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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.
Line 14: Line 13:
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 ===
{{harmonics in equal}}
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)
! Chemical Notation<br>(see below, if base note = 0)
! Chemical Notation<br />(see below, if {{nowrap|base note {{=}} 0}})
! [[Ups and downs notation]]
! [[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}}
| {{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}}
| {{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}}
| {{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}}
| {{UDnote|step=3}}
| SD
|-
|-
| 4
| 4
| 40.68
| 40.68
|  
|  
|[[50/49]]
| [[50/49]]
|  
|  
| beryllium
| beryllium
| {{UDnote|step=4}}
| {{UDnote|step=4}}
| OD, uEb
|-
|-
| 5
| 5
| 50.85
| 50.85
|  
|  
|[[36/35]]
| [[36/35]]
|  
|  
| boron
| boron
| {{UDnote|step=5}}
| {{UDnote|step=5}}
| UD, oEb
|-
|-
| 6
| 6
| 61.02
| 61.02
|  
|  
|[[28/27]]
| [[28/27]]
|  
|  
| carbon
| carbon
| {{UDnote|step=6}}
| {{UDnote|step=6}}
| sEb, uD#
|-
|-
| 7
| 7
| 71.19
| 71.19
|  
|  
|[[25/24]]
| [[25/24]]
|  
|  
| nitrogen
| nitrogen
| {{UDnote|step=7}}
| {{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}}
| {{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}}
| {{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}}
| {{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}}
| {{UDnote|step=11}}
| KEb, D#
|-
|-
| 12
| 12
| 122.03
| 122.03
|  
|  
|[[15/14]]
| [[15/14]]
|  
|  
| magnesium
| magnesium
| {{UDnote|step=12}}
| {{UDnote|step=12}}
| SEb
|-
|-
| 13
| 13
| 132.20
| 132.20
|  
|  
|[[27/25]]
| [[27/25]]
|  
|  
| aluminium
| aluminium
| {{UDnote|step=13}}
| {{UDnote|step=13}}
| OEb
|-
|-
| 14
| 14
| 142.37
| 142.37
|  
|  
|[[88/81]]
| [[88/81]]
|  
|  
| silicon
| silicon
| {{UDnote|step=14}}
| {{UDnote|step=14}}
| UEb
|-
|-
| 15
| 15
| 152.54
| 152.54
|  
|  
|[[12/11]]
| [[12/11]]
|  
|  
| phosphorus
| phosphorus
| {{UDnote|step=15}}
| {{UDnote|step=15}}
| uE
|-
|-
| 16
| 16
| 162.71
| 162.71
|  
|  
|[[11/10]]
| [[11/10]]
|  
|  
| sulphur
| sulphur
| {{UDnote|step=16}}
| {{UDnote|step=16}}
| oE
|-
|-
| 17
| 17
| 172.88
| 172.88
|  
|  
|[[21/19]]
| [[21/19]]
| diminished tone
| diminished tone
| chlorine
| chlorine
| {{UDnote|step=17}}
| {{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}}
| {{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}}
| {{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}}
| {{UDnote|step=20}}
| E
|-
|-
| 21
| 21
| 213.56
| 213.56
|  
|  
|[[17/15]]
| [[17/15]]
| augmented tone
| augmented tone
| scandium
| scandium
| {{UDnote|step=21}}
| {{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}}
| {{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}}
| {{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}}
| {{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}}
| {{UDnote|step=25}}
| UE, oF
|-
|-
| 26
| 26
| 260.41
| 260.41
|  
|  
|[[7/6]]
| [[7/6]]
| septimal third
| septimal third
