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{{Infobox ET}}
{{Infobox ET}}
 
{{ED intro}}
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.
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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]].
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}}
 
=== Octave stretch ===
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.  


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


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


* 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo. I 
* 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.   
* 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.   
* 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.   
* 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.
* 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 895: 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 902: 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 909: 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 916: 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 944: 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 8ve
! Periods<br>per 8ve
! Generator<br>(reduced)
! Generator*
! Cents<br>(reduced)
! Cents*
! Associated<br>ratio
! Associated<br>ratio*
! Temperaments
! Temperament
|-
|-
| 1
| 1
Line 994: Line 1,253:
| 498.31
| 498.31
| 4/3
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
| [[Helmholtz (temperament)|Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
|-
| 1
| 1
Line 1,006: Line 1,265:
| 20.34
| 20.34
| 81/80
| 81/80
| [[Commatic]]
| [[Bicommatic]]
|-
|-
| 2
| 2
Line 1,050: 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 ==
== Music ==
* [https://www.youtube.com/watch?v=eYnSsOnRZIs Pops] by [[Mercury Amalgam]]
; [[Mercury Amalgam]]
* [https://www.youtube.com/watch?v=eYnSsOnRZIs ''Pops''] (2022)


[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Gamelismic]]
[[Category:Gamelismic]]
[[Category:Guiron]]
[[Category:Guiron]]
[[Category:Listen]]
[[Category:Parakleismic]]
[[Category:Parakleismic]]
[[Category:Portent]]
[[Category:Portent]]
[[Category:Schismic]]
[[Category:Schismic]]

Latest revision as of 13:43, 19 June 2025

← 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

118 equal divisions of the octave (abbreviated 118edo or 118ed2), also called 118-tone equal temperament (118tet) or 118 equal temperament (118et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 118 equal parts of about 10.2 ¢ each. Each step represents a frequency ratio of 21/118, or the 118th root of 2.

Theory

118edo is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. It 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 comma, [-53 10 16⟩.

118edo is the 17th zeta peak edo, and it has decent approximations to harmonics 7, 11, 17, and 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 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.

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. In addition, one step of 118edo is close to the 2097152/2083725 (the bronzisma), 169/168, and 170/169.

Prime harmonics

Approximation of prime harmonics in 118edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -0.26 +0.13 -2.72 -2.17 +3.54 -3.26 -2.60 +2.23 -2.46 +4.12
Relative (%) +0.0 -2.6 +1.2 -26.8 -21.3 +34.8 -32.1 -25.5 +22.0 -24.2 +40.5
Steps
(reduced) 		118
(0) 		187
(69) 		274
(38) 		331
(95) 		408
(54) 		437
(83) 		482
(10) 		501
(29) 		534
(62) 		573
(101) 		585
(113)

Octave stretch

118edo's approximated harmonics 7, 11, 17 and 19 can be improved by employing a moderate 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 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

