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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 represents 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]].
 
=== Prime harmonics ===
{{Harmonics in equal|118}}


118edo is the 17th [[The Riemann Zeta Function and Tuning|zeta peak edo]].  
=== 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"
{| class="wikitable collapsible mw-collapsible mw-collapsed center-1 right-2 center-3"
|+ style=white-space:nowrap | Table of intervals in 118 EDO
|+ style="font-size: 105%; white-space: nowrap;" | Table of intervals in 118edo
!Step
|-
!Name
! Step
!Chemical notation
! Cents
<small>if base note = 0</small>
! Marks
!Associated ratio
! Approximate Ratios*
! Eliora's Naming System<br />(+Shruti 22 correspondence)
! Chemical Notation<br />(see below, if {{nowrap|base note {{=}} 0}})
! [[Ups and downs notation]]
! [[SKULO interval names|SKULO]] notation
|-
| 0
| 0.00
| P1
| [[1/1]]
| unison
| oganesson / neutronium
| {{UDnote|step=0}}
| D
|-
| 1
| 10.17
|
| [[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| hydrogen
| {{UDnote|step=1}}
| LD
|-
| 2
| 20.34
|
| [[81/80]], [[531441/524288]]
| comma
| helium
| {{UDnote|step=2}}
| KD
|-
| 3
| 30.51
|
| [[64/63]], [[49/48]]
| augmented comma
| lithium
| {{UDnote|step=3}}
| SD
|-
| 4
| 40.68
|
| [[50/49]]
|
| beryllium
| {{UDnote|step=4}}
| OD, uEb
|-
| 5
| 50.85
|
| [[36/35]]
|
| boron
| {{UDnote|step=5}}
| UD, oEb
|-
| 6
| 61.02
|
| [[28/27]]
|
| carbon
| {{UDnote|step=6}}
| sEb, uD#
|-
| 7
| 71.19
|
| [[25/24]]
|
| nitrogen
| {{UDnote|step=7}}
| kEb, oD#, (kkD#)
|-
| 8
| 81.36
|
| [[21/20]], [[22/21]]
|
| oxygen
| {{UDnote|step=8}}
| lEb, sD#
|-
| 9
| 91.53
| m2
| [[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| fluorine
| {{UDnote|step=9}}
| Eb, kD#
|-
| 10
| 101.69
|
| [[17/16]], [[18/17]]
| dodecaic semitone
| neon
| {{UDnote|step=10}}
| LEb, lD#
|-
| 11
| 111.86
|
| [[16/15]], [[2187/2048]]
| apotome, ranjani
| sodium
| {{UDnote|step=11}}
| KEb, D#
|-
| 12
| 122.03
|
| [[15/14]]
|
| magnesium
| {{UDnote|step=12}}
| SEb
|-
| 13
| 132.20
|
| [[27/25]]
|
| aluminium
| {{UDnote|step=13}}
| OEb
|-
| 14
| 142.37
|
| [[88/81]]
|
| silicon
| {{UDnote|step=14}}
| UEb
|-
| 15
| 152.54
|
| [[12/11]]
|
| phosphorus
| {{UDnote|step=15}}
| uE
|-
| 16
| 162.71
|
| [[11/10]]
|
| sulphur
| {{UDnote|step=16}}
| oE
|-
| 17
| 172.88
|
| [[21/19]]
| diminished tone
| chlorine
| {{UDnote|step=17}}
| sE
|-
| 18
| 183.05
|
| [[10/9]]
| minor tone, ratika
| argon
| {{UDnote|step=18}}
| kE
|-
| 19
| 193.22
|
| [[28/25]], [[19/17]]
| neutral tone, quasi-meantone
| potassium
| {{UDnote|step=19}}
| lE
|-
| 20
| 203.39
| M2
| [[9/8]]
| major tone, raudri
| calcium
| {{UDnote|step=20}}
| E
|-
| 21
| 213.56
|
| [[17/15]]
| augmented tone
| scandium
| {{UDnote|step=21}}
| LE
|-
| 22
| 223.73
|
| [[256/225]]
| minor slendric second
| titanium
