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{{Infobox ET
{{Infobox ET}}
| Prime factorization = 2 × 59
{{ED intro}}
| Step size = 10.169¢
| Fifth = 69\118 (701.695¢)
| Major 2nd = 20\118 (203¢)
| Semitones = 11:9 (112¢ : 92¢)
| Consistency = 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 [[cent]]s each.


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


In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank three [[Gamelismic family|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.
118edo is the 17th [[The Riemann zeta function and tuning|zeta peak edo]], and it has decent approximations to harmonics [[7/1|7]], [[11/1|11]], [[17/1|17]], and [[19/1|19]]. In the 7-limit, it is particularly notable for tempering out the [[gamelisma]], 1029/1024, and is an excellent tuning for the rank-3 [[Gamelismic family|gamelismic]] temperament, and for [[guiron]], the rank-2 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.
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.
Line 19: Line 11:
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 series of ten segments of twelve Pythagorean fifths minus the said comma.
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. In addition, one step of 118edo is close to the 2097152/2083725 (the [[bronzisma]]), [[169/168]], and [[170/169]].


118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]].
=== Prime harmonics ===
{{Harmonics in equal|118}}


=== Prime harmonics ===
=== Octave stretch ===
{{primes in edo|118}}
118edo's approximated harmonics 7, 11, 17 and 19 can be improved by employing a moderate [[stretched and compressed tuning|octave stretch]], using tunings such as [[69edf]] or [[187edt]], only at the cost of a little less accurate 5-limit part.
 
=== Subsets and supersets ===
Since 118 factors into primes as {{nowrap| 2 × 59 }}, 118edo contains [[2edo]] and [[59edo]] as subset edos. Its multiples, [[236edo]], [[354edo]] and [[472edo]] are all of various interests, each providing distinct interpretations of harmonics 7 and 11. See also [[118th-octave temperaments]].


== Intervals ==
== Intervals ==
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style=white-space:nowrap | Table of intervals in 118edo
|+ style="font-size: 105%; white-space: nowrap;" | Table of intervals in 118edo
|-
! Step
! Step
! Cents
! Cents
! Marks
! Marks
! Eliora's Naming System<br>(+Shruti 22 correspondence)
! Approximate Ratios*
! Eliora's Chemical Notation<br>(if base note = 0)
! Eliora's Naming System<br />(+Shruti 22 correspondence)
! Approximate Ratios *
! Chemical Notation<br />(see below, if {{nowrap|base note {{=}} 0}})
! [[Ups and downs notation]]
! [[SKULO interval names|SKULO]] notation
|-
|-
| 0
| 0
| 0.00
| 0.00
| P1
| P1
| [[1/1]]
| unison
| unison
| oganesson / neutronium
| oganesson / neutronium
| [[1/1]]
| {{UDnote|step=0}}
| D
|-
|-
| 1
| 1
| 10.17
| 10.17
|  
|  
| [[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| semicomma
| hydrogen
| hydrogen
| [[126/125]], [[225/224]], [[121/120]], [[243/242]]
| {{UDnote|step=1}}
| LD
|-
|-
| 2
| 2
| 20.34
| 20.34
