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398edo: Difference between revisions

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Created page with "{{Infobox ET}} {{EDO intro|398}} == Theory == 398et is only consistent to the 5-odd-limit. Using the patent val, it tempers out 1220703125/1219784832, 1280000000/12759898..."
 
m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"
 
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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|398}}
{{ED intro}}


== Theory ==
== Theory ==
398et is only consistent to the [[5-odd-limit]]. Using the patent val, it tempers out 1220703125/1219784832, 1280000000/1275989841, [[65625/65536]], 102760448/102515625 and 200120949/200000000 in the 7-limit; 1073741824/1071794405, 100663296/100656875, 161280/161051, 35156250/35153041, 2097152/2096325, [[4000/3993]], 2734375/2725888, 496125/495616, [[131072/130977]], [[6250/6237]], 9765625/9732096, 4302592/4296875, 352947/352000, 422576/421875, 184877/184320, [[3025/3024]], 9453125/9437184 and 456533/455625 in the 11-limit. It [[support]]s [[quartonic]], [[yarman]], [[bisupermajor]] and [[semiquindromeda]].  
398edo is only [[consistent]] to the [[5-odd-limit]], though it has a reasonable approximation to the full 13-limit using the [[patent val]], which [[tempering out|tempers out]] [[10976/10935]], [[65625/65536]], 1500625/1492992, 102760448/102515625, 102942875/102036672, and 200120949/200000000 in the 7-limit; [[3025/3024]], [[4000/3993]], [[6250/6237]], 59290/59049, 117649/117128, and [[131072/130977]] in the 11-limit; and [[625/624]], [[1575/1573]], [[2080/2079]], [[2200/2197]], [[4096/4095]], and [[4225/4224]] in the 13-limit. It [[support]]s [[yarman I]], [[bisupermajor]] and [[semiquindromeda]].  


=== Prime harmonics ===
=== Prime harmonics ===
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=== Subsets and supersets ===
=== Subsets and supersets ===
398 factors into 2 × 199, with [[2edo]] and [[199edo]] as its subset edos.
Since 398 factors into {{factorisation|398}}, 398edo has [[2edo]] and [[199edo]] as its subsets.


== 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|631 -398}}
! rowspan="2" | [[Comma list]]
|{{mapping|398 631}}
! rowspan="2" | [[Mapping]]
| -0.1759
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 631 -398 }}
| {{mapping| 398 631 }}
| −0.1759
| 0.1759
| 0.1759
| 5.83
| 5.83
|-
|-
|2.3.5
| 2.3.5
|390625000/387420489, {{monzo|-53 10 16}}
| 390625000/387420489, {{monzo| -53 10 16 }}
|{{mapping|398 631 924}}
| {{mapping| 398 631 924 }}
| -0.0622
| −0.0622
| 0.2157
| 0.2157
| 7.15
| 7.15
|-
|-
|2.3.5.7
| 2.3.5.7
|10976/10935, 65625/65536, 1280000000/1275989841
| 10976/10935, 65625/65536, 200120949/200000000
|{{mapping|398 631 924 1117}}
| {{mapping| 398 631 924 1117 }}
| +0.0412
| +0.0412
| 0.2588
| 0.2588
| 8.58
| 8.58
|-
|-
|2.3.5.7.11
| 2.3.5.7.11
|3025/3024, 6250/6237, 10976/10935, 496125/495616
| 3025/3024, 4000/3993, 10976/10935, 65625/65536
|{{mapping|398 631 924 1117 1377}}
| {{mapping| 398 631 924 1117 1377 }}
| +0.0075
| +0.0075
| 0.2411
| 0.2411
| 8.00
| 8.00
|-
|-
|2.3.5.7.11.13
| 2.3.5.7.11.13
|2080/2079, 625/624, 3025/3024, 4096/4095, 10976/10935
| 625/624, 1575/1573, 2080/2079, 2200/2197, 10976/10935
|{{mapping|398 631 924 1117 1377 1473}}
| {{mapping| 398 631 924 1117 1377 1473 }}
| -0.0243
| −0.0243
| 0.2313
| 0.2313
| 7.67
| 7.67
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=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+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
|-
|-
|1
| 1
|5\398
| 5\398
|15.08
| 15.08
|126/125
| 126/125
|[[Yarman]]
| [[Yarman I]]
|-
|-
|1
| 1
|183\398
| 183\398
|551.76
| 551.76
|11/8
| 11/8
|[[Emka]] / [[Emkay]]
| [[Emkay]]
|-
|-
|2
| 2
|54\398
| 54\398
|162.81
| 162.81
|11/10
| 11/10
|[[Kwazy]] / [[Bisupermajor]]
| [[Bisupermajor]]
|}
|}
 
<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
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
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