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Created page with "524 equal division divides the octave into steps of 2.29 cents each. == Theory == {{Harmonics in equal|524}} 524edo is excellent in the 2.7.13.19 subgroup, and good in the no..."
 
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524 equal division divides the octave into steps of 2.29 cents each.
{{Infobox ET}}
{{ED intro}}


== Theory ==
== 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 ===
{{Harmonics in equal|524}}
{{Harmonics in equal|524}}
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 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.
=== 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|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.5
| {{monzo| 1217 -524 }}
| {{mapping| 524 1217 }}
| −0.152
| 0.153
| 6.67
|-
| 2.5.7
| {{monzo| 33 -13 -1 }}, {{monzo| -4 -43 37 }}
| {{mapping| 524 1217 1471 }}
| −0.087
| 0.155
| 6.79
|-
| 2.5.7.11
| 1835008/1830125, {{monzo| 3 7 3 -8 }}, {{monzo| -13 -5 10 -1 }}
| {{mapping| 524 1217 1471 1813 }}
| −0.108
| 0.139
| 6.09
|-
| 2.5.7.11.13
| 1001/1000, 742586/741125, 2097152/2093663, 14201915/14172488
| {{mapping| 524 1217 1471 1813 1939 }}
| −0.082
| 0.135
| 5.88
|-
| 2.5.7.11.13.17
| 1001/1000, 6664/6655, 54080/54043, 147968/147875, 285719/285610
| {{mapping| 524 1217 1471 1813 1939 2142 }}
| −0.084
| 0.122
| 5.37
|}
 
== Scales ==
* Ostara[7]: 62 62 62 107 62 62 107 – [[2L 5s]]
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