|  
|  
| [[81/80]], [[531441/524288]]
| comma
| comma
| helium
| helium
| [[81/80]], [[531441/524288]]
| {{UDnote|step=2}}
| KD
|-
|-
| 3
| 3
| 30.51
| 30.51
|  
|  
|  
| [[64/63]], [[49/48]]
| augmented comma
| lithium
| lithium
| [[64/63]], [[49/48]]
| {{UDnote|step=3}}
| SD
|-
|-
| 4
| 4
| 40.68
| 40.68
|  
|  
| [[50/49]]
|  
|  
| beryllium
| beryllium
| [[50/49]]
| {{UDnote|step=4}}
| OD, uEb
|-
|-
| 5
| 5
| 50.85
| 50.85
|  
|  
| [[36/35]]
|  
|  
| boron
| boron
| [[36/35]]
| {{UDnote|step=5}}
| UD, oEb
|-
|-
| 6
| 6
| 61.02
| 61.02
|  
|  
| [[28/27]]
|  
|  
| carbon
| carbon
| [[28/27]]
| {{UDnote|step=6}}
| sEb, uD#
|-
|-
| 7
| 7
| 71.19
| 71.19
|  
|  
| [[25/24]]
|  
|  
| nitrogen
| nitrogen
| [[25/24]]
| {{UDnote|step=7}}
| kEb, oD#, (kkD#)
|-
|-
| 8
| 8
| 81.36
| 81.36
|  
|  
| [[21/20]], [[22/21]]
|  
|  
| oxygen
| oxygen
| [[21/20]], [[22/21]]
| {{UDnote|step=8}}
| lEb, sD#
|-
|-
| 9
| 9
| 91.53
| 91.53
| m2
| m2
| [[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| limma, dayavati
| fluorine
| fluorine
| [[19/18]], [[20/19]]
| {{UDnote|step=9}}
| Eb, kD#
|-
|-
| 10
| 10
| 101.69
| 101.69
|  
|  
| [[17/16]], [[18/17]]
| dodecaic semitone
| dodecaic semitone
| neon
| neon
| [[17/16]], [[18/17]]
| {{UDnote|step=10}}
| LEb, lD#
|-
|-
| 11
| 11
| 111.86
| 111.86
|  
|  
| [[16/15]], [[2187/2048]]
| apotome, ranjani
| apotome, ranjani
| sodium
| sodium
| [[16/15]]
| {{UDnote|step=11}}
| KEb, D#
|-
|-
| 12
| 12
| 122.03
| 122.03
|  
|  
| [[15/14]]
|  
|  
| magnesium
| magnesium
| [[15/14]]
| {{UDnote|step=12}}
| SEb
|-
|-
| 13
| 13
| 132.20
| 132.20
|  
|  
| [[27/25]]
|  
|  
| aluminium
| aluminium
| [[27/25]]
| {{UDnote|step=13}}
| OEb
|-
|-
| 14
| 14
| 142.37
| 142.37
|  
|  
| [[88/81]]
|  
|  
| silicon
| silicon
| [[88/81]]
| {{UDnote|step=14}}
| UEb
|-
|-
| 15
| 15
| 152.54
| 152.54
|  
|  
| [[12/11]]
|  
|  
| phosphorus
| phosphorus
| [[12/11]]
| {{UDnote|step=15}}
| uE
|-
|-
| 16
| 16
| 162.71
| 162.71
|  
|  
| [[11/10]]
|  
|  
| sulphur
| sulphur
| [[11/10]]
| {{UDnote|step=16}}
| oE
|-
|-
| 17
| 17
| 172.88
| 172.88
|  
|  
|  
| [[21/19]]
| diminished tone
| chlorine
| chlorine
| [[21/19]]
| {{UDnote|step=17}}
| sE
|-
|-
| 18
| 18
| 183.05
| 183.05
|  
|  
| diminished tone, ratika
| [[10/9]]
| minor tone, ratika
| argon
| argon
| [[10/9]]
| {{UDnote|step=18}}
| kE
|-
|-
| 19
| 19
| 193.22
| 193.22
|  
|  
| minor tone
| [[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
| potassium
| [[28/25]], [[19/17]]
| {{UDnote|step=19}}
| lE
|-
|-
| 20
| 20
| 203.39
| 203.39
| M2
| M2
| [[9/8]]
| major tone, raudri
| major tone, raudri
| calcium
| calcium
| [[9/8]]
| {{UDnote|step=20}}
| E
|-
|-
| 21
| 21
| 213.56
| 213.56
|  
|  
|  
| [[17/15]]
| augmented tone
| scandium
| scandium
| [[17/15]]
| {{UDnote|step=21}}
| LE
|-
|-
| 22
| 22
| 223.73
| 223.73
|  
|  
|  
| [[256/225]]
| minor slendric second
| titanium
| titanium
| [[256/225]]
| {{UDnote|step=22}}
| KE
|-
|-
| 23
| 23
| 233.90
| 233.90
|  
|  
| [[8/7]]
| septimal second, slendric 2
| septimal second, slendric 2
| vanadium
| vanadium
| [[8/7]]
