335edo: Difference between revisions

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== Theory ==

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

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.

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]].

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]].

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.

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=== Subsets and supersets ===

Since 335 factors into 5 × 67, 335edo has [[5edo]] and [[67edo]] as its subsets. [[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 ==

{~~{comma basis begin}}~~

{| class="wikitable center-4 center-5 center-6"

|-

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

! [[TE simple badness|Relative]] (%)

|-

| 2.3

| {{~~monzo~~| 531 -335 }}

| {{Monzo| 531 -335 }}

| {{~~mapping~~| 335 531 }}

| {{Mapping| 335 531 }}

| ~~−0~~.0424

| −0.0424

| 0.0424

| 1.18

|-

| 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

| 2.75

~~{{comma basis end}~~}

|}

=== Rank-2 temperaments ===

{{rank-2 ~~begin}}~~

{| class="wikitable center-all left-5"

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

|-

! Periods per 8ve

! Generator*

! Cents*

! Associated ratio*

! Temperaments

|-

| 1

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| 565.97

| 81920/59049

| [[~~Trident~~]] (335d) [[~~Trillium~~]] / pseudotrillium (335)

| [[Alphatrident]] (335d) [[Alphatrillium]] / pseudotrillium (335)

|-

| 5

| 103\335 (31\335)

| 103\335 (31\335)

| 368.96 (111.04)

| 368.96 (111.04)

| 99/80 (16/15)

| 99/80 (16/15)

| [[Quintosec]]

~~{{rank-2 end}~~}

|}

~~{{orf}}~~

<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|minimal form]] in parentheses if distinct

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|335}}
+{{ED intro}}
 == Theory ==
-edo only is [[consistent]] to the [[5-odd-limit]]. The equal temperament [[tempering out|tempers out]] {{monzo| 8 14 -13 }} ([[parakleisma]]) and {{monzo| 39 -29 3 }} ([[tricot comma]]), and is a quite efficient [[5-limit]] system.
+edo 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.
-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]].
+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]].
 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.
@@ Line 13: / Line 13: @@
 === Subsets and supersets ===
-Since 335 factors into 5 × 67, 335edo has [[5edo]] and [[67edo]] as its subsets. [[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 ==
-{{comma basis begin}}
+{| class="wikitable center-4 center-5 center-6"
+|-
+! rowspan="2" | [[Subgroup]]
+! rowspan="2" | [[Comma list]]
+! rowspan="2" | [[Mapping]]
+! rowspan="2" | Optimal<br>8ve stretch (¢)
+! colspan="2" | Tuning error
+|-
+! [[TE error|Absolute]] (¢)
+! [[TE simple badness|Relative]] (%)
 |-
 | 2.3
-| {{monzo| 531 -335 }}
+| {{Monzo| 531 -335 }}
-| {{mapping| 335 531 }}
+| {{Mapping| 335 531 }}
-| &minus;0.0424
+| −0.0424
 | 0.0424
 | 1.18
 |-
 | 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 }}
-| &minus;0.1075
+| −0.1075
 | 0.0984
 | 2.75
-{{comma basis end}}
+|}
 === Rank-2 temperaments ===
-{{rank-2 begin}}
+{| 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
@@ Line 52: / Line 68: @@
 | 565.97
 | 81920/59049
-| [[Trident]] (335d)<br />[[Trillium]] / pseudotrillium (335)
+| [[Alphatrident]] (335d)<br>[[Alphatrillium]] / pseudotrillium (335)
 |-
 | 5
-| 103\335<br />(31\335)
+| 103\335<br>(31\335)
-| 368.96<br />(111.04)
+| 368.96<br>(111.04)
-| 99/80<br />(16/15)
+| 99/80<br>(16/15)
 | [[Quintosec]]
-{{rank-2 end}}
+|}
-{{orf}}
+<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|minimal form]] in parentheses if distinct