{Infobox ET

~~| Prime factorization = 2<sup>2</sup> × 131~~

~~| Step size = 2.29008¢~~

~~| Fifth = 307\524 (703.0534¢)~~

}}

The '''524 equal divisions of the octave (524edo)''', or '''524(-tone) equal temperament (524-tet, 524et)''' when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 524 [[equal]] parts of about 2.29 [[cent]]s each.

== Theory ==

~~{{Harmonics in equal|524}}~~

524edo is only [[consistent]] to the [[5-odd-limit]] and [[3/1|harmonic 3]] is about halfway between its steps. Otherwise it has a good approximation to harmonics [[7/1|7]], [[9/1|9]], [[11/1|11]], [[13/1|13]], [[15/1|15]], [[17/1|17]], and [[19/1|19]], making it suitable for a 2.9.15.7.11.13.17.19 [[subgroup]] interpretation. The 2.9.7.13.19 subgroup is particularly good, where it [[support]]s [[Eliora]]'s [[ostara]], the 93 & 524 temperament. The generator 293\524 represents the [[28/19]] interval in the 2.7.19 subgroup and it serves as ostara's generator in the no-threes 19-limit. Being around 671 cents, can also be used to as a generator for [[mavila]] or [[pelog]].

524edo is ~~excellent in~~ the 2.7.13.19 subgroup, and ~~good~~ in the no-threes 19-limit. ~~In the 3-limit~~, ~~it is wise~~ to ~~treat 524edo~~ as a ~~dual-fifth system. The minor fifth, 306\524 reduces to 153\262, as such, on this val 524edo is contorted~~.

~~524 years is~~ the ~~length of a calendar leap week cycle with 93 leap weeks~~, ~~creating a 93~~ out ~~of 524 maximum evenness scale with the generator 293\524, represented by the 93 & 524 temperament~~.

In the 13-limit, 524edo tempers out 1001/1000 and 6664/6655.

~~In addition, both 93 and 524 represent well the 13:17:19~~ harmonics~~. The corresponding comma list~~ in ~~the 2.7.13.17.19 subgroup is 16807/16796, 157339/157216, 47071232/47045881. Eliora proposes that this temperament be named '''ostara''', after the feast of the spring equinox, which 93\~~524 ~~leap week rule approximates well. Other spring equinoctial temperaments, such as 41 & 231 (Dee leap week), 97 & 400 (Gregorian leap day), and 52 & 293 (Sym454) already have their identities and names.~~

=== Odd harmonics ===

~~The generator 293\~~524, ~~being around 671 cents~~, ~~can also be used to as a generator for [[mavila]] or [[pelog]].~~

=== Subsets and supersets ===

Since 524 factors into {{factorization|524}}, 524edo has subset edos {{EDOs| 2, 4, 131, and 262 }}.

~~In the 13-limit~~, ~~524edo tempers out 1001/1000~~ and ~~6664/6655~~.

== Regular temperament properties ==

Based on treating 524edo as a no-threes system:

{| class="wikitable center-4 center-5 center-6"

! rowspan="2" | Subgroup

|-

! rowspan="2" | [[Comma list]]

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

! rowspan="2" | [[Comma list|Comma List]]

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

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

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

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

! colspan="2" | Tuning Error

|-

! [[TE error|Absolute]] (¢)

Line 32:

Line 29:

| 2.5

| {{monzo| 1217 -524 }}

| [{{~~val~~| 524 1217 }}]

| {{mapping| 524 1217 }}

| -0.152

| −0.152

| 0.153

| 6.67

Line 39:

Line 36:

| 2.5.7

| {{monzo| 33 -13 -1 }}, {{monzo| -4 -43 37 }}

| [{{~~val~~|524 1217 1471}}]

| {{mapping| 524 1217 1471 }}

| -0.087

| −0.087

| 0.155

| 6.79

Line 46:

Line 43:

| 2.5.7.11

| 1835008/1830125, {{monzo| 3 7 3 -8 }}, {{monzo| -13 -5 10 -1 }}

| [{{~~val~~| 524 1217 1471 1813 }}]

| {{mapping| 524 1217 1471 1813 }}

| -0.108

| −0.108

| 0.139

| 6.09

Line 53:

Line 50:

| 2.5.7.11.13

| 1001/1000, 742586/741125, 2097152/2093663, 14201915/14172488

| [{{~~val~~| 524 1217 1471 1813 1939 }}]

| {{mapping| 524 1217 1471 1813 1939 }}

| -0.082

| −0.082

| 0.135

| 5.88

Line 60:

Line 57:

| 2.5.7.11.13.17

| 1001/1000, 6664/6655, 54080/54043, 147968/147875, 285719/285610

| [{{~~val~~| 524 1217 1471 1813 1939 2142 }}]

| {{mapping| 524 1217 1471 1813 1939 2142 }}

| -0.084

| −0.084

| 0.122

| 5.37

|}

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

== Scales ==

* Ostara[7]: 62 62 62 107 62 62 107 – [[2L 5s]]