| {{UDnote|step=23}}
| SE
|-
|-
| 24
| 24
| 244.07
| 244.07
|  
|  
|  
| [[144/125]], [[121/105]]
| major slendric second
| chromium
| chromium
| [[144/125]], [[121/105]]
| {{UDnote|step=24}}
| OE, uF
|-
|-
| 25
| 25
| 254.24
| 254.24
|  
|  
|  
| [[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
| manganese
| [[125/108]], [[81/70]], [[22/19]]
| {{UDnote|step=25}}
| UE, oF
|-
|-
| 26
| 26
| 260.41
| 260.41
|  
|  
| [[7/6]]
| septimal third
| septimal third
| iron
| iron
| [[7/6]]
| {{UDnote|step=26}}
| sF
|-
|-
| 27
| 27
| 274.58
| 274.58
|  
|  
|  
| [[75/64]]
| major septimal third
| cobalt
| cobalt
| [[75/64]]
| {{UDnote|step=27}}
| kF
|-
|-
| 28
| 28
| 284.75
| 284.75
|  
|  
| [[33/28]]
|  
|  
| nickel
| nickel
| [[33/28]]
| {{UDnote|step=28}}
| lF
|-
|-
| 29
| 29
| 294.92
| 294.92
| m3
| m3
| [[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| Pythagorean minor 3rd, krodha
| copper
| copper
| [[32/27]], [[19/16]]
| {{UDnote|step=29}}
| F
|-
|-
| 30
| 30
| 305.08
| 305.08
|  
|  
| [[25/21]]
|  
|  
| zinc
| zinc
| [[25/21]]
| {{UDnote|step=30}}
| LF
|-
|-
| 31
| 31
| 315.25
| 315.25
|  
|  
| [[6/5]]
| Classical minor 3rd, vajrika
| Classical minor 3rd, vajrika
| gallium
| gallium
| [[6/5]]
| {{UDnote|step=31}}
| KF
|-
|-
| 32
| 32
| 325.42
| 325.42
|  
|  
| [[98/81]]
|  
|  
| germanium
| germanium
| [[98/81]]
| {{UDnote|step=32}}
| SF
|-
|-
| 33
| 33
| 335.59
| 335.59
|  
|  
| [[40/33]], [[17/14]]
| Lesser tridecimal third
| Lesser tridecimal third
| arsenic
| arsenic
| [[40/33]], [[17/14]]
| {{UDnote|step=33}}
| OF
|-
|-
| 34
| 34
| 345.76
| 345.76
|  
|  
| [[11/9]]
| Minor-neutral third
| Minor-neutral third
| selenium
| selenium
| [[11/9]]
| {{UDnote|step=34}}
| UF
|-
|-
| 35
| 35
| 355.93
| 355.93
|  
|  
| [[27/22]], [[16/13]] I**
| Minor tridecimal neurtral third, "major-neutral" third
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
| bromine
| [[27/22]], 16/13 I**
| {{UDnote|step=35}}
| uF#
|-
|-
| 36
| 36
| 366.10
| 366.10
|  
|  
| [[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
| krypton
| [[99/80]], [[21/17]], 16/13 II**
| {{UDnote|step=36}}
| oF#
|-
|-
| 37
| 37
| 376.27
| 376.27
|  
|  
| [[56/45]]
|  
|  
| rubidium
| rubidium
| [[56/45]]
| {{UDnote|step=37}}
| sF#
|-
|-
| 38
| 38
| 386.44
| 386.44
|  
|  
| [[5/4]]
| Classical major 3rd, prasarini
| Classical major 3rd, prasarini
| strontium
| strontium
| [[5/4]]
| {{UDnote|step=38}}
| kF#
|-
|-
| 39
| 39
| 396.61
| 396.61
|  
|  
| [[63/50]]
|  
|  
| yttrium
| yttrium
| [[63/50]]
| {{UDnote|step=39}}
| lF#
|-
|-
| 40
| 40
| 406.78
| 406.78
| M3
| M3
| [[24/19]], [[19/15]]
| Pythagorean major 3rd
| Pythagorean major 3rd
| zirconium
| zirconium
| [[24/19]], [[19/15]]
| {{UDnote|step=40}}
| F#
|-
|-
| 41
| 41
| 416.95
| 416.95
|  
|  
| [[14/11]]
|  
|  
| niobium
| niobium
| [[14/11]]
| {{UDnote|step=41}}
| LF#
|-
|-
| 42
| 42
| 427.12
| 427.12
|  
|  
| [[77/60]]
|  
|  
| molybdenum
| molybdenum
| [[77/60]]
| {{UDnote|step=42}}
| KF#
|-
|-
| 43
| 43
| 437.29
| 437.29
|  
|  
| [[9/7]]
|  
|  
| technetium
| technetium
| [[9/7]]
