571edo: Difference between revisions

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m Text replacement - "Octave-reduced form, reduced to the first half-octave" to "Octave-reduced form, reduced to the first half-octave"
 
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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| -905 571 }}
| {{Monzo| -905 571 }}
| {{mapping| 571 905 }}
| {{Mapping| 571 905 }}
| +0.0090
| +0.0090
| 0.0090
| 0.0090
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|-
|-
| 2.3.5
| 2.3.5
| {{monzo| 8 14 -13 }}, {{monzo| -69 45 -1 }}
| {{Monzo| 8 14 -13 }}, {{monzo| -69 45 -1 }}
| {{mapping| 571 905 1326 }}
| {{Mapping| 571 905 1326 }}
| −0.0480
| −0.0480
| 0.0810
| 0.0810
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| 2.3.5.7
| 2.3.5.7
| 2401/2400, 14348907/14336000, 29360128/29296875
| 2401/2400, 14348907/14336000, 29360128/29296875
| {{mapping| 571 905 1326 1603 }}
| {{Mapping| 571 905 1326 1603 }}
| −0.0361
| −0.0361
| 0.0731
| 0.0731
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| 2.3.5.7.11
| 2.3.5.7.11
| 2401/2400, 3025/3024, 5632/5625, 14348907/14336000
| 2401/2400, 3025/3024, 5632/5625, 14348907/14336000
| {{mapping| 571 905 1326 1603 1975 }}
| {{Mapping| 571 905 1326 1603 1975 }}
| +0.0119
| +0.0119
| 0.1161
| 0.1161
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| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 1001/1000, 1716/1715, 3025/3024, 4096/4095, 107811/107653
| 1001/1000, 1716/1715, 3025/3024, 4096/4095, 107811/107653
| {{mapping| 571 905 1326 1603 1975 2113 }}
| {{Mapping| 571 905 1326 1603 1975 2113 }}
| +0.0053
| +0.0053
| 0.1070
| 0.1070
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| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 1001/1000, 1089/1088, 1716/1715, 2601/2600, 3025/3024, 4096/4095
| 1001/1000, 1089/1088, 1716/1715, 2601/2600, 3025/3024, 4096/4095
| {{mapping| 571 905 1326 1603 1975 2113 2334 }}
| {{Mapping| 571 905 1326 1603 1975 2113 2334 }}
| +0.0002
| +0.0002
| 0.0999
| 0.0999
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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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| 123\571
| 123\571
| 258.49
| 258.49
| {{monzo| -32 13 5 }}
| {{Monzo| -32 13 5 }}
| [[Lafa]]
| [[Lafa]]
|-
|-
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| 6/5
| 6/5
| [[Parakleismic]] (5-limit)
| [[Parakleismic]] (5-limit)
|-
| 1
| 187\571
| 392.99
| 2744/2187
| [[Emmthird]] (7-limit)
|-
|-
| 1
| 1
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| [[Maviloid]]
| [[Maviloid]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[Normal forms|minimal form]] in parentheses if distinct


== Music ==
== Music ==

Latest revision as of 13:34, 13 March 2026

← 570edo 571edo 572edo →
Prime factorization 571 (prime)
Step size 2.10158 ¢ 
Fifth 334\571 (701.926 ¢)
Semitones (A1:m2) 54:43 (113.5 ¢ : 90.37 ¢)
Consistency limit 9
Distinct consistency limit 9

571 equal divisions of the octave (abbreviated 571edo or 571ed2), also called 571-tone equal temperament (571tet) or 571 equal temperament (571et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 571 equal parts of about 2.1 ¢ each. Each step represents a frequency ratio of 21/571, or the 571st root of 2.

Theory

571et tempers out the parakleisma, [8 14 -13⟩, and the counterschisma, [-69 45 -1⟩, in the 5-limit, as well as the lafa comma, [77 -31 -12⟩; 2401/2400, 14348907/14336000, and 29360128/29296875 in the 7-limit; 3025/3024, 5632/5625, 41503/41472, and 17537553/17500000 in the 11-limit; 1001/1000, 1716/1715, 4096/4095, 17303/17280, and 107811/107653 in the 13-limit, supporting the 13-limit quasiorwell temperament; 1089/1088, 1701/1700, 2431/2430, 2601/2600, 5832/5831 and 7744/7735 in the 17-limit. The 7th harmonic is only 0.0007135 cents sharp in 571edo, as the denominator of a convergent to log27, after 109 and before 2393.

Prime harmonics

Approximation of prime harmonics in 571edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 -0.029 +0.376 +0.001 -0.705 +0.103 +0.123 +0.911 +0.097 +0.195 +0.323
Relative (%) +0.0 -1.4 +17.9 +0.0 -33.5 +4.9 +5.9 +43.3 +4.6 +9.3 +15.4
Steps
(reduced) 		571
(0) 		905
(334) 		1326
(184) 		1603
(461) 		1975
(262) 		2113
(400) 		2334
(50) 		2426
(142) 		2583
(299) 		2774
(490) 		2829
(545)

Subsets and supersets

571edo is the 105th prime edo.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-905 571⟩ [⟨571 905]] +0.0090 0.0090 0.43
2.3.5 [8 14 -13⟩, [-69 45 -1⟩ [⟨571 905 1326]] −0.0480 0.0810 3.85
2.3.5.7 2401/2400, 14348907/14336000, 29360128/29296875 [⟨571 905 1326 1603]] −0.0361 0.0731 3.48
2.3.5.7.11 2401/2400, 3025/3024, 5632/5625, 14348907/14336000 [⟨571 905 1326 1603 1975]] +0.0119 0.1161 5.53
2.3.5.7.11.13 1001/1000, 1716/1715, 3025/3024, 4096/4095, 107811/107653 [⟨571 905 1326 1603 1975 2113]] +0.0053 0.1070 5.09
2.3.5.7.11.13.17 1001/1000, 1089/1088, 1716/1715, 2601/2600, 3025/3024, 4096/4095 [⟨571 905 1326 1603 1975 2113 2334]] +0.0002 0.0999 4.75

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 123\571 258.49 [-32 13 5⟩ Lafa
1 129\571 271.10 90/77 Quasiorwell
1 147\571 315.24 6/5 Parakleismic (5-limit)
1 187\571 392.99 2744/2187 Emmthird (7-limit)
1 237\571 498.07 4/3 Counterschismic
1 248\571 521.19 875/648 Maviloid

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Music

Francium
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