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Created page with "{{Infobox ET}} {{EDO intro|335}} == Theory == 335et only is consistent to the 5-odd-limit. Using the patent val, it tempers out the parakleisma in the 5-limit; 4375/43..."
 
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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|335}}
{{ED intro}}


== Theory ==
== Theory ==
335edo only is [[consistent]] to the [[5-odd-limit]]. As an equal temperament, it [[tempering out|tempers out]] {{monzo| 8 14 -13 }} ([[parakleisma]]) and {{monzo| 39 -29 3 }} ([[alphatricot comma]]), and is a quite efficient [[5-limit]] system.


335et only is consistent to the [[5-odd-limit]].
The 335d [[val]] ({{val| 335 531 778 '''941''' 1159 1240 }}), which scores the best, tempers out [[6144/6125]], [[16875/16807]] and [[14348907/14336000]] in the [[7-limit]]; [[540/539]], [[1375/1372]], [[3025/3024]], [[5632/5625]] in the [[11-limit]]; and [[729/728]], [[2080/2079]], [[2200/2197]], and [[6656/6655]] in the [[13-limit]]. It [[support]]s [[grendel]].  
Using the patent val, it tempers out the parakleisma in the 5-limit; [[4375/4374]] and [[3136/3125]] in the 7-limit, [[support]]ing [[turan]], [[quintosec]], [[tricot]], [[pseudotrillium]], [[lifthrasir]], [[tritomere]], [[counterwürschmidt]] and [[gaster]].
 
Using the 335d val ({{val|335 531 778 ‚‘‘991‘‘‘}}), it tempers out [[6144/6125]], [[16875/16807]] and [[14348907/14336000]] in the 7-limit, [[support]]ing [[grendel]].
The [[patent val]] {{val| 335 531 778 940 }} tempers out the [[3136/3125]] and [[4375/4374]] and in the 7-limit, supporting septimal [[parakleismic]]. This extension tempers out [[441/440]], 5632/5625, and [[19712/19683]] in the 11-limit. The 13-limit version of this, {{val| 335 531 778 940 1159 1240 }}, tempers out [[847/845]], [[1001/1000]], [[1575/1573]], 2200/2197, [[4096/4095]], [[6656/6655]], and [[10648/10647]]. Another 13-limit extension is {{val| 335 531 778 940 1159 '''1239''' }} (335f), where it adds [[364/363]] and 2080/2079 to the comma list.  


=== Prime harmonics ===
=== Prime harmonics ===
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=== Subsets and supersets ===
=== Subsets and supersets ===
335 factors into 5 × 67 with [[5edo]] and [[67edo]] as its subset edos. [[670edo]], which doubles it, gives a good correction to the harmonic 7.
Since 335 factors into primes as {{nowrap| 5 × 67 }}, 335edo has [[5edo]] and [[67edo]] as its subsets. [[670edo]], which doubles it, gives a good correction to the harmonic 7.


== 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|531 -335}}
! rowspan="2" | [[Comma list]]
|{{mapping|335 531}}
! rowspan="2" | [[Mapping]]
| -0.0424
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| 531 -335 }}
| {{Mapping| 335 531 }}
| −0.0424
| 0.0424
| 0.0424
| 1.18
| 1.18
|-
|-
|2.3.5
| 2.3.5
|{{monzo|8 14 -13}}, {{monzo|47 -15 -10}}
| {{Monzo| 8 14 -13 }}, {{monzo| 47 -15 -10 }}
|{{mapping|335 531 778}}
| {{Mapping| 335 531 778 }}
| -0.1075
| −0.1075
| 0.0984
| 0.0984
| 2.75
| 2.75
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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<br>(reduced)*
! Generator*
! Cents<br>(reduced)*
! Cents*
! Associated<br>Ratio*
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
|-
|1
| 1
|88\335
| 88\335
|315.22
| 315.22
|6/5
| 6/5
|[[Parakleismic]]
| [[Parakleismic]] (335)
|-
|-
|1
| 1
|158\335
| 108\335
|565.97
| 386.87
|104/75
| 5/4
|[[Tricot]]
| [[Counterwürschmidt]]
|-
|-
|5
| 1
|232\335<br>(31\335)
| 158\335
|831.04<br>(111.04)
| 565.97
|80/49<br>(16/15)
| 81920/59049
|[[Qintosec]]
| [[Alphatrident]] (335d)<br>[[Alphatrillium]] / pseudotrillium (335)
|-
| 5
| 103\335<br>(31\335)
| 368.96<br>(111.04)
| 99/80<br>(16/15)
| [[Quintosec]]
|}
|}
 
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|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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