| {{UDnote|step=43}}
| SF#
|-
|-
| 44
| 44
| 447.46
| 447.46
|  
|  
| [[35/27]], [[22/17]]
|  
|  
| ruthenium
| ruthenium
| [[35/27]], [[22/17]]
| {{UDnote|step=44}}
| OF#, uG
|-
|-
| 45
| 45
| 457.63
| 457.63
|  
|  
| [[98/75]]
| Barbados 3rd
| Barbados 3rd
| rhodium
| rhodium
| [[98/75]]
| {{UDnote|step=45}}
| UF#, oG
|-
|-
| 46
| 46
| 467.80
| 467.80
|  
|  
| [[21/16]]
| Slendric 3
| Slendric 3
| palladium
| palladium
| [[21/16]]
| {{UDnote|step=46}}
| sG
|-
|-
| 47
| 47
| 477.97
| 477.97
|  
|  
| [[320/243]]
|  
|  
| silver
| silver
| [[320/243]]
| {{UDnote|step=47}}
| kG
|-
|-
| 48
| 48
| 488.14
| 488.14
|  
|  
| [[160/121]], [[85/64]]
|  
|  
| cadmium
| cadmium
| [[160/121]], [[85/64]]
| {{UDnote|step=48}}
| lG
|-
|-
| 49
| 49
| 498.31
| 498.31
| P4
| P4
| [[4/3]]
| perfect 4th
| perfect 4th
| indium
| indium
| [[4/3]]
| {{UDnote|step=49}}
| G
|-
|-
| 50
| 50
| 508.47
| 508.47
|  
|  
| [[75/56]], [[51/38]]
|  
|  
| tin
| tin
| [[75/56]], [[51/38]]
| {{UDnote|step=50}}
| LG
|-
|-
| 51
| 51
| 518.64
| 518.64
|  
|  
| [[27/20]]
| Kshiti
| Kshiti
| antimony
| antimony
| [[27/20]]
| {{UDnote|step=51}}
| KG
|-
|-
| 52
| 52
| 528.81
| 528.81
|  
|  
| [[49/36]], [[19/14]]
|  
|  
| tellurium
| tellurium
| [[49/36]], [[19/14]]
| {{UDnote|step=52}}
| SG
|-
|-
| 53
| 53
| 538.98
| 538.98
|  
|  
| [[15/11]]
| {{UDnote|step=53}}
|  
|  
| iodine
| iodine
| [[15/11]]
| OG, uGb
|-
|-
| 54
| 54
| 549.15
| 549.15
|  
|  
| [[48/35]], [[11/8]]
| {{UDnote|step=54}}
|  
|  
| xenon
| xenon
| [[48/35]], [[11/8]]
| UG, oAb
|-
|-
| 55
| 55
| 559.32
| 559.32
|  
|  
| [[112/81]]
|  
|  
| caesium
| caesium
| [[112/81]]
| {{UDnote|step=55}}
| uG#, sAb
|-
|-
| 56
| 56
| 569.49
| 569.49
|  
|  
| [[25/18]]
|  
|  
| barium
| barium
| [[25/18]]
| {{UDnote|step=56}}
| oG#, (kkG#), kAb
|-
|-
| 57
| 57
| 579.66
| 579.66
|  
|  
| [[7/5]]
|  
|  
| lanthanum
| lanthanum
| [[7/5]]
| {{UDnote|step=57}}
| sG#, lAb
|-
|-
| 58
| 58
| 589.83
| 589.83
| d5
| d5
| [[45/32]]
| Rakta
| Rakta
| cerium
| cerium
| [[45/32]]
| {{UDnote|step=58}}
| kG#, Ab
|-
|-
| 59
| 59
| 600.00
| 600.00
|  
|  
| [[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| symmetric tritone
| praseodymium
| praseodymium
| [[99/70]], [[140/99]], [[17/12]], [[24/17]]
| {{UDnote|step=59}}
| lG#, LAb
|-
|-
| 60
| 60
| 610.17
| 610.17
| A4
| A4
| [[64/45]], [[729/512]]
| Literal tritone, sandipani
| Literal tritone, sandipani
| neodymium
| neodymium
| [[64/45]]
| {{UDnote|step=60}}
| G#, KAb
|-
|-
| 61
| 61
| 620.34
| 620.34
|  
|  
| [[10/7]]
|  
|  
|  
| promethium
| [[10/7]]
| {{UDnote|step=61}}
| LG#, SAb
|-
|-
| 62
| 62
| 630.51
| 630.51
|  
|  
| [[36/25]]
|  
|  
|  
| samarium
| [[36/25]]
| {{UDnote|step=62}}
| KG#, OAb, (KKAb)
|-
|-
| 63
| 63
| 640.68
| 640.68
|  
|  
| [[81/56]]
|  
|  
|  
| europium
| [[81/56]]
| {{UDnote|step=63}}
| SG#, UAb
|-
|-
| 64
| 64
| 650.85
| 650.85
|  
|  
| [[35/24]], [[16/11]]
|  
|  
|  
| gadolinium
| [[35/24]], [[16/11]]
| {{UDnote|step=64}}
| OG#, uA
|-
|-
| 65
| 65
| 661.02
| 661.02
|  
|  
| [[22/15]]
|  
|  
|  
| terbium
| [[22/15]]