Table of intervals in 118edo
Step Cents Marks Approximate Ratios* Eliora's Naming System
(+Shruti 22 correspondence) 		Chemical Notation
(see below, if base note = 0) 		Ups and downs notation SKULO notation
0 0.00 P1 1/1 unison oganesson / neutronium D D
1 10.17 126/125, 225/224, 121/120, 243/242 semicomma hydrogen ^D, ^3E♭♭ LD
2 20.34 81/80, 531441/524288 comma helium ^^D, ^4E♭♭ KD
3 30.51 64/63, 49/48 augmented comma lithium ^3D, ^5E♭♭ SD
4 40.68 50/49 beryllium ^4D, v5E♭ OD, uEb
5 50.85 36/35 boron ^5D, v4E♭ UD, oEb
6 61.02 28/27 carbon v5D♯, v3E♭ sEb, uD#
7 71.19 25/24 nitrogen v4D♯, vvE♭ kEb, oD#, (kkD#)
8 81.36 21/20, 22/21 oxygen v3D♯, vE♭ lEb, sD#
9 91.53 m2 19/18, 20/19, 256/243 limma, dayavati fluorine vvD♯, E♭ Eb, kD#
10 101.69 17/16, 18/17 dodecaic semitone neon vD♯, ^E♭ LEb, lD#
11 111.86 16/15, 2187/2048 apotome, ranjani sodium D♯, ^^E♭ KEb, D#
12 122.03 15/14 magnesium ^D♯, ^3E♭ SEb
13 132.20 27/25 aluminium ^^D♯, ^4E♭ OEb
14 142.37 88/81 silicon ^3D♯, ^5E♭ UEb
15 152.54 12/11 phosphorus ^4D♯, v5E uE
16 162.71 11/10 sulphur ^5D♯, v4E oE
17 172.88 21/19 diminished tone chlorine v5D𝄪, v3E sE
18 183.05 10/9 minor tone, ratika argon v4D𝄪, vvE kE
19 193.22 28/25, 19/17 neutral tone, quasi-meantone potassium v3D𝄪, vE lE
20 203.39 M2 9/8 major tone, raudri calcium E E
21 213.56 17/15 augmented tone scandium ^E, ^3F♭ LE
22 223.73 256/225 minor slendric second titanium ^^E, ^4F♭ KE
23 233.90 8/7 septimal second, slendric 2 vanadium ^3E, ^5F♭ SE
24 244.07 144/125, 121/105 major slendric second chromium ^4E, v5F OE, uF
25 254.24 125/108, 81/70, 22/19 minor septimal third manganese ^5E, v4F UE, oF
26 260.41 7/6 septimal third iron v5E♯, v3F sF
27 274.58 75/64 major septimal third cobalt v4E♯, vvF kF
28 284.75 33/28 nickel v3E♯, vF lF
29 294.92 m3 32/27, 19/16 Pythagorean minor 3rd, krodha copper F F
30 305.08 25/21 zinc ^F, ^3G♭♭ LF
31 315.25 6/5 Classical minor 3rd, vajrika gallium ^^F, ^4G♭♭ KF
32 325.42 98/81 germanium ^3F, ^5G♭♭ SF
33 335.59 40/33, 17/14 Lesser tridecimal third arsenic ^4F, v5G♭ OF
34 345.76 11/9 Minor-neutral third selenium ^5F, v4G♭ UF
35 355.93 27/22, 16/13 I** Minor tridecimal neurtral third, "major-neutral" third bromine v5F♯, v3G♭ uF#
36 366.10 99/80, 21/17, 16/13 II** Golden ratio 3rd, major-tridecimal neutral third krypton v4F♯, vvG♭ oF#
37 376.27 56/45 rubidium v3F♯, vG♭ sF#
38 386.44 5/4 Classical major 3rd, prasarini strontium vvF♯, G♭ kF#
39 396.61 63/50 yttrium vF♯, ^G♭ lF#
40 406.78 M3 24/19, 19/15 Pythagorean major 3rd zirconium F♯, ^^G♭ F#
41 416.95 14/11 niobium ^F♯, ^3G♭ LF#
42 427.12 77/60 molybdenum ^^F♯, ^4G♭ KF#
43 437.29 9/7 technetium ^3F♯, ^5G♭ SF#
44 447.46 35/27, 22/17 ruthenium ^4F♯, v5G OF#, uG
45 457.63 98/75 Barbados 3rd rhodium ^5F♯, v4G UF#, oG
46 467.80 21/16 Slendric 3 palladium v5F𝄪, v3G sG
47 477.97 320/243 silver v4F𝄪, vvG kG
48 488.14 160/121, 85/64 cadmium v3F𝄪, vG lG
49 498.31 P4 4/3 perfect 4th indium G G
50 508.47 75/56, 51/38 tin ^G, ^3A♭♭ LG
51 518.64 27/20 Kshiti antimony ^^G, ^4A♭♭ KG
52 528.81 49/36, 19/14 tellurium ^3G, ^5A♭♭ SG
53 538.98 15/11 ^4G, v5A♭ iodine OG, uGb
54 549.15 48/35, 11/8 ^5G, v4A♭ xenon UG, oAb
55 559.32 112/81 caesium v5G♯, v3A♭ uG#, sAb
56 569.49 25/18 barium v4G♯, vvA♭ oG#, (kkG#), kAb
57 579.66 7/5 lanthanum v3G♯, vA♭ sG#, lAb
58 589.83 d5 45/32 Rakta cerium vvG♯, A♭ kG#, Ab
59 600.00 99/70, 140/99, 17/12, 24/17 symmetric tritone praseodymium vG♯, ^A♭ lG#, LAb
60 610.17 A4 64/45, 729/512 Literal tritone, sandipani neodymium G♯, ^^A♭ G#, KAb
61 620.34 10/7 promethium ^G♯, ^3A♭ LG#, SAb
62 630.51 36/25 samarium ^^G♯, ^4A♭ KG#, OAb, (KKAb)
63 640.68 81/56 europium ^3G♯, ^5A♭ SG#, UAb
64 650.85 35/24, 16/11 gadolinium ^4G♯, v5A OG#, uA
65 661.02 22/15 terbium ^5G♯, v4A oA
66 671.19 72/49, 28/19 dysprosium v5G𝄪, v3A sA
67 681.36 40/27 wolf 5th holmium v4G𝄪, vvA kA