| iron
| iron
| {{UDnote|step=26}}
| {{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}}
| {{UDnote|step=27}}
| kF
|-
|-
| 28
| 28
| 284.75
| 284.75
|  
|  
|[[33/28]]
| [[33/28]]
|  
|  
| nickel
| nickel
| {{UDnote|step=28}}
| {{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}}
| {{UDnote|step=29}}
| F
|-
|-
| 30
| 30
| 305.08
| 305.08
|  
|  
|[[25/21]]
| [[25/21]]
|  
|  
| zinc
| zinc
| {{UDnote|step=30}}
| {{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}}
| {{UDnote|step=31}}
| KF
|-
|-
| 32
| 32
| 325.42
| 325.42
|  
|  
|[[98/81]]
| [[98/81]]
|  
|  
| germanium
| germanium
| {{UDnote|step=32}}
| {{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}}
| {{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}}
| {{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}}
| {{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}}
| {{UDnote|step=36}}
| oF#
|-
|-
| 37
| 37
| 376.27
| 376.27
|  
|  
|[[56/45]]
| [[56/45]]
|  
|  
| rubidium
| rubidium
| {{UDnote|step=37}}
| {{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}}
| {{UDnote|step=38}}
| kF#
|-
|-
| 39
| 39
| 396.61
| 396.61
|  
|  
|[[63/50]]
| [[63/50]]
|  
|  
| yttrium
| yttrium
| {{UDnote|step=39}}
| {{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}}
| {{UDnote|step=40}}
| F#
|-
|-
| 41
| 41
| 416.95
| 416.95
|  
|  
|[[14/11]]
| [[14/11]]
|  
|  
| niobium
| niobium
| {{UDnote|step=41}}
| {{UDnote|step=41}}
| LF#
|-
|-
| 42
| 42
| 427.12
| 427.12
|  
|  
|[[77/60]]
| [[77/60]]
|  
|  
| molybdenum
| molybdenum
| {{UDnote|step=42}}
| {{UDnote|step=42}}
| KF#
|-
|-
| 43
| 43
| 437.29
| 437.29
|  
|  
|[[9/7]]
| [[9/7]]
|  
|  
| technetium
| technetium
| {{UDnote|step=43}}
| {{UDnote|step=43}}
| SF#
|-
|-
| 44
| 44
| 447.46
| 447.46
|  
|  
|[[35/27]], [[22/17]]
| [[35/27]], [[22/17]]
|  
|  
| ruthenium
| ruthenium
| {{UDnote|step=44}}
| {{UDnote|step=44}}
| OF#, uG
|-
|-
| 45
| 45
| 457.63
| 457.63
|  
|  
|[[98/75]]
| [[98/75]]
| Barbados 3rd
| Barbados 3rd
| rhodium
| rhodium
| {{UDnote|step=45}}
| {{UDnote|step=45}}
| UF#, oG
|-
|-
| 46
| 46
| 467.80
| 467.80
|  
|  
|[[21/16]]
| [[21/16]]
| Slendric 3
| Slendric 3
| palladium
| palladium
| {{UDnote|step=46}}
| {{UDnote|step=46}}
| sG
|-
|-
| 47
| 47
| 477.97
| 477.97
|  
|  
|[[320/243]]
| [[320/243]]
|  
|  
| silver
| silver
| {{UDnote|step=47}}
| {{UDnote|step=47}}
| kG
|-
|-
| 48
| 48
| 488.14
| 488.14
|  
|  
|[[160/121]], [[85/64]]
| [[160/121]], [[85/64]]
|  
|  
| cadmium
| cadmium
| {{UDnote|step=48}}
| {{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}}
| {{UDnote|step=49}}
| G
|-
|-
| 50
| 50
| 508.47
| 508.47
|  
|  
|[[75/56]], [[51/38]]
| [[75/56]], [[51/38]]
|  
|  
| tin
| tin
| {{UDnote|step=50}}
| {{UDnote|step=50}}
| LG
|-
|-
| 51
| 51
| 518.64
| 518.64
|  
|  
|[[27/20]]
| [[27/20]]
| Kshiti
| Kshiti
| antimony
| antimony
| {{UDnote|step=51}}
| {{UDnote|step=51}}
| KG
|-
|-
| 52
| 52
| 528.81
| 528.81
|  
|  
|[[49/36]], [[19/14]]
| [[49/36]], [[19/14]]
|  
|  
| tellurium
| tellurium
| {{UDnote|step=52}}
| {{UDnote|step=52}}