| {{UDnote|step=65}}
| oA
|-
|-
| 66
| 66
| 671.19
| 671.19
|  
|  
| [[72/49]], [[28/19]]
|  
|  
|  
| dysprosium
| [[72/49]], [[28/19]]
| {{UDnote|step=66}}
| sA
|-
|-
| 67
| 67
| 681.36
| 681.36
|  
|  
| [[40/27]]
| wolf 5th
| wolf 5th
| holmium
| holmium
| [[40/27]]
| {{UDnote|step=67}}
| kA
|-
|-
| 68
| 68
| 691.53
| 691.53
|  
|  
| [[112/75]], [[76/51]]
| wolf cub 5th
| wolf cub 5th
| erbium
| erbium
| [[112/75]], [[76/51]]
| {{UDnote|step=68}}
| lA
|-
|-
| 69
| 69
| 701.69
| 701.69
| P5
| P5
| perfect 5th
| [[3/2]]
| perfect 5th, slendric 4
| thulium
| thulium
| [[3/2]]
| {{UDnote|step=69}}
| A
|-
|-
| 70
| 70
| 711.86
| 711.86
|  
|  
|  
| [[121/80]], [[128/85]]
| sheep 5th
| ytterbium
| ytterbium
| [[121/80]], [[128/85]]
| {{UDnote|step=70}}
| lA
|-
|-
| 71
| 71
| 722.03
| 722.03
|  
|  
|  
| [[243/160]]
| lamb 5th
| lutetium
| lutetium
| [[243/160]]
| {{UDnote|step=71}}
| KA
|-
|-
| 72
| 72
| 732.20
| 732.20
|  
|  
| [[32/21]]
|  
|  
| hafnium
| hafnium
| [[32/21]]
| {{UDnote|step=72}}
| SA
|-
|-
| 73
| 73
| 742.37
| 742.37
|  
|  
| [[75/49]]
|  
|  
| tantalum
| tantalum
| [[75/49]]
| {{UDnote|step=73}}
| OA, uBb
|-
|-
| 74
| 74
| 752.54
| 752.54
|  
|  
| [[54/35]], [[17/11]]
|  
|  
| tungsten
| tungsten
| [[54/35]], [[17/11]]
| {{UDnote|step=74}}
| UA, oBb
|-
|-
| 75
| 75
| 762.71
| 762.71
|  
|  
| [[14/9]]
|  
|  
| rhenium
| rhenium
| [[14/9]]
| {{UDnote|step=75}}
| sBb
|-
|-
| 76
| 76
| 772.88
| 772.88
|  
|  
| [[120/77]]
|  
|  
| osmium
| osmium
| [[120/77]]
| {{UDnote|step=76}}
| kBb
|-
|-
| 77
| 77
| 783.05
| 783.05
|  
|  
| [[11/7]]
|  
|  
| iridium
| iridium
| [[11/7]]
| {{UDnote|step=77}}
| lBb
|-
|-
| 78
| 78
| 793.22
| 793.22
| m6
| m6
| [[19/12]], [[30/19]]
| Pythagorean minor 6th
| Pythagorean minor 6th
| platinum
| platinum
| [[19/12]], [[30/19]]
| {{UDnote|step=78}}
| Bb
|-
|-
| 79
| 79
| 803.39
| 803.39
|  
|  
| [[100/63]]
|  
|  
| gold
| gold
| [[100/63]]
| {{UDnote|step=79}}
| LBb
|-
|-
| 80
| 80
| 813.56
| 813.56
|  
|  
| [[8/5]]
| Classical minor 6th
| Classical minor 6th
| mercury
| mercury
| [[8/5]]
| {{UDnote|step=80}}
| KBb
|-
|-
| 81
| 81
| 823.73
| 823.73
|  
|  
| [[45/28]]
|  
|  
| thallium
| thallium
| [[45/28]]
| {{UDnote|step=81}}
| SBb
|-
|-
| 82
| 82
| 833.90
| 833.90
|  
|  
| [[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
| lead
| [[160/99]], [[34/21]], [[13/8]] I**
| {{UDnote|step=82}}
| OBb
|-
|-
| 83
| 83
| 844.07
| 844.07
|  
|  
| [[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
| bismuth
| [[44/27]], 13/8 II**
| {{UDnote|step=83}}
| UBb
|-
|-
| 84
| 84
| 854.24
| 854.24
|  
|  
| [[18/11]]
| Major-neutral sixth
| Major-neutral sixth
| polonium
| polonium
| [[18/11]]
| {{UDnote|step=84}}
| uB
|-
|-
| 85
| 85
| 864.41
| 864.41
|  
|  
| [[28/17]]
|  
|  
|  
| astatine
| [[28/17]]
| {{UDnote|step=85}}
| oB
|-
|-
| 86
| 86
| 874.58
| 874.58
|  
|  
| [[81/49]]
|  
|  
|  
| radon
| [[81/49]]
| {{UDnote|step=86}}
| sB
|-
|-
| 87
| 87
| 884.75
| 884.75
|  
|  
| [[5/3]]
| Classical major 6th
| Classical major 6th
| francium
| francium
| [[5/3]]
| {{UDnote|step=87}}
