523edo: Difference between revisions

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! rowspan="2" | [[Comma list]]
! rowspan="2" | [[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
|-
|-
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|-
|-
| 2.3
| 2.3
| {{monzo|829 -523}}
| {{Monzo| 829 -523 }}
| {{mapping|523 829}}
| {{Mapping| 523 829 }}
| −0.0468
| −0.0468
| 0.0468
| 0.0468
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|-
|-
| 2.3.5
| 2.3.5
| 1600000/1594323, {{monzo|-101 11 36}}
| 1600000/1594323, {{monzo| -101 11 36 }}
| {{mapping|523 829 1214}}
| {{Mapping| 523 829 1214 }}
| +0.0901
| +0.0901
| 0.1974
| 0.1974
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| 2.3.5.7
| 2.3.5.7
| 2401/2400, 1600000/1594323, 341796875/339738624
| 2401/2400, 1600000/1594323, 341796875/339738624
| {{mapping|523 829 1214 1468}}
| {{Mapping| 523 829 1214 1468 }}
| +0.1180
| +0.1180
| 0.1776
| 0.1776
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|-
|-
| 2.3.5.7.11
| 2.3.5.7.11
| 6250/6237, 2401/2400, 19712/19683, 166375/165888
| 2401/2400, 6250/6237, 19712/19683, 166375/165888
| {{mapping|523 829 1214 1468 1809}}
| {{Mapping| 523 829 1214 1468 1809 }}
| +0.1319
| +0.1319
| 0.1613
| 0.1613
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|-
|-
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 2080/2079, 625/624, 4459/4455, 19712/19683, 50193/50176
| 625/624, 2080/2079, 2401/2400, 19712/19683, 50193/50176
| {{mapping|523 829 1214 1468 1809 1935}}
| {{Mapping| 523 829 1214 1468 1809 1935 }}
| +0.1440
| +0.1440
| 0.1497
| 0.1497
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|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|-
! Periods<br />per 8ve
! Periods<br>per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br />ratio*
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
|-
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| 339.58
| 339.58
| 243/200
| 243/200
| [[Amity]]
| [[Amity]] (5-limit)
|}
|}
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]]


== Music ==
== Music ==

Latest revision as of 07:57, 20 August 2026

← 522edo 523edo 524edo →
Prime factorization 523 (prime)
Step size 2.29446 ¢ 
Fifth 306\523 (702.103 ¢)
Semitones (A1:m2) 50:39 (114.7 ¢ : 89.48 ¢)
Consistency limit 15
Distinct consistency limit 15

523 equal divisions of the octave (abbreviated 523edo or 523ed2), also called 523-tone equal temperament (523tet) or 523 equal temperament (523et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 523 equal parts of about 2.29 ¢ each. Each step represents a frequency ratio of 21/523, or the 523rd root of 2.

Theory

523et tempers out 2401/2400 in the 7-limit; 56723625/56689952, 26214400/26198073, 78675968/78594219, 759375/758912, 496125/495616, 6250/6237, 42875/42768, 19712/19683, 1479016/1476225, 16808715/16777216, 3294225/3294172, 43923/43904, 102487/102400, and 322102/321489 in the 11-limit.

Prime harmonics

Approximation of prime harmonics in 523edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.15 -0.85 -0.57 -0.65 -0.76 +0.59 +0.77 +0.41 +0.63 -0.10
Relative (%) +0.0 +6.5 -36.8 -24.7 -28.3 -33.0 +25.7 +33.4 +17.7 +27.6 -4.5
Steps
(reduced) 		523
(0) 		829
(306) 		1214
(168) 		1468
(422) 		1809
(240) 		1935
(366) 		2138
(46) 		2222
(130) 		2366
(274) 		2541
(449) 		2591
(499)

Subsets and supersets

523edo is the 99th prime edo.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [829 -523⟩ [⟨523 829]] −0.0468 0.0468 2.04
2.3.5 1600000/1594323, [-101 11 36⟩ [⟨523 829 1214]] +0.0901 0.1974 8.60
2.3.5.7 2401/2400, 1600000/1594323, 341796875/339738624 [⟨523 829 1214 1468]] +0.1180 0.1776 7.74
2.3.5.7.11 2401/2400, 6250/6237, 19712/19683, 166375/165888 [⟨523 829 1214 1468 1809]] +0.1319 0.1613 7.03
2.3.5.7.11.13 625/624, 2080/2079, 2401/2400, 19712/19683, 50193/50176 [⟨523 829 1214 1468 1809 1935]] +0.1440 0.1497 6.52

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 37\523 84.89 21/20 Amicable
1 52\523 119.31 15/14 Septidiasemi
1 148\523 339.58 243/200 Amity (5-limit)

* In minimal-generator form

Music

Francium
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