68 691.53 112/75, 76/51 wolf cub 5th erbium v3G𝄪, vA lA
69 701.69 P5 3/2 perfect 5th, slendric 4 thulium A A
70 711.86 121/80, 128/85 sheep 5th ytterbium ^A, ^3B♭♭ lA
71 722.03 243/160 lamb 5th lutetium ^^A, ^4B♭♭ KA
72 732.20 32/21 hafnium ^3A, ^5B♭♭ SA
73 742.37 75/49 tantalum ^4A, v5B♭ OA, uBb
74 752.54 54/35, 17/11 tungsten ^5A, v4B♭ UA, oBb
75 762.71 14/9 rhenium v5A♯, v3B♭ sBb
76 772.88 120/77 osmium v4A♯, vvB♭ kBb
77 783.05 11/7 iridium v3A♯, vB♭ lBb
78 793.22 m6 19/12, 30/19 Pythagorean minor 6th platinum vvA♯, B♭ Bb
79 803.39 100/63 gold vA♯, ^B♭ LBb
80 813.56 8/5 Classical minor 6th mercury A♯, ^^B♭ KBb
81 823.73 45/28 thallium ^A♯, ^3B♭ SBb
82 833.90 160/99, 34/21, 13/8 I** Golden ratio sixth, minor-neutral tridecimal sixth lead ^^A♯, ^4B♭ OBb
83 844.07 44/27, 13/8 II** Major tridecimal neutral sixth, "minor-neutral" sixth bismuth ^3A♯, ^5B♭ UBb
84 854.24 18/11 Major-neutral sixth polonium ^4A♯, v5B uB
85 864.41 28/17 astatine ^5A♯, v4B oB
86 874.58 81/49 radon v5A𝄪, v3B sB
87 884.75 5/3 Classical major 6th francium v4A𝄪, vvB kB
88 894.92 42/25 radium v3A𝄪, vB lB
89 905.08 M6 27/16, 32/19 Pythagorean major 6th actinium B B
90 915.25 56/33 thorium ^B, ^3C♭ LB
91 925.42 128/75 protactinium ^^B, ^4C♭ KB
92 935.59 12/7 Septimal supermajor 6th, slendric 5 uranium ^3B, ^5C♭ SB
93 945.76 216/125, 140/81, 121/70, 19/11 neptunium ^4B, v5C OB, uC
94 955.93 125/72 plutonium ^5B, v4C UB, oC
95 966.10 7/4 Harmonic 7th americium v5B♯, v3C sC
96 976.27 225/128 curium v4B♯, vvC kC
97 986.44 30/17 berkelium v3B♯, vC lC
98 996.61 m7 16/9 Pythagorean minor 7th californium C C
99 1006.78 25/14 einsteinium ^C, ^3D♭♭ LC
100 1016.95 9/5 Tivra fermium ^^C, ^4D♭♭ KC
101 1027.12 38/21 mendelevium ^3C, ^5D♭♭ SC
102 1037.29 20/11 nobelium ^4C, v5D♭ OC, uDb
103 1047.46 11/6 lawrencium ^5C, v4D♭ UC, oDb
104 1057.63 81/44 rutherfordium v5C♯, v3D♭ uC#, sDb
105 1067.80 50/27 dubnium v4C♯, vvD♭ oC#, kDb
106 1077.97 28/15 seaborgium v3C♯, vD♭ sC#, lDb
107 1088.14 15/8 bohrium vvC♯, D♭ kC#, Db
108 1098.31 32/17, 17/9 hassium vC♯, ^D♭ lC#, LDb
109 1108.47 M7 36/19, 19/10, 243/128 Pythagorean major 7th meitnerium C♯, ^^D♭ C#, KDb
110 1118.64 40/21, 21/11 darmstadtium ^C♯, ^3D♭ LC#, SDb
111 1128.81 48/25 roentgenium ^^C♯, ^4D♭ KC#, ODb, (kkDb)
112 1138.98 27/14 copernicium ^3C♯, ^5D♭ SC#, UDb
113 1149.15 35/18, 64/33 nihonium ^4C♯, v5D OC#, uD
114 1159.32 49/25 flerovium ^5C♯, v4D UC#, oD
115 1169.49 63/32, 96/49 moscovium v5C𝄪, v3D sD
116 1179.66 160/81 Comma supermajor 7th livermorium v4C𝄪, vvD kD
117 1189.83 125/63, 448/225, 240/121, 484/243 Semicomma supermajor 7th tenessine v3C𝄪, vD lD
118 1200.00 P8 2/1 perfect 8ve oganesson / neutronium D D

* Based on treating 2.3.5.7.11.17.19 subgroup as a 5-limit temperament; other approaches are also possible.

** Based on a dual-interval interpretation for the 13th harmonic

Notation

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. 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.

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

  • 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

Tuning Strength Octave (cents) Integer limit
ZPI Steps
per 8ve 		Step size
(cents) 		Height Integral Gap Size Stretch Consistent Distinct
Tempered Pure
706zpi 117.969514 10.17212 9.850823 8.968412 1.54428 18.861062 1200.310112 0.310112 12 12

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 equal temperaments in the 5-limit. Not until 171 do we find a better one 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 8ve 		Generator* Cents* Associated
ratio* 		Temperament
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 Bicommatic
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

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

Instruments

Lumatone mapping for 118edo

Music

Mercury Amalgam
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