574edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|574}}
{{EDO intro|574}}
== Theory ==
== Theory ==
574et tempers out 67108864/66976875, [[78125000/78121827]] and [[2100875/2097152]] in the 7-limit; 161280/161051, 1019215872/1019046875, 166698/166375, [[131072/130977]], 107495424/107421875, [[5632/5625]], 1890625/1889568, 160083/160000, [[532400/531441]] and 391314/390625 in the 11-limit.
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.
 
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.  
 
=== Prime harmonics ===
{{Harmonics in equal|574}}
{{Harmonics in equal|574}}
==Regular temperament properties==
 
=== Subsets and supersets ===
Since 574 factors into 2 × 7 × 41, 574edo has subset edos {{EDOs| 2, 7, 14, 41, 82, and 287 }}.
 
== 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|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 simple badness|Relative]] (%)
|-
|-
|2.3
! [[TE error|Absolute]] (¢)
|{{monzo|65 -41}}
! [[TE simple badness|Relative]] (%)
|{{val|574 910}}
| -0.1527
| 0.1527
| 7.30
|-
|-
|2.3.5
| 2.3.5
|{{monzo|23 6 -14}}, {{monzo|42 -47 14}}
| {{monzo| 23 6 -14 }}, {{monzo| 42 -47 14 }}
|{{val|574 910 1333}}
| {{mapping| 574 910 1333 }}
| -0.1658
| -0.1658
| 0.1260
| 0.1260
| 6.03
| 6.03
|-
|-
|2.3.5.7
| 2.3.5.7
|10976/10935, 2100875/2097152, 78125000/78121827
| 10976/10935, 2100875/2097152, 49009212/48828125
|{{val|574 910 1333 1611}}
| {{mapping| 574 910 1333 1611 }} (574)
| -0.0459
| -0.0459
| 0.2347
| 0.2347
| 11.23
| 11.23
|-
|-
|2.3.5.7.11
| 2.3.5.7.11
|5632/5625, 10976/10935, 160083/160000, 166698/166375
| 5632/5625, 10976/10935, 131072/130977, 166698/166375
|{{val|574 910 1333 1611 1986}}
| {{mapping| 574 910 1333 1611 1986 }} (574)
| -0.0713
| -0.0713
| 0.2160
| 0.2160
| 10.33
| 10.33
|-
|-
|2.3.5.7.11.13
| 2.3.5.7.11.13
|2080/2079, 1001/1000, 4096/4095, 10976/10935, 166698/166375
| 676/675, 1001/1000, 4096/4095, 10976/10935, 166698/166375
|{{val|574 910 1333 1611 1986 2124}}
| {{mapping| 574 910 1333 1611 1986 2124 }} (574)
| -0.0544
| -0.0544
| 0.2007
| 0.2007
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|+Table of rank-2 temperaments by generator
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator<br>(reduced)
! Generator*
! Cents<br>(reduced)
! Cents*
! Associated<br>ratio
! Associated<br>Ratio*
! Temperaments
! Temperaments
|-
|-
|2
| 2
|34\571
| 34\571
|71.454
| 71.454
|25/24
| 25/24
|[[Vishnu]]
| [[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

Revision as of 08:33, 29 October 2023

← 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

Template:EDO intro

Theory

574 = 14 × 41, and 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), 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.

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 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 it is distinct

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