| {{UDnote|step=22}}
| KE
|-
| 23
| 233.90
|
| [[8/7]]
| septimal second, slendric 2
| vanadium
| {{UDnote|step=23}}
| SE
|-
| 24
| 244.07
|
| [[144/125]], [[121/105]]
| major slendric second
| chromium
| {{UDnote|step=24}}
| OE, uF
|-
| 25
| 254.24
|
| [[125/108]], [[81/70]], [[22/19]]
| minor septimal third
| manganese
| {{UDnote|step=25}}
| UE, oF
|-
| 26
| 260.41
|
| [[7/6]]
| septimal third
| iron
| {{UDnote|step=26}}
| sF
|-
| 27
| 274.58
|
| [[75/64]]
| major septimal third
| cobalt
| {{UDnote|step=27}}
| kF
|-
| 28
| 284.75
|
| [[33/28]]
|
| nickel
| {{UDnote|step=28}}
| lF
|-
| 29
| 294.92
| m3
| [[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| copper
| {{UDnote|step=29}}
| F
|-
| 30
| 305.08
|
| [[25/21]]
|
| zinc
| {{UDnote|step=30}}
| LF
|-
| 31
| 315.25
|
| [[6/5]]
| Classical minor 3rd, vajrika
| gallium
| {{UDnote|step=31}}
| KF
|-
| 32
| 325.42
|
| [[98/81]]
|
| germanium
| {{UDnote|step=32}}
| SF
|-
| 33
| 335.59
|
| [[40/33]], [[17/14]]
| Lesser tridecimal third
| arsenic
| {{UDnote|step=33}}
| OF
|-
| 34
| 345.76
|
| [[11/9]]
| Minor-neutral third
| selenium
| {{UDnote|step=34}}
| UF
|-
| 35
| 355.93
|
| [[27/22]], [[16/13]] I**
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
| {{UDnote|step=35}}
| uF#
|-
| 36
| 366.10
|
| [[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
| {{UDnote|step=36}}
| oF#
|-
| 37
| 376.27
|
| [[56/45]]
|
| rubidium
| {{UDnote|step=37}}
| sF#
|-
| 38
| 386.44
|
| [[5/4]]
| Classical major 3rd, prasarini
| strontium
| {{UDnote|step=38}}
| kF#
|-
| 39
| 396.61
|
| [[63/50]]
|
| yttrium
| {{UDnote|step=39}}
| lF#
|-
| 40
| 406.78
| M3
| [[24/19]], [[19/15]]
| Pythagorean major 3rd
| zirconium
| {{UDnote|step=40}}
| F#
|-
| 41
| 416.95
|
| [[14/11]]
|
| niobium
| {{UDnote|step=41}}
| LF#
|-
| 42
| 427.12
|
| [[77/60]]
|
| molybdenum
| {{UDnote|step=42}}
| KF#
|-
| 43
| 437.29
|
| [[9/7]]
|
| technetium
| {{UDnote|step=43}}
| SF#
|-
| 44
| 447.46
|
| [[35/27]], [[22/17]]
|
| ruthenium
| {{UDnote|step=44}}
| OF#, uG
|-
| 45
| 457.63
|
| [[98/75]]
| Barbados 3rd
| rhodium
| {{UDnote|step=45}}
| UF#, oG
|-
| 46
| 467.80
|
| [[21/16]]
| Slendric 3
| palladium
| {{UDnote|step=46}}
| sG
|-
| 47
| 477.97
|
| [[320/243]]
|
| silver
| {{UDnote|step=47}}
| kG
|-
| 48
| 488.14
|
| [[160/121]], [[85/64]]
|
| cadmium
| {{UDnote|step=48}}
| lG
|-
| 49
| 498.31
| P4
| [[4/3]]
| perfect 4th
| indium
| {{UDnote|step=49}}
| G
|-
| 50
| 508.47
|
| [[75/56]], [[51/38]]
|
| tin
| {{UDnote|step=50}}
| LG
|-
| 51
| 518.64
|
| [[27/20]]
| Kshiti
| antimony
| {{UDnote|step=51}}
| KG
|-
| 52
| 528.81
|
| [[49/36]], [[19/14]]
|
| tellurium
| {{UDnote|step=52}}
| SG
|-
| 53
| 538.98
|
| [[15/11]]
| {{UDnote|step=53}}
|
| iodine
| OG, uGb
|-
| 54
| 549.15
|
| [[48/35]], [[11/8]]
| {{UDnote|step=54}}
|
| xenon
| UG, oAb
|-
| 55
| 559.32
|
| [[112/81]]
|
| caesium
| {{UDnote|step=55}}
| uG#, sAb
|-
| 56
