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]].
468edo is [[consistent]] to the [[13-odd-limit]] with a sharp tuning tendency and a [[2.3.7.13 subgroup]] inherited from [[234edo]].
 
As an equal temperament, it [[tempering out|tempers out]] the [[undim comma]] and the [[ennealimma]] in the [[5-limit]]; [[2401/2400]] and [[4375/4374]] in the [[7-limit]], [[support]]ing [[ennealimmal]]; [[3025/3024]] and [[9801/9800]] in the [[11-limit]], supporting [[hemiennealimmal]]; and [[676/675]], [[1001/1000]], [[1716/1715]], [[2080/2079]] in the [[13-limit]], supporting 13-limit hemiennealimmal. It also provides the [[optimal patent val]] for 13-limit [[unlit]].


=== Odd harmonics ===
=== Odd harmonics ===
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=== Subsets and supersets ===
=== Subsets and supersets ===
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}}.
Since 468 factors into primes as {{nowrap| 2<sup>2</sup> × 3<sup>2</sup> × 13 }}, 468edo has 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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! rowspan="2" | [[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]] (¢)
! [[TE simple badness|Relative]] (%)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo|371 -234}}
| {{mapping|468 742}}
| −0.1922
| 0.1921
| 7.49
|-
|-
| 2.3.5
| 2.3.5
| 2199023255552/2179240250625, 7629394531250/7625597484987
| {{Monzo| 41 -20 -4 }}, {{monzo| 1 -27 18 }}
| {{mapping|468 742 1087}}
| {{Mapping| 468 742 1087 }}
| −0.2524
| −0.2524
| 0.1785
| 0.1785
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|-
|-
| 2.3.5.7
| 2.3.5.7
| 4375/4374, 2401/2400, 47141561040896/46708681640625
| 2401/2400, 4375/4374, {{monzo| 34 -14 -10 3 }}
| {{mapping|468 742 1087 1314}}
| {{Mapping| 468 742 1087 1314 }}
| −0.2253
| −0.2253
| 0.1615
| 0.1615
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|-
|-
| 2.3.5.7.11
| 2.3.5.7.11
| 3025/3024, 4375/4374, 2401/2400, 5767168/5740875
| 2401/2400, 3025/3024, 4375/4374, 5767168/5740875
| {{mapping|468 742 1087 1314 1619}}
| {{Mapping| 468 742 1087 1314 1619 }}
| −0.1782
| −0.1782
| 0.1725
| 0.1725
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|-
|-
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 2080/2079, 1001/1000, 3025/3024, 1716/1715, 1982464/1974375
| 676/675, 1001/1000, 1716/1715, 3025/3024, 495616/494325
| {{mapping|468 742 1087 1314 1619 1732}}
| {{Mapping| 468 742 1087 1314 1619 1732 }}
| −0.1709
| −0.1709
| 0.1583
| 0.1583
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|-
|-
| 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
| 676/675, 1001/1000, 1156/1155, 1716/1715, 3025/3024, 7616/7605
| {{mapping|468 742 1087 1314 1619 1732 1913}}
| {{Mapping| 468 742 1087 1314 1619 1732 1913 }}
| −0.1526
| −0.1526
| 0.1533
| 0.1533
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|+ style="font-size: 105%;" | 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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|-
|-
| 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]]
| [[Unlit]]
|-
|-
| 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]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

Latest revision as of 04:26, 9 June 2026

← 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

468 equal divisions of the octave (abbreviated 468edo or 468ed2), also called 468-tone equal temperament (468tet) or 468 equal temperament (468et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 468 equal parts of about 2.56 ¢ each. Each step represents a frequency ratio of 21/468, or the 468th root of 2.

Theory

468edo is consistent to the 13-odd-limit with a sharp tuning tendency and a 2.3.7.13 subgroup inherited from 234edo.

As an equal temperament, it tempers out the undim comma and the ennealimma in the 5-limit; 2401/2400 and 4375/4374 in the 7-limit, supporting ennealimmal; 3025/3024 and 9801/9800 in the 11-limit, supporting hemiennealimmal; and 676/675, 1001/1000, 1716/1715, 2080/2079 in the 13-limit, supporting 13-limit hemiennealimmal. It also provides the optimal patent val for 13-limit 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

Since 468 factors into primes as 22 × 32 × 13, 468edo has 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 (%)
2.3.5 [41 -20 -4⟩, [1 -27 18⟩ [⟨468 742 1087]] −0.2524 0.1785 6.96
2.3.5.7 2401/2400, 4375/4374, [34 -14 -10 3⟩ [⟨468 742 1087 1314]] −0.2253 0.1615 6.30
2.3.5.7.11 2401/2400, 3025/3024, 4375/4374, 5767168/5740875 [⟨468 742 1087 1314 1619]] −0.1782 0.1725 6.73
2.3.5.7.11.13 676/675, 1001/1000, 1716/1715, 3025/3024, 495616/494325 [⟨468 742 1087 1314 1619 1732]] −0.1709 0.1583 6.17
2.3.5.7.11.13.17 676/675, 1001/1000, 1156/1155, 1716/1715, 3025/3024, 7616/7605 [⟨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) 		Unlit
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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