| kB
|-
|-
| 88
| 88
| 894.92
| 894.92
|  
|  
| [[42/25]]
|  
|  
|  
| radium
| [[42/25]]
| {{UDnote|step=88}}
| lB
|-
|-
| 89
| 89
| 905.08
| 905.08
| M6
| M6
| [[27/16]], [[32/19]]
| Pythagorean major 6th
| Pythagorean major 6th
| actinium
| actinium
| [[27/16]], [[32/19]]
| {{UDnote|step=89}}
| B
|-
|-
| 90
| 90
| 915.25
| 915.25
|  
|  
| [[56/33]]
|  
|  
|  
| thorium
| [[56/33]]
| {{UDnote|step=90}}
| LB
|-
|-
| 91
| 91
| 925.42
| 925.42
|  
|  
| [[128/75]]
|  
|  
|  
| protactinium
| [[128/75]]
| {{UDnote|step=91}}
| KB
|-
|-
| 92
| 92
| 935.59
| 935.59
|  
|  
| Septimal supermajor 6th, slendro 5
| [[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
| uranium
| [[12/7]]
| {{UDnote|step=92}}
| SB
|-
|-
| 93
| 93
| 945.76
| 945.76
|  
|  
| [[216/125]], [[140/81]], [[121/70]], [[19/11]]
|  
|  
| neptunium
| neptunium
| [[216/125]], [[140/81]], [[121/70]], [[19/11]]
| {{UDnote|step=93}}
| OB, uC
|-
|-
| 94
| 94
| 955.93
| 955.93
|  
|  
| [[125/72]]
|  
|  
| plutonium
| plutonium
| [[125/72]]
| {{UDnote|step=94}}
| UB, oC
|-
|-
| 95
| 95
| 966.10
| 966.10
|  
|  
| [[7/4]]
| Harmonic 7th
| Harmonic 7th
| americium
| americium
| [[7/4]]
| {{UDnote|step=95}}
| sC
|-
|-
| 96
| 96
| 976.27
| 976.27
|  
|  
| [[225/128]]
|  
|  
| curium
| curium
| [[225/128]]
| {{UDnote|step=96}}
| kC
|-
|-
| 97
| 97
| 986.44
| 986.44
|  
|  
| [[30/17]]
|  
|  
| berkelium
| berkelium
| [[30/17]]
| {{UDnote|step=97}}
| lC
|-
|-
| 98
| 98
| 996.61
| 996.61
| m7
| m7
|
|
| [[16/9]]
| [[16/9]]
| Pythagorean minor 7th
| californium
| {{UDnote|step=98}}
| C
|-
|-
| 99
| 99
| 1006.78
| 1006.78
|  
|  
| [[25/14]]
|  
|  
|  
| einsteinium
| [[25/14]]
| {{UDnote|step=99}}
| LC
|-
|-
| 100
| 100
| 1016.95
| 1016.95
|  
|  
| [[9/5]]
| Tivra
| Tivra
| fermium
| fermium
| [[9/5]]
| {{UDnote|step=100}}
| KC
|-
|-
| 101
| 101
| 1027.12
| 1027.12
|  
|  
| [[38/21]]
|  
|  
|  
| mendelevium
| [[38/21]]
| {{UDnote|step=101}}
| SC
|-
|-
| 102
| 102
| 1037.29
| 1037.29
|  
|  
| [[20/11]]
|  
|  
|  
| nobelium
| [[20/11]]
| {{UDnote|step=102}}
| OC, uDb
|-
|-
| 103
| 103
| 1047.46
| 1047.46
|  
|  
| [[11/6]]
|  
|  
|  
| lawrencium
| [[11/6]]
| {{UDnote|step=103}}
| UC, oDb
|-
|-
| 104
| 104
| 1057.63
| 1057.63
|  
|  
| [[81/44]]
|  
|  
|  
| rutherfordium
| [[81/44]]
| {{UDnote|step=104}}
| uC#, sDb
|-
|-
| 105
| 105
| 1067.80
| 1067.80
|  
|  
| [[50/27]]
|  
|  
|  
| dubnium
| [[50/27]]
| {{UDnote|step=105}}
| oC#, kDb
|-
|-
| 106
| 106
| 1077.97
| 1077.97
|  
|  
| [[28/15]]
|  
|  
|  
| seaborgium
| [[28/15]]
| {{UDnote|step=106}}
| sC#, lDb
|-
|-
| 107
| 107
| 1088.14
| 1088.14
|  
|  
| [[15/8]]
|  
|  
|  
| bohrium
| [[15/8]]
| {{UDnote|step=107}}
| kC#, Db
|-
|-
| 108
| 108
| 1098.31
| 1098.31
|  
|  
| [[32/17]], [[17/9]]
|  
|  
|  
| hassium
| [[32/17]], [[17/9]]
| {{UDnote|step=108}}
| lC#, LDb
|-
|-
| 109
| 109
| 1108.47
| 1108.47
| M7
| M7
| [[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| Pythagorean major 7th
| meitnerium
| meitnerium
| [[36/19]], [[19/10]]
| {{UDnote|step=109}}
| C#, KDb
|-
|-
| 110
| 110
| 1118.64
| 1118.64
