574edo: Difference between revisions

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== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{{comma basis begin}}
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
|-
| 2.3.5
| 2.3.5
| {{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
| &minus;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
| &minus;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 1986 }} (574d)
| {{mapping| 574 910 1333 1612 1986 }} (574d)
| -0.2202
| &minus;0.2202
| 0.1435
| 0.1435
| 6.87
| 6.87
Line 50: Line 42:
| 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 1986 2124 }} (574d)
| {{mapping| 574 910 1333 1612 1986 2124 }} (574d)
| -0.1786
| &minus;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
| &minus;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
| &minus;0.0713
| 0.2160
| 0.2160
| 10.33
| 10.33
Line 71: Line 63:
| 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
| &minus;0.0544
| 0.2007
| 0.2007
| 9.60
| 9.60
|}
{{comma basis end}}


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{{rank-2 begin}}
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>Ratio*
! Temperaments
|-
|-
| 2
| 2
Line 90: Line 76:
| 25/24
| 25/24
| [[Vishnu]] (574d)
| [[Vishnu]] (574d)
|}
{{rank-2 end}}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
{{orf}}

Revision as of 04:43, 16 November 2024

← 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), 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

Template:Comma basis begin |- | 2.3.5 | [23 6 -14⟩, [42 -47 14⟩ | [⟨574 910 1333]] | −0.1658 | 0.1260 | 6.03 |- style="border-top: double;" | 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 |- style="border-top: double;" | 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 Template:Comma basis end

Rank-2 temperaments

Template:Rank-2 begin |- | 2 | 34\571 | 71.454 | 25/24 | Vishnu (574d) Template:Rank-2 end Template:Orf

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