574edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|574}}
{{ED intro}}


== Theory ==
== Theory ==
574 = 14 × 41, and 574edo shares the [[harmonic]] [[3/1|3]] with [[41edo]]. Unfortunately, it is only [[consistent]] to the [[5-odd-limit]], and the [[patent val]] and the 574d [[val]] are about equally viable.  
574edo shares the [[harmonic]] [[3/1|3]] with [[41edo]]. Unfortunately, it is only [[consistent]] to the [[5-odd-limit]], and the [[patent val]] and the 574d [[val]] are about equally viable.  


The 574d val [[tempering out|tempers out]] 4375/4374 ([[ragisma]]), 29360128/29296875 (quasiorwellisma), and 40500000/40353607 in the 7-limit; [[3025/3024]], [[5632/5625]], [[9801/9800]], and [[19712/19683]] in the 11-limit; [[676/675]], [[4225/4224]], and [[10648/10647]] in the 13-limit.  
The 574d val [[tempering out|tempers out]] 4375/4374 ([[ragisma]]), 29360128/29296875 (quasiorwellisma), and 40500000/40353607 in the 7-limit; [[3025/3024]], [[5632/5625]], [[9801/9800]], and [[19712/19683]] in the 11-limit; [[676/675]], [[4225/4224]], and [[10648/10647]] in the 13-limit.  


The patent val tempers out 2100875/2097152 ([[rainy comma]]), 67108864/66976875 ([[vishnuzma]]), 49009212/48828125, and 78125000/78121827 ([[euzenius comma]]) in the 7-limit; 5632/5625, 42875/42768, 117649/117128, [[131072/130977]], 160083/160000, 161280/161051, 166698/166375, 391314/390625, and [[532400/531441]] in the 11-limit; 676/675, [[1001/1000]], [[2080/2079]], [[4096/4095]], and 4225/4224 in the 13-limit.  
The patent val tempers out 2100875/2097152 ([[rainy comma]]), 6115295232/6103515625 ([[vishnuzma]]), 49009212/48828125, and 78125000/78121827 ([[euzenius comma]]) in the 7-limit; 5632/5625, 42875/42768, 117649/117128, [[131072/130977]], 160083/160000, 161280/161051, 166698/166375, 391314/390625, and [[532400/531441]] in the 11-limit; 676/675, [[1001/1000]], [[2080/2079]], [[4096/4095]], and 4225/4224 in the 13-limit.  


=== Prime harmonics ===
=== Prime harmonics ===
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=== Subsets and supersets ===
=== Subsets and supersets ===
Since 574 factors into 2 × 7 × 41, 574edo has subset edos {{EDOs| 2, 7, 14, 41, 82, and 287 }}.  
Since 574 factors into 2 × 7 × 41, 574edo has subset edos {{EDOs| 2, 7, 14, 41, 82, and 287 }}.


== Regular temperament properties ==
== Regular temperament properties ==
{| 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|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
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
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| {{monzo| 23 6 -14 }}, {{monzo| 42 -47 14 }}
| {{monzo| 23 6 -14 }}, {{monzo| 42 -47 14 }}
| {{mapping| 574 910 1333 }}
| {{mapping| 574 910 1333 }}
| -0.1658
| −0.1658
| 0.1260
| 0.1260
| 6.03
| 6.03
|-
|- style="border-top: double;"
| style="border-top: double;" | 2.3.5.7
| 2.3.5.7
| style="border-top: double;" | 4375/4374, 29360128/29296875, 40500000/40353607
| 4375/4374, 29360128/29296875, 40500000/40353607
| style="border-top: double;" | {{mapping| 574 910 1333 1612 }} (574d)
| {{mapping| 574 910 1333 1612 }} (574d)
| style="border-top: double;" | -0.2320
| −0.2320
| style="border-top: double;" | 0.1583
| 0.1583
| style="border-top: double;" | 7.57
| 7.57
|-
|-
| 2.3.5.7.11
| 2.3.5.7.11
| 3025/3024, 4375/4374, 5632/5625, 40500000/40353607
| 3025/3024, 4375/4374, 5632/5625, 40500000/40353607
| {{mapping| 574 910 1333 1612 }} (574d)
| {{mapping| 574 910 1333 1612 1986 }} (574d)
| -0.2202
| −0.2202
| 0.1435
| 0.1435
| 6.87
| 6.87
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| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 676/675, 3025/3024, 4225/4224, 4375/4374, 41600/41503
| 676/675, 3025/3024, 4225/4224, 4375/4374, 41600/41503
| {{mapping| 574 910 1333 1612 }} (574d)
| {{mapping| 574 910 1333 1612 1986 2124 }} (574d)
| -0.1786
| −0.1786
| 0.1607
| 0.1607
| 7.69
| 7.69
|-
|- style="border-top: double;"
| style="border-top: double;" | 2.3.5.7
| 2.3.5.7
| style="border-top: double;" | 10976/10935, 2100875/2097152, 49009212/48828125
| 10976/10935, 2100875/2097152, 49009212/48828125
| style="border-top: double;" | {{mapping| 574 910 1333 1611 }} (574)
| {{mapping| 574 910 1333 1611 }} (574)
| style="border-top: double;" | -0.0459
| −0.0459
| style="border-top: double;" | 0.2347
| 0.2347
| style="border-top: double;" | 11.23
| 11.23
|-
|-
| 2.3.5.7.11
| 2.3.5.7.11
| 5632/5625, 10976/10935, 131072/130977, 166698/166375
| 5632/5625, 10976/10935, 131072/130977, 166698/166375
| {{mapping| 574 910 1333 1611 1986 }} (574)
| {{mapping| 574 910 1333 1611 1986 }} (574)
| -0.0713
| −0.0713
| 0.2160
| 0.2160
| 10.33
| 10.33
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| 676/675, 1001/1000, 4096/4095, 10976/10935, 166698/166375
| 676/675, 1001/1000, 4096/4095, 10976/10935, 166698/166375
| {{mapping| 574 910 1333 1611 1986 2124 }} (574)
| {{mapping| 574 910 1333 1611 1986 2124 }} (574)
| -0.0544
| −0.0544
| 0.2007
| 0.2007
| 9.60
| 9.60
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=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+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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| [[Vishnu]] (574d)
| [[Vishnu]] (574d)
|}
|}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