|  
|  
| [[40/21]], [[21/11]]
|  
|  
| darmstadtium
| darmstadtium
| [[40/21]], [[21/11]]
| {{UDnote|step=110}}
| LC#, SDb
|-
|-
| 111
| 111
| 1128.81
| 1128.81
|  
|  
| [[48/25]]
|  
|  
| roentgenium
| roentgenium
| [[48/25]]
| {{UDnote|step=111}}
| KC#, ODb, (kkDb)
|-
|-
| 112
| 112
| 1138.98
| 1138.98
|  
|  
| [[27/14]]
|  
|  
| copernicium
| copernicium
| [[27/14]]
| {{UDnote|step=112}}
| SC#, UDb
|-
|-
| 113
| 113
| 1149.15
| 1149.15
|  
|  
| [[35/18]], [[64/33]]
|  
|  
| nihonium
| nihonium
| [[35/18]], [[64/33]]
| {{UDnote|step=113}}
| OC#, uD
|-
|-
| 114
| 114
| 1159.32
| 1159.32
|  
|  
| [[49/25]]
|  
|  
| flerovium
| flerovium
| [[49/25]]
| {{UDnote|step=114}}
| UC#, oD
|-
|-
| 115
| 115
| 1169.49
| 1169.49
|  
|  
| [[63/32]], [[96/49]]
|  
|  
| moscovium
| moscovium
| [[63/32]], [[96/49]]
| {{UDnote|step=115}}
| sD
|-
|-
| 116
| 116
| 1179.66
| 1179.66
|  
|  
| Comma 7th
| [[160/81]]
| Comma supermajor 7th
| livermorium
| livermorium
| [[160/81]]
| {{UDnote|step=116}}
| kD
|-
|-
| 117
| 117
| 1189.83
| 1189.83
|  
|  
| [[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| Semicomma supermajor 7th
| tenessine
| tenessine
| [[125/63]], [[448/225]], [[240/121]], [[484/243]]
| {{UDnote|step=117}}
| lD
|-
|-
| 118
| 118
| 1200.00
| 1200.00
| P8
| P8
| [[2/1]]
| perfect 8ve
| perfect 8ve
| oganesson / neutronium
| oganesson / neutronium
| [[2/1]]
| {{UDnote|step=118}}
| D
|}
|}
<nowiki>*</nowiki> treated as a 2.3.5.7.11.17.19 system
<nowiki />* {{sg|2.3.5.7.11.17.19 subgroup}}


<nowiki>**</nowiki> based on a dual-interval interpretation for the 13th harmonic
<nowiki />** Based on a dual-interval interpretation for the 13th harmonic


== Notation ==
== Notation ==
=== Possible chemical notation ===
=== Chemical notation ===
This notation was proposed by Eliora in November 2021.  
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.
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. Chemical notation's properties can be a disadvantage - it requires memorizing the names of the elements of the periodic table. However, the notation is succinct and some people prefer this kind of notation for edosteps, as unlike MOS or JI-based notations, it is entirely based on 118edo alone and does not imply a preference of one edo over another.


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.
The following are the correspondences of the periodic table structure with 118edo: 


This being said, there are a few correspondences - 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo. In addition, 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively. Mercury, ending the 6d-block, corresponds to 8/5. The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.   
* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo.
* 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively.  
* Mercury, ending the 6d-block, corresponds to 8/5.  
* The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.   
* 6\118, the width of the p-block, corresponds to one small step of the maximally even parakleismic scale, created by stacking 6/5.
