524edo: Difference between revisions

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~~524 equal division divides the octave into steps of 2.29 cents each.~~

== Theory ==

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]].

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

=== Odd harmonics ===

~~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.~~

524 years is the length of a calendar leap week cycle with 93 leap weeks, creating a 93 out of 524 maximum evenness scale, represented by the 93 & 524 temperament. 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, 97 & 400, and 52 & 293 already have their identities and names.

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

=== Subsets and supersets ===

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

== 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" |[[Mapping]]~~

~~! rowspan="2" |Optimal~~

~~8ve stretch (¢)~~

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

|-

![[TE error|Absolute]] (¢)

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

![[TE simple badness|Relative]] (%)

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

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

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

! colspan="2" | Tuning Error

|-

! [[TE error|Absolute]] (¢)

! [[TE simple badness|Relative]] (%)

|-

|2.5

| 2.5

|{{~~Monzo~~|1217 0 -524}}

| {{monzo| 1217 -524 }}

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

| {{mapping| 524 1217 }}

| -0.152

| −0.152

|0.153

| 0.153

|6.66

| 6.67

|-

|2.5.7

| 2.5.7

|{{~~Monzo~~|33 0 -13 -1}}, {{~~Monzo~~|-4 0 -43 37}}

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

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

| {{mapping| 524 1217 1471 }}

| -0.087

| −0.087

|0.155

| 0.155

|6.79

| 6.79

|-

|2.5.7.11

| 2.5.7.11

|1835008/1830125, {{~~Monzo~~|3 0 7 3 -8}}, {{~~Monzo~~|-13 0 -5 10 -1}}

| 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

| 0.139

|6.07

| 6.09

|-

|2.5.7.11.13

| 2.5.7.11.13

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

| 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

| 0.135

|5.87

| 5.88

|-

|2.5.7.11.13.17

| 2.5.7.11.13.17

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

| 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

| 0.122

|

| 5.37

|}

== Scales ==

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

@@ Line 1: / Line 1: @@
-equal division divides the octave into steps of 2.29 cents each.
+{{Infobox ET}}
+{{ED intro}}
 == Theory ==
+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]].
+In the 13-limit, 524edo tempers out 1001/1000 and 6664/6655.
+=== Odd harmonics ===
 {{Harmonics in equal|524}}
-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.
-years is the length of a calendar leap week cycle with 93 leap weeks, creating a 93 out of 524 maximum evenness scale, represented by the 93 & 524 temperament. 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, 97 & 400, and 52 & 293 already have their identities and names.
-In the 13-limit, 524edo tempers out 1001/1000 and 6664/6655.
+=== Subsets and supersets ===
+Since 524 factors into {{factorization|524}}, 524edo has subset edos {{EDOs| 2, 4, 131, and 262 }}.
 == 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" |[[Mapping]]
-! rowspan="2" |Optimal
-ve stretch (¢)
-! colspan="2" |Tuning error
 |-
-![[TE error|Absolute]] (¢)
+! rowspan="2" | [[Subgroup]]
-![[TE simple badness|Relative]] (%)
+! rowspan="2" | [[Comma list|Comma List]]
+! rowspan="2" | [[Mapping]]
+! rowspan="2" | Optimal<br>8ve Stretch (¢)
+! colspan="2" | Tuning Error
+|-
+! [[TE error|Absolute]] (¢)
+! [[TE simple badness|Relative]] (%)
 |-
-|2.5
+| 2.5
-|{{Monzo|1217 0 -524}}
+| {{monzo| 1217 -524 }}
-|[{{val|524 1217}}]
+| {{mapping| 524 1217 }}
-| -0.152
+| −0.152
-|0.153
+| 0.153
-|6.66
+| 6.67
 |-
-|2.5.7
+| 2.5.7
-|{{Monzo|33 0 -13 -1}}, {{Monzo|-4 0 -43 37}}
+| {{monzo| 33 -13 -1 }}, {{monzo| -4 -43 37 }}
-|[{{val|524 1217 1471}}]
+| {{mapping| 524 1217 1471 }}
-| -0.087
+| −0.087
-|0.155
+| 0.155
-|6.79
+| 6.79
 |-
-|2.5.7.11
+| 2.5.7.11
-|1835008/1830125, {{Monzo|3 0 7 3 -8}}, {{Monzo|-13 0 -5 10 -1}}
+| 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
+| 0.139
-|6.07
+| 6.09
 |-
-|2.5.7.11.13
+| 2.5.7.11.13
-|1001/1000, 742586/741125, 2097152/2093663, 14201915/14172488
+| 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
+| 0.135
-|5.87
+| 5.88
 |-
-|2.5.7.11.13.17
+| 2.5.7.11.13.17
-|1001/1000, 6664/6655, 54080/54043, 147968/147875, 285719/285610
+| 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
+| 0.122
-|
+| 5.37
 |}
+== Scales ==
+* Ostara[7]: 62 62 62 107 62 62 107 – [[2L 5s]]