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== Intervals ==
== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed"
{| class="wikitable collapsible mw-collapsible mw-collapsed"
|+ style=white-space:nowrap | Table of intervals in 118 EDO
|+ style=white-space:nowrap | Table of intervals in 118edo
!Step
! Step
!Name
! Eliora's Naming System
!Chemical notation
! Eliora's Chemical Notation<br>(if base note = 0)
<small>if base note = 0</small>
! Approximate Ratios
!Associated ratio
|-
|-
|0
| 0
|unison
| unison
|oganesson / neutronium
| oganesson / neutronium
|1/1 exact
| 1/1 exact
|-
|-
|1
| 1
|semicomma
| semicomma
|hydrogen
| hydrogen
|[[243/242]], many others
| [[243/242]], many others
|-
|-
|2
| 2
|comma
| comma
|helium
| helium
|[[531441/524288]], [[81/80]]
| [[531441/524288]], [[81/80]]
|-
|-
|9
| 9
|limma, dayavati
| limma, dayavati
|fluorine
| fluorine
|[[256/243]]
| [[256/243]]
|-
|-
|10
| 10
|dodecaic semitone
| dodecaic semitone
|neon
| neon
|[[17/16]]
| [[17/16]]
|-
|-
|11
| 11
|apotome, ranjani
| apotome, ranjani
|sodium
| sodium
|[[16/15]], [[2187/2048]]
| [[16/15]], [[2187/2048]]
|-
|-
|18
| 18
|diminished tone, ratika
| diminished tone, ratika
|argon
| argon
|[[10/9]]
| [[10/9]]
|-
|-
|19
| 19
|minor tone
| minor tone
|potassium
| potassium
|[[19/17]]
| [[19/17]]
|-
|-
|20
| 20
|major tone, raudri
| major tone, raudri
|calcium
| calcium
|[[9/8]]
| [[9/8]]
|-
|-
|23
| 23
|septimal second, slendric 2
| septimal second, slendric 2
|vanadium
| vanadium
|[[8/7]]
| [[8/7]]
|-
|-
|26
| 26
|septimal third
| septimal third
|iron
| iron
|[[7/6]]
| [[7/6]]
|-
|-
|29
| 29
|Pythagorean minor 3rd, krodha
| Pythagorean minor 3rd, krodha
|copper
| copper
|[[32/27]]
| [[32/27]]
|-
|-
|31
| 31
|Classical minor 3rd, vajrika
| Classical minor 3rd, vajrika
|gallium
| gallium
|[[6/5]]
| [[6/5]]
|-
|-
|33
| 33
|Lesser tridecimal third
| Lesser tridecimal third
|germanium
| germanium
|[[39/32]]
| [[39/32]]
|-
|-
|34
| 34
|Minor-neutral third
| Minor-neutral third
|selenium
| selenium
|[[11/9]]
| [[11/9]]
|-
|-
|35
| 35
|Minor tridecimal neurtral third, "major-neutral" third
| Minor tridecimal neurtral third, "major-neutral" third
|bromine
| bromine
|[[16/13]], 70/57
| [[16/13]], 70/57
|-
|-
|36
| 36
|Golden ratio 3rd, major-tridecimal neutral third
| Golden ratio 3rd, major-tridecimal neutral third
|krypton
| krypton
|[[16/13]], [[26/21]], [[21/17]]
| [[16/13]], [[26/21]], [[21/17]]
|-
|-
|38
| 38
|Classical major 3rd, prasarini
| Classical major 3rd, prasarini
|strontium
| strontium
|[[5/4]]
| [[5/4]]
|-
|-
|40
| 40
|Pythagorean major 3rd
| Pythagorean major 3rd
|zirconium
| zirconium
|[[81/64]]
| [[81/64]]
|-
|-
|45
| 45
|Barbados 3rd
| Barbados 3rd
|rhodium
| rhodium
|[[13/10]],
| [[13/10]],
|-
|-
|46
| 46
|Slendric 3
| Slendric 3
|palladium
| palladium
|[[21/16]],
| [[21/16]],
|-
|-
|49
| 49
|perfect 4th
| perfect 4th
|indium
| indium
|[[4/3]]
| [[4/3]]
|-
|-
|51
| 51
|Kshiti
| Kshiti
|antimony
| antimony
|[[27/20]]
| [[27/20]]
|-
|-
|58
| 58
|Rakta
| Rakta
|cerium
| cerium
|[[45/32]]
| [[45/32]]
|-
|-
|59
| 59
|symmetric tritone
| symmetric tritone
|praseodymium
| praseodymium
|[[99/70]], [[140/99]]
| [[99/70]], [[140/99]]
|-
|-
|60
| 60
|Literal tritone, sandipani
| Literal tritone, sandipani
|neodymium
| neodymium
|[[729/512]]
| [[729/512]]
|-
|-
|69
| 69
|perfect 5th
| perfect 5th
|thulium
| thulium
|[[3/2]]
| [[3/2]]
|-
|-
|78
| 78
|Pythagorean minor 6th
| Pythagorean minor 6th
|platinum
| platinum
|[[128/81]]
| [[128/81]]
|-
|-
|80
| 80
|Classical minor 6th
| Classical minor 6th
|mercury
| mercury
|[[8/5]]
| [[8/5]]
|-
|-
|82
| 82
|Golden ratio sixth, minor-neutral tridecimal sixth
| Golden ratio sixth, minor-neutral tridecimal sixth
|lead
| lead
|13/8 I, [[21/13]], [[34/21]], [[Acoustic phi]]
| 13/8 I, [[21/13]], [[34/21]], [[Acoustic phi]]
|-
|-
|83
| 83
|Major tridecimal neutral sixth, "minor-neutral" sixth
| Major tridecimal neutral sixth, "minor-neutral" sixth
|bismuth
| bismuth
|13/8 II, 57/35
| 13/8 II, 57/35
|-
|-
|84
| 84
|Major-neutral sixth
| Major-neutral sixth
|polonium
| polonium
|[[18/11]]
| [[18/11]]
|-
|-
|87
| 87
|Classical major 6th
| Classical major 6th
|francium
| francium
|[[5/3]]
| [[5/3]]
|-
|-
|89
| 89
|Pythagorean major 6th
| Pythagorean major 6th
|actinium
| actinium
|[[27/16]]
| [[27/16]]
|-
|-
|92
| 92
|Septimal supermajor 6th, slendro 5
| Septimal supermajor 6th, slendro 5
|uranium
| uranium
|[[12/7]]
| [[12/7]]
|-
|-
|95
| 95
|Harmonic 7th
| Harmonic 7th
|americium
| americium
|[[7/4]]
| [[7/4]]
|-
|-
|100
| 100
|Tivra
| Tivra
|fermium
| fermium
|[[9/5]]
| [[9/5]]
|-
|-
|109
| 109
|Pythagorean major 7th
| Pythagorean major 7th
|meitnerium
| meitnerium
|[[243/128]]
| [[243/128]]
|-
|-
|116
| 116
|Comma 7th
| Comma 7th
|livermorium
| livermorium
|[[160/81]]
| [[160/81]]
|-
|-
|117
| 117
|Semicomma supermajor 7th
| Semicomma supermajor 7th
|tenessine
| tenessine
|multiple
| multiple
|-
|-
|118
| 118
|perfect 8ve
| perfect 8ve
|oganesson / neutronium
| oganesson / neutronium
|2/1 exact
| 2/1 exact
|}
|}