| SG
|-
|-
| 53
| 53
| 538.98
| 538.98
|  
|  
|[[15/11]]
| [[15/11]]
| {{UDnote|step=53}}
| {{UDnote|step=53}}
|  
|  
| iodine
| iodine
| OG, uGb
|-
|-
| 54
| 54
| 549.15
| 549.15
|  
|  
|[[48/35]], [[11/8]]
| [[48/35]], [[11/8]]
| {{UDnote|step=54}}
| {{UDnote|step=54}}
|  
|  
| xenon
| xenon
| UG, oAb
|-
|-
| 55
| 55
| 559.32
| 559.32
|  
|  
|[[112/81]]
| [[112/81]]
|  
|  
| caesium
| caesium
| {{UDnote|step=55}}
| {{UDnote|step=55}}
| uG#, sAb
|-
|-
| 56
| 56
| 569.49
| 569.49
|  
|  
|[[25/18]]
| [[25/18]]
|  
|  
| barium
| barium
| {{UDnote|step=56}}
| {{UDnote|step=56}}
| oG#, (kkG#), kAb
|-
|-
| 57
| 57
| 579.66
| 579.66
|  
|  
|[[7/5]]
| [[7/5]]
|  
|  
| lanthanum
| lanthanum
| {{UDnote|step=57}}
| {{UDnote|step=57}}
| sG#, lAb
|-
|-
| 58
| 58
| 589.83
| 589.83
| d5
| d5
|[[45/32]]
| [[45/32]]
| Rakta
| Rakta
| cerium
| cerium
| {{UDnote|step=58}}
| {{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}}
| {{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}}
| {{UDnote|step=60}}
| G#, KAb
|-
|-
| 61
| 61
| 620.34
| 620.34
|  
|  
|[[10/7]]
| [[10/7]]
|  
|  
| promethium
| promethium
| {{UDnote|step=61}}
| {{UDnote|step=61}}
| LG#, SAb
|-
|-
| 62
| 62
| 630.51
| 630.51
|  
|  
|[[36/25]]
| [[36/25]]
|  
|  
| samarium
| samarium
| {{UDnote|step=62}}
| {{UDnote|step=62}}
| KG#, OAb, (KKAb)
|-
|-
| 63
| 63
| 640.68
| 640.68
|  
|  
|[[81/56]]
| [[81/56]]
|  
|  
| europium
| europium
| {{UDnote|step=63}}
| {{UDnote|step=63}}
| SG#, UAb
|-
|-
| 64
| 64
| 650.85
| 650.85
|  
|  
|[[35/24]], [[16/11]]
| [[35/24]], [[16/11]]
|  
|  
| gadolinium
| gadolinium
| {{UDnote|step=64}}
| {{UDnote|step=64}}
| OG#, uA
|-
|-
| 65
| 65
| 661.02
| 661.02
|  
|  
|[[22/15]]
| [[22/15]]
|  
|  
| terbium
| terbium
| {{UDnote|step=65}}
| {{UDnote|step=65}}
| oA
|-
|-
| 66
| 66
| 671.19
| 671.19
|  
|  
|[[72/49]], [[28/19]]
| [[72/49]], [[28/19]]
|  
|  
| dysprosium
| dysprosium
| {{UDnote|step=66}}
| {{UDnote|step=66}}
| sA
|-
|-
| 67
| 67
| 681.36
| 681.36
|  
|  
|[[40/27]]
| [[40/27]]
| wolf 5th
| wolf 5th
| holmium
| holmium
| {{UDnote|step=67}}
| {{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}}
| {{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}}
| {{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}}
| {{UDnote|step=70}}
| lA
|-
|-
| 71
| 71
| 722.03
| 722.03
|  
|  
|[[243/160]]
| [[243/160]]
| lamb 5th
| lamb 5th
| lutetium
| lutetium
| {{UDnote|step=71}}
| {{UDnote|step=71}}
| KA
|-
|-
| 72
| 72
| 732.20
| 732.20
|  
|  
|[[32/21]]
| [[32/21]]
|  
|  
| hafnium
| hafnium
| {{UDnote|step=72}}
| {{UDnote|step=72}}
| SA
|-
|-
| 73
| 73
| 742.37
| 742.37
|  
|  
|[[75/49]]
| [[75/49]]
|  
|  
| tantalum
| tantalum
| {{UDnote|step=73}}
| {{UDnote|step=73}}
| OA, uBb
|-
|-
| 74
| 74
| 752.54
| 752.54
|  
|  
|[[54/35]], [[17/11]]
| [[54/35]], [[17/11]]
|  
|  
| tungsten
| tungsten
| {{UDnote|step=74}}
| {{UDnote|step=74}}
| UA, oBb
|-
|-
| 75
| 75
| 762.71
| 762.71
|  
|  
|[[14/9]]
| [[14/9]]
|  
|  
| rhenium
| rhenium
| {{UDnote|step=75}}
| {{UDnote|step=75}}