| 569.49
|
| [[25/18]]
|
| barium
| {{UDnote|step=56}}
| oG#, (kkG#), kAb
|-
| 57
| 579.66
|
| [[7/5]]
|
| lanthanum
| {{UDnote|step=57}}
| sG#, lAb
|-
| 58
| 589.83
| d5
| [[45/32]]
| Rakta
| cerium
| {{UDnote|step=58}}
| kG#, Ab
|-
| 59
| 600.00
|
| [[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| praseodymium
| {{UDnote|step=59}}
| lG#, LAb
|-
| 60
| 610.17
| A4
| [[64/45]], [[729/512]]
| Literal tritone, sandipani
| neodymium
| {{UDnote|step=60}}
| G#, KAb
|-
| 61
| 620.34
|
| [[10/7]]
|
| promethium
| {{UDnote|step=61}}
| LG#, SAb
|-
| 62
| 630.51
|
| [[36/25]]
|
| samarium
| {{UDnote|step=62}}
| KG#, OAb, (KKAb)
|-
| 63
| 640.68
|
| [[81/56]]
|
| europium
| {{UDnote|step=63}}
| SG#, UAb
|-
| 64
| 650.85
|
| [[35/24]], [[16/11]]
|
| gadolinium
| {{UDnote|step=64}}
| OG#, uA
|-
| 65
| 661.02
|
| [[22/15]]
|
| terbium
| {{UDnote|step=65}}
| oA
|-
| 66
| 671.19
|
| [[72/49]], [[28/19]]
|
| dysprosium
| {{UDnote|step=66}}
| sA
|-
| 67
| 681.36
|
| [[40/27]]
| wolf 5th
| holmium
| {{UDnote|step=67}}
| kA
|-
| 68
| 691.53
|
| [[112/75]], [[76/51]]
| wolf cub 5th
| erbium
| {{UDnote|step=68}}
| lA
|-
| 69
| 701.69
| P5
| [[3/2]]
| perfect 5th, slendric 4
| thulium
| {{UDnote|step=69}}
| A
|-
| 70
| 711.86
|
| [[121/80]], [[128/85]]
| sheep 5th
| ytterbium
| {{UDnote|step=70}}
| lA
|-
| 71
| 722.03
|
| [[243/160]]
| lamb 5th
| lutetium
| {{UDnote|step=71}}
| KA
|-
| 72
| 732.20
|
| [[32/21]]
|
| hafnium
| {{UDnote|step=72}}
| SA
|-
|-
|0
| 73
|unison
| 742.37
|oganesson / neutronium
|  
|1/1 exact
| [[75/49]]
|
| tantalum
| {{UDnote|step=73}}
| OA, uBb
|-
|-
|1
| 74
|semicomma
| 752.54
|hydrogen
|  
|[[243/242]], many others
| [[54/35]], [[17/11]]
|
| tungsten
| {{UDnote|step=74}}
| UA, oBb
|-
|-
|2
| 75
|comma
| 762.71
|helium
|  
|[[531441/524288]], [[81/80]]
| [[14/9]]
|
| rhenium
| {{UDnote|step=75}}
| sBb
|-
|-
|9
| 76
|limma, dayavati
| 772.88
|fluorine
|  
|[[256/243]]
| [[120/77]]
|
| osmium
| {{UDnote|step=76}}
| kBb
|-
|-
|10
| 77
|dodecaic semitone
| 783.05
|neon
|  
|[[17/16]]
| [[11/7]]
|
| iridium
| {{UDnote|step=77}}
| lBb
|-
|-
|11
| 78
|apotome, ranjani
| 793.22
|sodium
| m6
|[[16/15]], [[2187/2048]]
| [[19/12]], [[30/19]]
| Pythagorean minor 6th
| platinum
| {{UDnote|step=78}}
| Bb
|-
|-
|18
| 79
|diminished tone, ratika
| 803.39
|argon
|  
|[[10/9]]
| [[100/63]]
|
| gold
| {{UDnote|step=79}}
| LBb
|-
|-
|19
| 80
|minor tone
| 813.56
|potassium
|  
|[[19/17]]
| [[8/5]]
| Classical minor 6th
| mercury
| {{UDnote|step=80}}
| KBb
|-
|-
|20
| 81
|major tone, raudri
| 823.73
|calcium
|  
|[[9/8]]
| [[45/28]]
|
| thallium
| {{UDnote|step=81}}
| SBb
|-
|-
|23
| 82
|septimal second, slendro gulu
| 833.90
|vanadium
|  
|[[8/7]]
| [[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
| {{UDnote|step=82}}
| OBb
|-
|-
|26
| 83
|septimal third
| 844.07
|iron
|  
|[[7/6]]
| [[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
