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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
| 2.3
|{{monzo|371 -234}}
| {{monzo|371 -234}}
|{{mapping|468 742}}
| {{mapping|468 742}}
| -0.1922
| &minus;0.1922
| 0.1921
| 0.1921
| 7.49
| 7.49
|-
|-
|2.3.5
| 2.3.5
|2199023255552/2179240250625, 7629394531250/7625597484987
| 2199023255552/2179240250625, 7629394531250/7625597484987
|{{mapping|468 742 1087}}
| {{mapping|468 742 1087}}
| -0.2524
| &minus;0.2524
| 0.1785
| 0.1785
| 6.96
| 6.96
|-
|-
|2.3.5.7
| 2.3.5.7
|4375/4374, 2401/2400, 47141561040896/46708681640625
| 4375/4374, 2401/2400, 47141561040896/46708681640625
|{{mapping|468 742 1087 1314}}
| {{mapping|468 742 1087 1314}}
| -0.2253
| &minus;0.2253
| 0.1615
| 0.1615
| 6.30
| 6.30
|-
|-
|2.3.5.7.11
| 2.3.5.7.11
|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}}
| -0.1782
| &minus;0.1782
| 0.1725
| 0.1725
| 6.73
| 6.73
|-
|-
|2.3.5.7.11.13
| 2.3.5.7.11.13
|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}}
| -0.1709
| &minus;0.1709
| 0.1583
| 0.1583
| 6.17
| 6.17
|-
|-
|2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
|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}}
| -0.1526
| &minus;0.1526
| 0.1533
| 0.1533
| 5.98
| 5.98
|}
{{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<br>(reduced)*
! Cents<br>(reduced)*
! Associated<br>Ratio*
! Temperaments
|-
|-
|1
| 1
|179\468
| 179\468
|458.97
| 458.97
|125/96
| 125/96
|[[Majvam]]
| [[Majvam]]
|-
|-
|4
| 4
|194\468<br>(40\468)
| 194\468<br />(40\468)
|497.44<br>(102.56)
| 497.44<br />(102.56)
|4/3<br>(35/33)
| 4/3<br />(35/33)
|[[Undim]]
| [[Undim]]
|-
|-
|9
| 9
|123\468<br>(19\468)
| 123\468<br />(19\468)
|315.38<br>(48.72)
| 315.38<br />(48.72)
|6/5<br>(36/35)
| 6/5<br />(36/35)
|[[Ennealimmal]]
| [[Ennealimmal]]
|-
|-
|18
| 18
|97\468<br>(7\468)
| 97\468<br />(7\468)
|248.72<br>(17.95)
| 248.72<br />(17.95)
|231/200<br>(99/98)
| 231/200<br />(99/98)
|[[Hemiennealimmal]]
| [[Hemiennealimmal]]
|}
{{rank-2 end}}
 
{{orf}}
<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 02:02, 16 November 2024

← 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

Template:Comma basis begin |- | 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 Template:Comma basis end

Rank-2 temperaments

Template:Rank-2 begin |- | 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 Template:Rank-2 end Template:Orf

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