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468edo: Difference between revisions

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Created page with "{{Infobox ET}} {{EDO intro|468}} == Theory == 468et tempers out 78125000/78121827, 4375/4374, 250047/250000, 2401/2400, 420175/419904, 200120949/200000000 and..."
 
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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|468}}
{{ED intro}}


== 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 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}}.
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 ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! 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
! rowspan="2" | [[Subgroup]]
|{{monzo|371 -234}}
! rowspan="2" | [[Comma list]]
|{{mapping|468 742}}
! rowspan="2" | [[Mapping]]
| -0.1922
! rowspan="2" | Optimal<br>8ve stretch (¢)
| 0.1921
! colspan="2" | Tuning error
| 7.49
|-
|-
|2.3.5
! [[TE error|Absolute]] (¢)
|2199023255552/2179240250625, 7629394531250/7625597484987
! [[TE simple badness|Relative]] (%)
|{{mapping|468 742 1087}}
|-
| -0.2524
| 2.3.5
| {{Monzo| 41 -20 -4 }}, {{monzo| 1 -27 18 }}
| {{Mapping| 468 742 1087 }}
| −0.2524
| 0.1785
| 0.1785
| 6.96
| 6.96
|-
|-
|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
| 6.30
| 6.30
|-
|-
|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
| 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
| 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
| 6.17
| 6.17
|-
| 2.3.5.7.11.13.17
| 676/675, 1001/1000, 1156/1155, 1716/1715, 3025/3024, 7616/7605
| {{Mapping| 468 742 1087 1314 1619 1732 1913 }}
| −0.1526
| 0.1533
| 5.98
|}
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>ratio*
! Temperaments
|-
| 1
| 179\468
| 458.97
| 125/96
| [[Majvam]]
|-
| 4
| 194\468<br>(40\468)
| 497.44<br>(102.56)
| 4/3<br>(35/33)
| [[Unlit]]
|-
| 9
| 123\468<br>(19\468)
| 315.38<br>(48.72)
| 6/5<br>(36/35)
| [[Ennealimmal]]
|-
| 18
| 97\468<br>(7\468)
| 248.72<br>(17.95)
| 231/200<br>(99/98)
| [[Hemiennealimmal]]
|}
|}
<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
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