| sBb
|-
|-
| 76
| 76
| 772.88
| 772.88
|  
|  
|[[120/77]]
| [[120/77]]
|  
|  
| osmium
| osmium
| {{UDnote|step=76}}
| {{UDnote|step=76}}
| kBb
|-
|-
| 77
| 77
| 783.05
| 783.05
|  
|  
|[[11/7]]
| [[11/7]]
|  
|  
| iridium
| iridium
| {{UDnote|step=77}}
| {{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}}
| {{UDnote|step=78}}
| Bb
|-
|-
| 79
| 79
| 803.39
| 803.39
|  
|  
|[[100/63]]
| [[100/63]]
|  
|  
| gold
| gold
| {{UDnote|step=79}}
| {{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}}
| {{UDnote|step=80}}
| KBb
|-
|-
| 81
| 81
| 823.73
| 823.73
|  
|  
|[[45/28]]
| [[45/28]]
|  
|  
| thallium
| thallium
| {{UDnote|step=81}}
| {{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}}
| {{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}}
| {{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}}
| {{UDnote|step=84}}
| uB
|-
|-
| 85
| 85
| 864.41
| 864.41
|  
|  
|[[28/17]]
| [[28/17]]
|  
|  
| astatine
| astatine
| {{UDnote|step=85}}
| {{UDnote|step=85}}
| oB
|-
|-
| 86
| 86
| 874.58
| 874.58
|  
|  
|[[81/49]]
| [[81/49]]
|  
|  
| radon
| radon
| {{UDnote|step=86}}
| {{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}}
| {{UDnote|step=87}}
| kB
|-
|-
| 88
| 88
| 894.92
| 894.92
|  
|  
|[[42/25]]
| [[42/25]]
|  
|  
| radium
| radium
| {{UDnote|step=88}}
| {{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}}
| {{UDnote|step=89}}
| B
|-
|-
| 90
| 90
| 915.25
| 915.25
|  
|  
|[[56/33]]
| [[56/33]]
|  
|  
| thorium
| thorium
| {{UDnote|step=90}}
| {{UDnote|step=90}}
| LB
|-
|-
| 91
| 91
| 925.42
| 925.42
|  
|  
|[[128/75]]
| [[128/75]]
|  
|  
| protactinium
| protactinium
| {{UDnote|step=91}}
| {{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}}
| {{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}}
| {{UDnote|step=93}}
| OB, uC
|-
|-
| 94
| 94
| 955.93
| 955.93
|  
|  
|[[125/72]]
| [[125/72]]
|  
|  
| plutonium
| plutonium
| {{UDnote|step=94}}
| {{UDnote|step=94}}
| UB, oC
|-
|-
| 95
| 95
| 966.10
| 966.10
|  
|  
|[[7/4]]
| [[7/4]]
| Harmonic 7th
| Harmonic 7th
| americium
| americium
| {{UDnote|step=95}}
| {{UDnote|step=95}}
| sC
|-
|-
| 96
| 96
| 976.27
| 976.27
|  
|  
|[[225/128]]
| [[225/128]]
|  
|  
| curium
| curium
| {{UDnote|step=96}}
| {{UDnote|step=96}}
| kC
|-
|-
| 97
| 97
| 986.44
| 986.44
|  
|  
|[[30/17]]
| [[30/17]]
|  
|  
| berkelium
| berkelium
| {{UDnote|step=97}}
| {{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}}
| {{UDnote|step=98}}
| C
|-
|-
| 99
| 99
| 1006.78
| 1006.78
|  
|  
|[[25/14]]
| [[25/14]]
|  
|  
| einsteinium
| einsteinium
| {{UDnote|step=99}}
| {{UDnote|step=99}}
| LC
|-
|-
| 100
| 100
| 1016.95
| 1016.95
|  
|  
|[[9/5]]
| [[9/5]]
| Tivra
| Tivra
| fermium
| fermium
| {{UDnote|step=100}}
| {{UDnote|step=100}}
| KC
|-
|-
| 101
| 101
| 1027.12
| 1027.12
|  
|  
|[[38/21]]
| [[38/21]]
|  
|  
| mendelevium
| mendelevium
| {{UDnote|step=101}}
| {{UDnote|step=101}}
| SC
|-
|-
| 102
| 102
| 1037.29
| 1037.29
|  
|  
|[[20/11]]
| [[20/11]]
|  
|  
| nobelium
| nobelium
| {{UDnote|step=102}}