@@ Line 1: / Line 1: @@
-{{Infobox ET
+{{Infobox ET}}
-| Prime factorization = 2<sup>2</sup> × 131
+{{ED intro}}
-| Step size = 2.29008¢
-| Fifth = 307\524 (703.0534¢)
-}}
-The '''524 equal divisions of the octave (524edo)''', or '''524(-tone) equal temperament (524-tet, 524et)''' when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 524 [[equal]] parts of about 2.29 [[cent]]s each.
 == Theory ==
-{{Harmonics in equal|524}}
+edo is only [[consistent]] to the [[5-odd-limit]] and [[3/1|harmonic 3]] is about halfway between its steps. Otherwise it has a good approximation to harmonics [[7/1|7]], [[9/1|9]], [[11/1|11]], [[13/1|13]], [[15/1|15]], [[17/1|17]], and [[19/1|19]], making it suitable for a 2.9.15.7.11.13.17.19 [[subgroup]] interpretation. The 2.9.7.13.19 subgroup is particularly good, where it [[support]]s [[Eliora]]'s [[ostara]], the 93 & 524 temperament. The generator 293\524 represents the [[28/19]] interval in the 2.7.19 subgroup and it serves as ostara's generator in the no-threes 19-limit. Being around 671 cents, can also be used to as a generator for [[mavila]] or [[pelog]].
-edo is excellent in the 2.7.13.19 subgroup, and good in the no-threes 19-limit. In the 3-limit, it is wise to treat 524edo as a dual-fifth system. The minor fifth, 306\524 reduces to 153\262, as such, on this val 524edo is contorted.
-years is the length of a calendar leap week cycle with 93 leap weeks, creating a 93 out of 524 maximum evenness scale with the generator 293\524, represented by the 93 & 524 temperament.
+In the 13-limit, 524edo tempers out 1001/1000 and 6664/6655.
-In addition, both 93 and 524 represent well the 13:17:19 harmonics. The corresponding comma list in the 2.7.13.17.19 subgroup is 16807/16796, 157339/157216, 47071232/47045881. Eliora proposes that this temperament be named '''ostara''', after the feast of the spring equinox, which 93\524 leap week rule approximates well. Other spring equinoctial temperaments, such as 41 & 231 (Dee leap week), 97 & 400 (Gregorian leap day), and 52 & 293 (Sym454) already have their identities and names.
+=== Odd harmonics ===
+{{Harmonics in equal|524}}
-The generator 293\524, being around 671 cents, can also be used to as a generator for [[mavila]] or [[pelog]].
+=== Subsets and supersets ===
+Since 524 factors into {{factorization|524}}, 524edo has subset edos {{EDOs| 2, 4, 131, and 262 }}.
-In the 13-limit, 524edo tempers out 1001/1000 and 6664/6655.
 == Regular temperament properties ==
 Based on treating 524edo as a no-threes system:
 {| class="wikitable center-4 center-5 center-6"
-! rowspan="2" | Subgroup
+|-
-! rowspan="2" | [[Comma list]]
+! rowspan="2" | [[Subgroup]]
+! rowspan="2" | [[Comma list|Comma List]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br>8ve stretch (¢)
+! rowspan="2" | Optimal<br>8ve Stretch (¢)
-! colspan="2" | Tuning error
+! colspan="2" | Tuning Error
 |-
 ! [[TE error|Absolute]] (¢)
@@ Line 32: / Line 29: @@
 | 2.5
 | {{monzo| 1217 -524 }}
-| [{{val| 524 1217 }}]
+| {{mapping| 524 1217 }}
-| -0.152
+| −0.152
 | 0.153
 | 6.67
@@ Line 39: / Line 36: @@
 | 2.5.7
 | {{monzo| 33 -13 -1 }}, {{monzo| -4 -43 37 }}
-| [{{val|524 1217 1471}}]
+| {{mapping| 524 1217 1471 }}
-| -0.087
+| −0.087
 | 0.155
 | 6.79
@@ Line 46: / Line 43: @@
 | 2.5.7.11
 | 1835008/1830125, {{monzo| 3 7 3 -8 }}, {{monzo| -13 -5 10 -1 }}
-| [{{val| 524 1217 1471 1813 }}]
+| {{mapping| 524 1217 1471 1813 }}
-| -0.108
+| −0.108
 | 0.139
 | 6.09
@@ Line 53: / Line 50: @@
 | 2.5.7.11.13
 | 1001/1000, 742586/741125, 2097152/2093663, 14201915/14172488
-| [{{val| 524 1217 1471 1813 1939 }}]
+| {{mapping| 524 1217 1471 1813 1939 }}
-| -0.082
+| −0.082
 | 0.135
 | 5.88
@@ Line 60: / Line 57: @@
 | 2.5.7.11.13.17
 | 1001/1000, 6664/6655, 54080/54043, 147968/147875, 285719/285610
-| [{{val| 524 1217 1471 1813 1939 2142 }}]
+| {{mapping| 524 1217 1471 1813 1939 2142 }}
-| -0.084
+| −0.084
 | 0.122
 | 5.37
 |}
-[[Category:Equal divisions of the octave]]
+== Scales ==
+* Ostara[7]: 62 62 62 107 62 62 107 – [[2L 5s]]