| {{UDnote|step=83}}
| UBb
|-
|-
|29
| 84
|Pythagorean minor 3rd, krodha
| 854.24
|copper
|  
|[[32/27]]
| [[18/11]]
| Major-neutral sixth
| polonium
| {{UDnote|step=84}}
| uB
|-
|-
|31
| 85
|Classical minor 3rd, vajrika
| 864.41
|gallium
|  
|[[6/5]]
| [[28/17]]
|
| astatine
| {{UDnote|step=85}}
| oB
|-
|-
|33
| 86
|Lesser tridecimal third
| 874.58
|germanium
|  
|[[39/32]]
| [[81/49]]
|
| radon
| {{UDnote|step=86}}
| sB
|-
|-
|34
| 87
|Minor-neutral third
| 884.75
|selenium
|  
|[[11/9]]
| [[5/3]]
| Classical major 6th
| francium
| {{UDnote|step=87}}
| kB
|-
|-
|35
| 88
|Minor tridecimal neurtral third, "major-neutral" third
| 894.92
|bromine
|  
|[[16/13]], 70/57
| [[42/25]]
|
| radium
| {{UDnote|step=88}}
| lB
|-
|-
|36
| 89
|Golden ratio 3rd, major-tridecimal neutral third
| 905.08
|krypton
| M6
|[[16/13]], [[26/21]], [[21/17]]
| [[27/16]], [[32/19]]
| Pythagorean major 6th
| actinium
| {{UDnote|step=89}}
| B
|-
|-
|38
| 90
|Classical major 3rd, prasarini
| 915.25
|strontium
|  
|[[5/4]]
| [[56/33]]
|
| thorium
| {{UDnote|step=90}}
| LB
|-
|-
|40
| 91
|Pythagorean major 3rd
| 925.42
|zirconium
|  
|[[81/64]]
| [[128/75]]
|
| protactinium
| {{UDnote|step=91}}
| KB
|-
|-
|49
| 92
|perfect 4th
| 935.59
|indium
|  
|[[4/3]]
| [[12/7]]
| Septimal supermajor 6th, slendric 5
| uranium
| {{UDnote|step=92}}
| SB
|-
|-
|51
| 93
|Kshiti
| 945.76
|antimony
|  
|[[27/20]]
| [[216/125]], [[140/81]], [[121/70]], [[19/11]]
|
| neptunium
| {{UDnote|step=93}}
| OB, uC
|-
|-
|58
| 94
|Rakta
| 955.93
|cerium
|  
|[[45/32]]
| [[125/72]]
|
| plutonium
| {{UDnote|step=94}}
| UB, oC
|-
|-
|59
| 95
|symmetric tritone
| 966.10
|praseodymium
|  
|[[99/70]], [[140/99]]
| [[7/4]]
| Harmonic 7th
| americium
| {{UDnote|step=95}}
| sC
|-
|-
|60
| 96
|Literal tritone, sandipani
| 976.27
|neodymium
|  
|[[729/512]]
| [[225/128]]
|
| curium
| {{UDnote|step=96}}
| kC
|-
|-
|69
| 97
|perfect 5th
| 986.44
|thulium
|  
|[[3/2]]
| [[30/17]]
|
| berkelium
| {{UDnote|step=97}}
| lC
|-
|-
|78
| 98
|Pythagorean minor 6th
| 996.61
|platinum
| m7
|[[128/81]]
| [[16/9]]
| Pythagorean minor 7th
| californium
| {{UDnote|step=98}}
| C
|-
|-
|80
| 99
|Classical minor 6th
| 1006.78
|mercury
|  
|[[8/5]]
| [[25/14]]
|
| einsteinium
| {{UDnote|step=99}}
| LC
|-
|-
|82
| 100
|Golden ratio sixth, minor-neutral tridecimal sixth
| 1016.95
|lead
|  
|13/8, [[21/13]], [[34/21]], [[Acoustic phi]]
| [[9/5]]
| Tivra
| fermium
| {{UDnote|step=100}}
| KC
|-
|-
|83
| 101
|Major tridecimal neutral sixth, "minor-neutral" sixth
| 1027.12
|bismuth
|  
|13/8, 57/35
| [[38/21]]
|
| mendelevium
| {{UDnote|step=101}}
| SC
|-
|-
|84
| 102
|Major-neutral sixth
| 1037.29
|polonium
|  
|[[18/11]]
| [[20/11]]
|
| nobelium
| {{UDnote|step=102}}
| OC, uDb
|-
|-
|87
| 103
|Classical major 6th
| 1047.46
|francium
|  
|[[5/3]]
| [[11/6]]
|
| lawrencium
| {{UDnote|step=103}}
| UC, oDb
|-
|-
|89