| {{UDnote|step=102}}
| OC, uDb
|-
|-
| 103
| 103
| 1047.46
| 1047.46
|  
|  
|[[11/6]]
| [[11/6]]
|  
|  
| lawrencium
| lawrencium
| {{UDnote|step=103}}
| {{UDnote|step=103}}
| UC, oDb
|-
|-
| 104
| 104
| 1057.63
| 1057.63
|  
|  
|[[81/44]]
| [[81/44]]
|  
|  
| rutherfordium
| rutherfordium
| {{UDnote|step=104}}
| {{UDnote|step=104}}
| uC#, sDb
|-
|-
| 105
| 105
| 1067.80
| 1067.80
|  
|  
|[[50/27]]
| [[50/27]]
|  
|  
| dubnium
| dubnium
| {{UDnote|step=105}}
| {{UDnote|step=105}}
| oC#, kDb
|-
|-
| 106
| 106
| 1077.97
| 1077.97
|  
|  
|[[28/15]]
| [[28/15]]
|  
|  
| seaborgium
| seaborgium
| {{UDnote|step=106}}
| {{UDnote|step=106}}
| sC#, lDb
|-
|-
| 107
| 107
| 1088.14
| 1088.14
|  
|  
|[[15/8]]
| [[15/8]]
|  
|  
| bohrium
| bohrium
| {{UDnote|step=107}}
| {{UDnote|step=107}}
| kC#, Db
|-
|-
| 108
| 108
| 1098.31
| 1098.31
|  
|  
|[[32/17]], [[17/9]]
| [[32/17]], [[17/9]]
|  
|  
| hassium
| hassium
| {{UDnote|step=108}}
| {{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}}
| {{UDnote|step=109}}
| C#, KDb
|-
|-
| 110
| 110
| 1118.64
| 1118.64
|  
|  
|[[40/21]], [[21/11]]
| [[40/21]], [[21/11]]
|  
|  
| darmstadtium
| darmstadtium
| {{UDnote|step=110}}
| {{UDnote|step=110}}
| LC#, SDb
|-
|-
| 111
| 111
| 1128.81
| 1128.81
|  
|  
|[[48/25]]
| [[48/25]]
|  
|  
| roentgenium
| roentgenium
| {{UDnote|step=111}}
| {{UDnote|step=111}}
| KC#, ODb, (kkDb)
|-
|-
| 112
| 112
| 1138.98
| 1138.98
|  
|  
|[[27/14]]
| [[27/14]]
|  
|  
| copernicium
| copernicium
| {{UDnote|step=112}}
| {{UDnote|step=112}}
| SC#, UDb
|-
|-
| 113
| 113
| 1149.15
| 1149.15
|  
|  
|[[35/18]], [[64/33]]
| [[35/18]], [[64/33]]
|  
|  
| nihonium
| nihonium
| {{UDnote|step=113}}
| {{UDnote|step=113}}
| OC#, uD
|-
|-
| 114
| 114
| 1159.32
| 1159.32
|  
|  
|[[49/25]]
| [[49/25]]
|  
|  
| flerovium
| flerovium
| {{UDnote|step=114}}
| {{UDnote|step=114}}
| UC#, oD
|-
|-
| 115
| 115
| 1169.49
| 1169.49
|  
|  
|[[63/32]], [[96/49]]
| [[63/32]], [[96/49]]
|  
|  
| moscovium
| moscovium
| {{UDnote|step=115}}
| {{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}}
| {{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}}
| {{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}}
| {{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.  


Line 994: Line 1,118:
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 1,013: 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 1,020: 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 1,027: 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 1,034: 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 1,062: 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 1,112: Line 1,253:
| 498.31
| 498.31
| 4/3
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
| [[Helmholtz (temperament)|Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
|-
| 1
| 1
Line 1,124: Line 1,265:
| 20.34
| 20.34
| 81/80
| 81/80
| [[Commatic]]
| [[Bicommatic]]
|-
|-
| 2
| 2
Line 1,168: 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]]
Retrieved from "https://en.xen.wiki/w/118edo"