468edo: Difference between revisions

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== Theory ==
== Theory ==
468et tempers out [[78125000/78121827]], [[4375/4374]], [[250047/250000]], [[2401/2400]], 420175/419904, 200120949/200000000 and [[40353607/40310784]] in the 7-limit; 21437500/21434787, 151263/151250, 117649/117612, 514714375/514434888, 47265625/47258883, [[9801/9800]], [[3025/3024]], 1890625/1889568, 160083/160000, [[41503/41472]], 3294225/3294172, 43923/43904, 102487/102400 and [[1771561/1769472]] in the 11-limit. It provides the optimal patent val for [[unlit]].
468et tempers out [[78125000/78121827]], [[4375/4374]], [[250047/250000]], [[2401/2400]], 420175/419904, 200120949/200000000, and [[40353607/40310784]] in the 7-limit; 21437500/21434787, 151263/151250, 117649/117612, 514714375/514434888, 47265625/47258883, [[9801/9800]], [[3025/3024]], 1890625/1889568, 160083/160000, [[41503/41472]], 3294225/3294172, 43923/43904, 102487/102400, and [[1771561/1769472]] in the 11-limit. It provides the optimal patent val for [[unlit]].


=== Odd harmonics ===
=== Odd harmonics ===
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=== Subsets and supersets ===
=== Subsets and supersets ===
468 factors into 2<sup>2</sup> × 3<sup>2</sup> × 13, with subset edos {{EDOs|2, 3, 4, 6, 9, 12, 13, 18, 26, 36, 39, 52, 78, 117, 156, and 234}}.
468 factors into {{factorisation|468}}, with subset edos {{EDOs|2, 3, 4, 6, 9, 12, 13, 18, 26, 36, 39, 52, 78, 117, 156, and 234}}.


== Regular temperament properties ==
== Regular temperament properties ==
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| {{monzo|371 -234}}
| {{monzo|371 -234}}
| {{mapping|468 742}}
| {{mapping|468 742}}
| &minus;0.1922
| −0.1922
| 0.1921
| 0.1921
| 7.49
| 7.49
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| 2199023255552/2179240250625, 7629394531250/7625597484987
| 2199023255552/2179240250625, 7629394531250/7625597484987
| {{mapping|468 742 1087}}
| {{mapping|468 742 1087}}
| &minus;0.2524
| −0.2524
| 0.1785
| 0.1785
| 6.96
| 6.96
Line 40: Line 40:
| 4375/4374, 2401/2400, 47141561040896/46708681640625
| 4375/4374, 2401/2400, 47141561040896/46708681640625
| {{mapping|468 742 1087 1314}}
| {{mapping|468 742 1087 1314}}
| &minus;0.2253
| −0.2253
| 0.1615
| 0.1615
| 6.30
| 6.30
Line 47: Line 47:
| 3025/3024, 4375/4374, 2401/2400, 5767168/5740875
| 3025/3024, 4375/4374, 2401/2400, 5767168/5740875
| {{mapping|468 742 1087 1314 1619}}
| {{mapping|468 742 1087 1314 1619}}
| &minus;0.1782
| −0.1782
| 0.1725
| 0.1725
| 6.73
| 6.73
Line 54: Line 54:
| 2080/2079, 1001/1000, 3025/3024, 1716/1715, 1982464/1974375
| 2080/2079, 1001/1000, 3025/3024, 1716/1715, 1982464/1974375
| {{mapping|468 742 1087 1314 1619 1732}}
| {{mapping|468 742 1087 1314 1619 1732}}
| &minus;0.1709
| −0.1709
| 0.1583
| 0.1583
| 6.17
| 6.17
Line 61: Line 61:
| 2080/2079, 1001/1000, 3025/3024, 1716/1715, 17920/17901, 30464/30375
| 2080/2079, 1001/1000, 3025/3024, 1716/1715, 17920/17901, 30464/30375
| {{mapping|468 742 1087 1314 1619 1732 1913}}
| {{mapping|468 742 1087 1314 1619 1732 1913}}
| &minus;0.1526
| −0.1526
| 0.1533
| 0.1533
| 5.98
| 5.98
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| [[Hemiennealimmal]]
| [[Hemiennealimmal]]
|}
|}
<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

Revision as of 19:54, 15 January 2025

← 467edo 468edo 469edo →
Prime factorization 22 × 32 × 13
Step size 2.5641 ¢ 
Fifth 274\468 (702.564 ¢) (→ 137\234)
Semitones (A1:m2) 46:34 (117.9 ¢ : 87.18 ¢)
Consistency limit 13
Distinct consistency limit 13

Template:EDO intro

Theory

468et tempers out 78125000/78121827, 4375/4374, 250047/250000, 2401/2400, 420175/419904, 200120949/200000000, and 40353607/40310784 in the 7-limit; 21437500/21434787, 151263/151250, 117649/117612, 514714375/514434888, 47265625/47258883, 9801/9800, 3025/3024, 1890625/1889568, 160083/160000, 41503/41472, 3294225/3294172, 43923/43904, 102487/102400, and 1771561/1769472 in the 11-limit. It provides the optimal patent val for unlit.

Odd harmonics

Approximation of odd harmonics in 468edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +0.61 +0.87 +0.40 +1.22 -0.04 +0.50 -1.09 +0.17 -0.08 +1.01 -0.07
Relative (%) +23.8 +33.8 +15.8 +47.5 -1.4 +19.4 -42.5 +6.7 -3.0 +39.5 -2.7
Steps
(reduced) 		742
(274) 		1087
(151) 		1314
(378) 		1484
(80) 		1619
(215) 		1732
(328) 		1828
(424) 		1913
(41) 		1988
(116) 		2056
(184) 		2117
(245)

Subsets and supersets

468 factors into 22 × 32 × 13, with subset edos 2, 3, 4, 6, 9, 12, 13, 18, 26, 36, 39, 52, 78, 117, 156, and 234.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [371 -234⟩ [⟨468 742]] −0.1922 0.1921 7.49
2.3.5 2199023255552/2179240250625, 7629394531250/7625597484987 [⟨468 742 1087]] −0.2524 0.1785 6.96
2.3.5.7 4375/4374, 2401/2400, 47141561040896/46708681640625 [⟨468 742 1087 1314]] −0.2253 0.1615 6.30
2.3.5.7.11 3025/3024, 4375/4374, 2401/2400, 5767168/5740875 [⟨468 742 1087 1314 1619]] −0.1782 0.1725 6.73
2.3.5.7.11.13 2080/2079, 1001/1000, 3025/3024, 1716/1715, 1982464/1974375 [⟨468 742 1087 1314 1619 1732]] −0.1709 0.1583 6.17
2.3.5.7.11.13.17 2080/2079, 1001/1000, 3025/3024, 1716/1715, 17920/17901, 30464/30375 [⟨468 742 1087 1314 1619 1732 1913]] −0.1526 0.1533 5.98

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 179\468 458.97 125/96 Majvam
4 194\468
(40\468) 		497.44
(102.56) 		4/3
(35/33) 		Undim
9 123\468
(19\468) 		315.38
(48.72) 		6/5
(36/35) 		Ennealimmal
18 97\468
(7\468) 		248.72
(17.95) 		231/200
(99/98) 		Hemiennealimmal

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

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