| 104
|Pythagorean major 6th
| 1057.63
|actinium
|  
|[[27/16]]
| [[81/44]]
|
| rutherfordium
| {{UDnote|step=104}}
| uC#, sDb
|-
|-
|100
| 105
|Tivra
| 1067.80
|fermium
|  
|[[9/5]]
| [[50/27]]
|
| dubnium
| {{UDnote|step=105}}
| oC#, kDb
|-
|-
|109
| 106
|Pythagorean major 7th
| 1077.97
|meitnerium
|  
|[[243/128]]
| [[28/15]]
|
| seaborgium
| {{UDnote|step=106}}
| sC#, lDb
|-
|-
|118
| 107
|perfect 8ve
| 1088.14
|oganesson / neutronium
|
|2/1 exact
| [[15/8]]
|
| bohrium
| {{UDnote|step=107}}
| kC#, Db
|-
| 108
| 1098.31
|
| [[32/17]], [[17/9]]
|
| hassium
| {{UDnote|step=108}}
| lC#, LDb
|-
| 109
| 1108.47
| M7
| [[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| meitnerium
| {{UDnote|step=109}}
| C#, KDb
|-
| 110
| 1118.64
|
| [[40/21]], [[21/11]]
|
| darmstadtium
| {{UDnote|step=110}}
| LC#, SDb
|-
| 111
| 1128.81
|
| [[48/25]]
|
| roentgenium
| {{UDnote|step=111}}
| KC#, ODb, (kkDb)
|-
| 112
| 1138.98
|
| [[27/14]]
|
| copernicium
| {{UDnote|step=112}}
| SC#, UDb
|-
| 113
| 1149.15
|
| [[35/18]], [[64/33]]
|
| nihonium
| {{UDnote|step=113}}
| OC#, uD
|-
| 114
| 1159.32
|
| [[49/25]]
|
| flerovium
| {{UDnote|step=114}}
| UC#, oD
|-
| 115
| 1169.49
|
| [[63/32]], [[96/49]]
|
| moscovium
| {{UDnote|step=115}}
| sD
|-
| 116
| 1179.66
|
| [[160/81]]
| Comma supermajor 7th
| livermorium
| {{UDnote|step=116}}
| kD
|-
| 117
| 1189.83
|
| [[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| tenessine
| {{UDnote|step=117}}
| lD
|-
| 118
| 1200.00
| P8
| [[2/1]]
| perfect 8ve
| oganesson / neutronium
| {{UDnote|step=118}}
| D
|}
|}
<nowiki />* {{sg|2.3.5.7.11.17.19 subgroup}}
<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 and provides a fine one-to-one correspondence between notes and their names, as opposed to extending small scales into large EDOs which create excessive labels. Some may argue that other notations. like [[Ups and Downs Notation|ups and downs]] favor [[12edo]] or [[5L 2s|the diatonic scale]], while the chemical notation system has no such issue.  
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, 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: 
 
* 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 232: 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 239: 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 246: 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 253: 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 281: 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 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 330: Line 1,253:
| 498.31
| 498.31
| 4/3
| 4/3
| [[Helmholtz]] / [[pontiac]] / helenoid / pontic
| [[Helmholtz (temperament)|Helmholtz]] / [[pontiac]] / helenoid / pontic
|-
|-
| 1
| 1
Line 342: Line 1,265:
| 20.34
| 20.34
| 81/80
| 81/80
| [[Commatic]]
| [[Bicommatic]]
|-
|-
| 2
| 2
Line 386: 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"