Revision as of 18:21, 4 December 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 addition, 118edo excellently approximates the 22 Shruti scale.

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 series of ten segments of twelve Pythagorean fifths minus the said comma.

118edo is the 17th zeta peak edo.

Prime harmonics

Script error: No such module "primes_in_edo".

Intervals

Table of intervals in 118edo
Step Eliora's Naming System Eliora's Chemical Notation
(if base note = 0) 		Approximate Ratios
0 unison oganesson / neutronium 1/1 exact
1 semicomma hydrogen 243/242, many others
2 comma helium 531441/524288, 81/80
9 limma, dayavati fluorine 256/243
10 dodecaic semitone neon 17/16
11 apotome, ranjani sodium 16/15, 2187/2048
18 diminished tone, ratika argon 10/9
19 minor tone potassium 19/17
20 major tone, raudri calcium 9/8
23 septimal second, slendric 2 vanadium 8/7
26 septimal third iron 7/6
29 Pythagorean minor 3rd, krodha copper 32/27
31 Classical minor 3rd, vajrika gallium 6/5
33 Lesser tridecimal third germanium 39/32
34 Minor-neutral third selenium 11/9
35 Minor tridecimal neurtral third, "major-neutral" third bromine 16/13, 70/57
36 Golden ratio 3rd, major-tridecimal neutral third krypton 16/13, 26/21, 21/17
38 Classical major 3rd, prasarini strontium 5/4
40 Pythagorean major 3rd zirconium 81/64
45 Barbados 3rd rhodium 13/10,
46 Slendric 3 palladium 21/16,
49 perfect 4th indium 4/3
51 Kshiti antimony 27/20
58 Rakta cerium 45/32
59 symmetric tritone praseodymium 99/70, 140/99
60 Literal tritone, sandipani neodymium 729/512
69 perfect 5th thulium 3/2
78 Pythagorean minor 6th platinum 128/81
80 Classical minor 6th mercury 8/5
82 Golden ratio sixth, minor-neutral tridecimal sixth lead 13/8 I, 21/13, 34/21, Acoustic phi
83 Major tridecimal neutral sixth, "minor-neutral" sixth bismuth 13/8 II, 57/35
84 Major-neutral sixth polonium 18/11
87 Classical major 6th francium 5/3
89 Pythagorean major 6th actinium 27/16
92 Septimal supermajor 6th, slendro 5 uranium 12/7
95 Harmonic 7th americium 7/4
100 Tivra fermium 9/5
109 Pythagorean major 7th meitnerium 243/128
116 Comma 7th livermorium 160/81
117 Semicomma supermajor 7th tenessine multiple
118 perfect 8ve oganesson / neutronium 2/1 exact

Notation

Possible chemical notation

This notation was proposed by Eliora in November 2021.

118 is the number of chemical elements in the first 7 periods of the periodic table, and it is the number of elements which are ever expected to be most useful to humans. As a result, chemical element names can be used as note names in 118edo. In addition, such a notation is succinct as each pitch class is unique, and also it doesn't favor any other temperament or tuning besides 118edo.

However, chemical notation's properties can also be a disadvantage - it requires memorizing the names of the elements of the periodic table. In addition, uniqueness of pitch class is a disadvantage as well - since all the notes are separately named, it does not reflect the harmonic structure of 118edo.

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
  • 118et is lower in relative error than any previous ETs in the 5-limit. Not until 171 do we find a better ET in terms of absolute error, and not until 441 do we find one in terms of relative error.

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
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