335edo: Difference between revisions

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== 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 }} ([[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.  
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 ===
=== 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 ==
== 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
|-
|-
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|-
|-
| 2.3
| 2.3
| {{monzo| 531 -335 }}
| {{Monzo| 531 -335 }}
| {{mapping| 335 531 }}
| {{Mapping| 335 531 }}
| −0.0424
| −0.0424
| 0.0424
| 0.0424
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|-
|-
| 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
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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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| 565.97
| 565.97
| 81920/59049
| 81920/59049
| [[Trident]] (335d)<br />[[Trillium]] / pseudotrillium (335)
| [[Alphatrident]] (335d)<br>[[Alphatrillium]] / pseudotrillium (335)
|-
|-
| 5
| 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]]
| [[Quintosec]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|minimal form]] in parentheses if distinct

Latest revision as of 15:19, 16 March 2025

← 334edo 335edo 336edo →
Prime factorization 5 × 67
Step size 3.58209 ¢ 
Fifth 196\335 (702.09 ¢)
Semitones (A1:m2) 32:25 (114.6 ¢ : 89.55 ¢)
Consistency limit 5
Distinct consistency limit 5

335 equal divisions of the octave (abbreviated 335edo or 335ed2), also called 335-tone equal temperament (335tet) or 335 equal temperament (335et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 335 equal parts of about 3.58 ¢ each. Each step represents a frequency ratio of 21/335, or the 335th root of 2.

Theory

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

The 335d 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 supports grendel.

The patent 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, ⟨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 ⟨335 531 778 940 1159 1239] (335f), where it adds 364/363 and 2080/2079 to the comma list.

Prime harmonics

Approximation of prime harmonics in 335edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.13 +0.55 -1.66 +0.32 +1.26 -1.07 -0.20 -1.41 -1.52 +1.23
Relative (%) +0.0 +3.8 +15.4 -46.4 +9.0 +35.3 -30.0 -5.6 -39.3 -42.4 +34.4
Steps
(reduced) 		335
(0) 		531
(196) 		778
(108) 		940
(270) 		1159
(154) 		1240
(235) 		1369
(29) 		1423
(83) 		1515
(175) 		1627
(287) 		1660
(320)

Subsets and supersets

Since 335 factors into primes as 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

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [531 -335⟩ [⟨335 531]] −0.0424 0.0424 1.18
2.3.5 [8 14 -13⟩, [47 -15 -10⟩ [⟨335 531 778]] −0.1075 0.0984 2.75

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 88\335 315.22 6/5 Parakleismic (335)
1 108\335 386.87 5/4 Counterwürschmidt
1 158\335 565.97 81920/59049 Alphatrident (335d)
Alphatrillium / pseudotrillium (335)
5 103\335
(31\335) 		368.96
(111.04) 		99/80
(16/15) 		Quintosec

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

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