524edo: Difference between revisions

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{{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 ==
== 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 ===
{{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 ==
== Regular temperament properties ==
Based on treating 524edo as a no-threes system:
Based on treating 524edo as a no-threes system:
{| class="wikitable center-4 center-5 center-6"
{| 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" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning Error
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
Line 32: Line 29:
| 2.5
| 2.5
| {{monzo| 1217 -524 }}
| {{monzo| 1217 -524 }}
| [{{val| 524 1217 }}]
| {{mapping| 524 1217 }}
| -0.152
| −0.152
| 0.153
| 0.153
| 6.67
| 6.67
Line 39: Line 36:
| 2.5.7
| 2.5.7
| {{monzo| 33 -13 -1 }}, {{monzo| -4 -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
Line 46: Line 43:
| 2.5.7.11
| 2.5.7.11
| 1835008/1830125, {{monzo| 3 7 3 -8 }}, {{monzo| -13 -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
| 6.09
| 6.09
Line 53: Line 50:
| 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
| 5.88
| 5.88
Line 60: Line 57:
| 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
| 5.37
|}
|}


[[Category:Equal divisions of the octave]]
== Scales ==
* Ostara[7]: 62 62 62 107 62 62 107 – [[2L 5s]]

Latest revision as of 22:59, 20 February 2025

← 523edo 524edo 525edo →
Prime factorization 22 × 131
Step size 2.29008 ¢ 
Fifth 307\524 (703.053 ¢)
Semitones (A1:m2) 53:37 (121.4 ¢ : 84.73 ¢)
Dual sharp fifth 307\524 (703.053 ¢)
Dual flat fifth 306\524 (700.763 ¢) (→ 153\262)
Dual major 2nd 89\524 (203.817 ¢)
Consistency limit 5
Distinct consistency limit 5

524 equal divisions of the octave (abbreviated 524edo or 524ed2), also called 524-tone equal temperament (524tet) or 524 equal temperament (524et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 524 equal parts of about 2.29 ¢ each. Each step represents a frequency ratio of 21/524, or the 524th root of 2.

Theory

524edo is only consistent to the 5-odd-limit and harmonic 3 is about halfway between its steps. Otherwise it has a good approximation to harmonics 7, 9, 11, 13, 15, 17, and 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 supports 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

Approximation of odd harmonics in 524edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +1.10 +0.71 -0.12 -0.09 +0.59 -0.07 -0.48 +0.39 +0.20 +0.97 -0.79
Relative (%) +48.0 +31.0 -5.4 -4.1 +25.8 -3.0 -21.1 +16.9 +8.6 +42.6 -34.6
Steps
(reduced) 		831
(307) 		1217
(169) 		1471
(423) 		1661
(89) 		1813
(241) 		1939
(367) 		2047
(475) 		2142
(46) 		2226
(130) 		2302
(206) 		2370
(274)

Subsets and supersets

Since 524 factors into 22 × 131, 524edo has subset edos 2, 4, 131, and 262.

Regular temperament properties

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

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.5 [1217 -524⟩ [⟨524 1217]] −0.152 0.153 6.67
2.5.7 [33 -13 -1⟩, [-4 -43 37⟩ [⟨524 1217 1471]] −0.087 0.155 6.79
2.5.7.11 1835008/1830125, [3 7 3 -8⟩, [-13 -5 10 -1⟩ [⟨524 1217 1471 1813]] −0.108 0.139 6.09
2.5.7.11.13 1001/1000, 742586/741125, 2097152/2093663, 14201915/14172488 [⟨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 [⟨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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