 
== Approximation to JI ==
=== Zeta peak index ===
{{ZPI
| zpi = 706
| steps = 117.969513574257
| step size = 10.1721195895637
| tempered height = 9.850823
| pure height = 8.968412
| integral = 1.544280
| gap = 18.861062
| octave = 1200.31011156852
| consistent = 12
| distinct = 12
}}


== 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" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
Line 896: Line 1,153:
| 2.3
| 2.3
| {{monzo| -187 118 }}
| {{monzo| -187 118 }}
| [{{val| 118 187 }}]
| {{mapping| 118 187 }}
| -0.119
| &minus;0.119
| 0.082
| 0.082
| 0.81
| 0.81
Line 903: Line 1,160:
| 2.3.5
| 2.3.5
| 32805/32768, {{monzo| 8 14 -13 }}
| 32805/32768, {{monzo| 8 14 -13 }}
| [{{val| 118 187 274 }}]
| {{mapping| 118 187 274 }}
| +0.036
| +0.036
| 0.093
| 0.093
Line 910: Line 1,167:
| 2.3.5.7
| 2.3.5.7
| 1029/1024, 3136/3125, 4375/4374
| 1029/1024, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 }}]
| {{mapping| 118 187 274 331 }}
| +0.270
| +0.270
| 0.412
| 0.412
Line 917: Line 1,174:
| 2.3.5.7.11
| 2.3.5.7.11
| 385/384, 441/440, 3136/3125, 4375/4374
| 385/384, 441/440, 3136/3125, 4375/4374
| [{{val| 118 187 274 331 408 }}]
| {{mapping| 118 187 274 331 408 }}
| +0.341
| +0.341
| 0.370
| 0.370
| 3.89
| 3.89
|-
|- style="border-top: double;"
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 196/195, 352/351, 384/384, 625/624, 729/728
| 196/195, 352/351, 384/384, 625/624, 729/728
| [{{val| 118 187 274 331 408 437 }}] (118)
| {{mapping| 118 187 274 331 408 437 }} (118)
| +0.125
| +0.125
| 0.604
| 0.604
| 5.93
| 5.93
|-
|- style="border-top: double;"
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| 169/168, 325/324, 364/363, 385/384, 3136/3125
| [{{val| 118 187 274 331 408 436 }}] (118f)
| {{mapping| 118 187 274 331 408 436 }} (118f)
| +0.583
| +0.583
| 0.650
| 0.650
| 6.39
| 6.39
|-
|- style="border-top: double;"
| 2.3.5.7.11.17
| 2.3.5.7.11.17
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| 289/288, 385/384, 441/440, 561/560, 3136/3125
| [{{val| 118 187 274 331 408 482 }}]
| {{mapping| 118 187 274 331 408 482 }}
| +0.417
| +0.417
| 0.399
| 0.399
Line 945: Line 1,202:
| 2.3.5.7.11.17.19
| 2.3.5.7.11.17.19
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968
| [{{val| 118 187 274 331 408 482 501 }}]
| {{mapping| 118 187 274 331 408 482 501 }}
| +0.445
| +0.445
| 0.376
| 0.376
| 3.69
| 3.69
|}
|}
* 118et is lower in relative error than any previous ETs in the 5-limit. Not until [[171edo|171]] do we find a better ET in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.
* 118et is lower in relative error than any previous equal temperaments in the 5-limit. Not until [[171edo|171]] do we find a better one in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.


=== 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 octave
|-
! Generator<br>(reduced)
! Periods<br>per 8ve
! Cents<br>(reduced)
! Generator*
! Associated<br>ratio
! Cents*
! Temperaments
! Associated<br>ratio*
! Temperament
|-
|-
| 1
| 1
Line 995: Line 1,253:
| 498.31
| 498.31
| 4/3
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
| [[Helmholtz (temperament)|Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
|-
| 1
| 1
Line 1,007: Line 1,265:
| 20.34
| 20.34
| 81/80
| 81/80
| [[Commatic]]
| [[Bicommatic]]
|-
|-
| 2
| 2
Line 1,051: Line 1,309:
| [[Semiparakleismic]]
| [[Semiparakleismic]]
|}
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
== Instruments ==
[[Lumatone mapping for 118edo]]
== Music ==
; [[Mercury Amalgam]]
* [https://www.youtube.com/watch?v=eYnSsOnRZIs ''Pops''] (2022)


[[Category:Equal divisions of the octave]]
[[Category:Theory]]
[[Category:Gamelismic]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Guiron]]
[[Category:Listen]]
[[Category:Parakleismic]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Portent]]
[[Category:Schismic]]
[[Category:Schismic]]
Retrieved from "https://en.xen.wiki/w/118edo"