571edo: Difference between revisions

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== Theory ==
== Theory ==
571edo [[tempering out|tempers out]] the [[parakleisma]], {{monzo| 8 14 -13 }}, and the [[counterschisma]], {{monzo| -69 45 -1 }}, in the [[5-limit]], as well as the lafa comma, {{monzo| 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 log<sub>2</sub>7, after [[109edo|109]] and before [[2393edo|2393]].
The equal temperament [[tempering out|tempers out]] the [[parakleisma]], {{monzo| 8 14 -13 }}, and the [[counterschisma]], {{monzo| -69 45 -1 }}, in the [[5-limit]], as well as the lafa comma, {{monzo| 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 log<sub>2</sub>7, after [[109edo|109]] and before [[2393edo|2393]].
 
571edo is the 105th [[prime edo]].


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|571|columns=11}}
{{Harmonics in equal|571|columns=11}}
=== Subsets and supersets ===
571edo is the 105th [[prime edo]].


== Regular temperament properties ==
== Regular temperament properties ==
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| 2.3
| 2.3
| {{monzo| -905 571 }}
| {{monzo| -905 571 }}
| [{{val| 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 }}
| [{{val| 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
| [{{val| 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
| [{{val| 571 905 1326 1603 1975 }}]
| {{mapping| 571 905 1326 1603 1975 }}
| +0.0119
| +0.0119
| 0.1161
| 0.1161
Line 51: Line 52:
| 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
| [{{val| 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
| [{{val| 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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{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+Table of rank-2 temperaments by generator
! Periods<br>per Octave
! Periods<br>per 8ve
! Generator<br>(Reduced)
! Generator*
! Cents<br>(Reduced)
! Cents*
! Associated<br>Ratio
! Associated<br>Ratio*
! Temperaments
! Temperaments
|-
|-
Line 103: Line 104:
| [[Maviloid]]
| [[Maviloid]]
|}
|}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Prime EDO]]
[[Category:Quasiorwell]]
[[Category:Quasiorwell]]

Revision as of 08:54, 29 October 2023

← 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

Template:EDO intro

Theory

The equal temperament 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)
1 237\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 it is distinct

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