Latest revision as of 06:31, 21 February 2025

← 573edo 574edo 575edo →
Prime factorization 2 × 7 × 41
Step size 2.09059 ¢ 
Fifth 336\574 (702.439 ¢) (→ 24\41)
Semitones (A1:m2) 56:42 (117.1 ¢ : 87.8 ¢)
Consistency limit 5
Distinct consistency limit 5

574 equal divisions of the octave (abbreviated 574edo or 574ed2), also called 574-tone equal temperament (574tet) or 574 equal temperament (574et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 574 equal parts of about 2.09 ¢ each. Each step represents a frequency ratio of 21/574, or the 574th root of 2.

Theory

574edo shares the harmonic 3 with 41edo. Unfortunately, it is only consistent to the 5-odd-limit, and the patent val and the 574d val are about equally viable.

The 574d val tempers out 4375/4374 (ragisma), 29360128/29296875 (quasiorwellisma), and 40500000/40353607 in the 7-limit; 3025/3024, 5632/5625, 9801/9800, and 19712/19683 in the 11-limit; 676/675, 4225/4224, and 10648/10647 in the 13-limit.

The patent val tempers out 2100875/2097152 (rainy comma), 6115295232/6103515625 (vishnuzma), 49009212/48828125, and 78125000/78121827 (euzenius comma) in the 7-limit; 5632/5625, 42875/42768, 117649/117128, 131072/130977, 160083/160000, 161280/161051, 166698/166375, 391314/390625, and 532400/531441 in the 11-limit; 676/675, 1001/1000, 2080/2079, 4096/4095, and 4225/4224 in the 13-limit.

Prime harmonics

Approximation of prime harmonics in 574edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 +0.484 +0.446 -0.882 +0.598 -0.110 -0.426 -0.649 +0.994 -1.006 +0.609
Relative (%) +0.0 +23.2 +21.3 -42.2 +28.6 -5.2 -20.4 -31.0 +47.5 -48.1 +29.1
Steps
(reduced) 		574
(0) 		910
(336) 		1333
(185) 		1611
(463) 		1986
(264) 		2124
(402) 		2346
(50) 		2438
(142) 		2597
(301) 		2788
(492) 		2844
(548)

Subsets and supersets

Since 574 factors into 2 × 7 × 41, 574edo has subset edos 2, 7, 14, 41, 82, and 287.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3.5 [23 6 -14⟩, [42 -47 14⟩ [⟨574 910 1333]] −0.1658 0.1260 6.03
2.3.5.7 4375/4374, 29360128/29296875, 40500000/40353607 [⟨574 910 1333 1612]] (574d) −0.2320 0.1583 7.57
2.3.5.7.11 3025/3024, 4375/4374, 5632/5625, 40500000/40353607 [⟨574 910 1333 1612 1986]] (574d) −0.2202 0.1435 6.87
2.3.5.7.11.13 676/675, 3025/3024, 4225/4224, 4375/4374, 41600/41503 [⟨574 910 1333 1612 1986 2124]] (574d) −0.1786 0.1607 7.69
2.3.5.7 10976/10935, 2100875/2097152, 49009212/48828125 [⟨574 910 1333 1611]] (574) −0.0459 0.2347 11.23
2.3.5.7.11 5632/5625, 10976/10935, 131072/130977, 166698/166375 [⟨574 910 1333 1611 1986]] (574) −0.0713 0.2160 10.33
2.3.5.7.11.13 676/675, 1001/1000, 4096/4095, 10976/10935, 166698/166375 [⟨574 910 1333 1611 1986 2124]] (574) −0.0544 0.2007 9.60

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
2 34\571 71.454 25/24 Vishnu (574d)

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

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