118edo: Difference between revisions

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It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]].  
It has two reasonable mappings for 13. The [[patent val]] tempers out [[196/195]], [[352/351]], [[625/624]], [[729/728]], [[1001/1000]], [[1575/1573]] and [[4096/4095]]. The 118f val tempers out [[169/168]], [[325/324]], [[351/350]], [[364/363]], [[1573/1568]], [[1716/1715]] and [[2080/2079]]. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the [[21-odd-limit]].  


Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a stack of 10 [[12edo]]<nowiki/>s minus the said comma.  
Since the [[Pythagorean comma]] maps to 2 steps of 118edo, it can be interpreted as a stack of 10 [[12edo]]<nowiki/>s minus the said comma.


118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]].  
118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]].  
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{{primes in edo|118}}
{{primes in edo|118}}
==Table of intervals==
==Table of intervals==
{| class="wikitable"
|+Selected 118 EDO intervals
!Step
!Name
!Associated ratio
|-
|0
|unison
|1/1 exact
|-
|2
|comma
|[[531441/524288]], [[81/80]]
|-
|9
|limma
|[[256/243]]
|-
|11
|apotome
|[[2187/2048]]
|-
|20
|whole tone
|[[9/8]]
|-
|23
|septimal second
|[[8/7]]
|-
|26
|septimal third
|[[7/6]]
|-
|29
|Pythagorean minor 3rd
|[[32/27]]
|-
|31
|Classical minor 3rd
|[[6/5]]
|-
|38
|Classical major 3rd
|[[5/4]]
|-
|40
|Pythagorean major 3rd
|[[81/64]]
|-
|49
|perfect 4th
|[[4/3]]
|-
|59
|symmetric tritone
|
|-
|69
|perfect 5th
|[[3/2]]
|-
|78
|Pythagorean minor 6th
|[[128/81]]
|-
|80
|Classical minor 6th
|[[8/5]]
|-
|87
|Classical major 6th
|[[5/3]]
|-
|89
|Pythagorean major 6th
|[[27/16]]
|-
|118
|perfect 8ve
|2/1 exact
|}


== Regular temperament properties ==
== Regular temperament properties ==

Revision as of 11:04, 7 November 2021

← 117edo 118edo 119edo →
Prime factorization 2 × 59
Step size 10.1695 ¢ 
Fifth 69\118 (701.695 ¢)
Semitones (A1:m2) 11:9 (111.9 ¢ : 91.53 ¢)
Consistency limit 11
Distinct consistency limit 11

The 118 equal divisions of the octave (118edo), or the 118(-tone) equal temperament (118tet, 118et) when viewed from a regular temperament perspective, is the equal division of the octave into 118 parts of about 10.2 cents each.

Theory

118edo represents the intersection of the 5-limit schismatic and parakleismic temperaments, tempering out both the schisma, [-15 8 1⟩ and the parakleisma, [8 14 -13⟩, as well as the vishnuzma, [23 6 -14⟩, the hemithirds comma, [38 -2 -15⟩, and the kwazy, [-53 10 16⟩. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent.

In the 7-limit, it is particularly notable for tempering out the gamelisma, 1029/1024, and is an excellent tuning for the rank three gamelan temperament, and for guiron, the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but 99edo does better with that.

In the 11-limit, it tempers out 385/384 and 441/440, and is an excellent tuning for portent, the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The patent val tempers out 196/195, 352/351, 625/624, 729/728, 1001/1000, 1575/1573 and 4096/4095. The 118f val tempers out 169/168, 325/324, 351/350, 364/363, 1573/1568, 1716/1715 and 2080/2079. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the 21-odd-limit.

Since the Pythagorean comma maps to 2 steps of 118edo, it can be interpreted as a stack of 10 12edos minus the said comma.

118edo is the 17th zeta peak edo.

Prime harmonics

Script error: No such module "primes_in_edo".

Table of intervals

Selected 118 EDO intervals
Step Name Associated ratio
0 unison 1/1 exact
2 comma 531441/524288, 81/80
9 limma 256/243
11 apotome 2187/2048
20 whole tone 9/8
23 septimal second 8/7
26 septimal third 7/6
29 Pythagorean minor 3rd 32/27
31 Classical minor 3rd 6/5
38 Classical major 3rd 5/4
40 Pythagorean major 3rd 81/64
49 perfect 4th 4/3
59 symmetric tritone
69 perfect 5th 3/2
78 Pythagorean minor 6th 128/81
80 Classical minor 6th 8/5
87 Classical major 6th 5/3
89 Pythagorean major 6th 27/16
118 perfect 8ve 2/1 exact

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-187 118⟩ [⟨118 187]] -0.119 0.082 0.81
2.3.5 32805/32768, [8 14 -13⟩ [⟨118 187 274]] +0.036 0.093 0.91
2.3.5.7 1029/1024, 3136/3125, 4375/4374 [⟨118 187 274 331]] +0.270 0.412 4.05
2.3.5.7.11 385/384, 441/440, 3136/3125, 4375/4374 [⟨118 187 274 331 408]] +0.341 0.370 3.89
2.3.5.7.11.13 196/195, 352/351, 384/384, 625/624, 729/728 [⟨118 187 274 331 408 437]] (118) +0.125 0.604 5.93
2.3.5.7.11.13 169/168, 325/324, 364/363, 385/384, 3136/3125 [⟨118 187 274 331 408 436]] (118f) +0.583 0.650 6.39
2.3.5.7.11.17 289/288, 385/384, 441/440, 561/560, 3136/3125 [⟨118 187 274 331 408 482]] +0.417 0.399 3.92
2.3.5.7.11.17.19 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968 [⟨118 187 274 331 408 482 501]] +0.445 0.376 3.69

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per octave 		Generator
(reduced) 		Cents
(reduced) 		Associated
ratio 		Temperaments
1 11\118 111.86 16/15 Vavoom
1 19\118 193.22 28/25 Luna / hemithirds / lunatic
1 23\118 233.90 8/7 Slendric / guiron
1 31\118 315.25 6/5 Parakleismic / paralytic
1 39\118 396.61 44/35 Squarschmidt
1 49\118 498.31 4/3 Helmholtz / pontiac / helenoid / pontic
1 55\118 559.32 242/175 Tritriple
2 2\118 20.34 81/80 Commatic
2 5\118 50.85 33/32~36/35 Kleischismic
2 7\118 71.19 25/24 Vishnu / ananta (118) / acyuta (118f)
2 10\118 101.69 35/33 Bischismic / bipont (118) / counterbipont (118f)
2 16\118 162.71 11/10 Kwazy / bisupermajor
2 18\118 183.05 10/9 Unidec / ekadash (118) / hendec (118f)
2 19\118 193.22 121/108 Semiluna
2 31\118
(28\118) 		315.25
(284.75) 		6/5
(33/28) 		Semiparakleismic

Trivia

118 is the number of chemical elements in the first 7 periods of the periodic table, meaning that the element names can be used as note names.

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