524edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
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.
{{EDO intro|524}}


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


== Scales ==
=== Odd harmonics ===
 
{{Harmonics in equal|524}}
* Ostara[7]: 62 62 62 107 62 62 107 - [[2L 5s]]


== 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" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! 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 24:
| 2.5
| 2.5
| {{monzo| 1217 -524 }}
| {{monzo| 1217 -524 }}
| [{{val| 524 1217 }}]
| {{mapping| 524 1217 }}
| -0.152
| -0.152
| 0.153
| 0.153
Line 39: Line 31:
| 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
Line 46: Line 38:
| 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
Line 53: Line 45:
| 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
Line 60: Line 52:
| 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
Line 66: Line 58:
|}
|}


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
== Scales ==
* Ostara[7]: 62 62 62 107 62 62 107 - [[2L 5s]]

Revision as of 11:46, 29 October 2023

← 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

Template:EDO intro

Theory

524edo is only consistent to the 5-odd-limit and harmonic 3 is about halfway between its steps, but 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)

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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