118edo: Difference between revisions

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m →Intervals: the "marks" column is just diatonic degrees and doesn't really involve FJS. And plz save from text sizes
→Intervals: move the ratios column before naming attempts
Line 32: Line 32:
! Cents
! Cents
! Marks
! Marks
! 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)
! Eliora's Chemical Notation<br>(if base note = 0)
! Approximate Ratios *
|-
|-
| 0
| 0
| 0.00
| 0.00
| P1
| P1
|[[1/1]]
| unison
| unison
| oganesson / neutronium
| oganesson / neutronium
| [[1/1]]
|-
|-
| 1
| 1
| 10.17
| 10.17
|  
|  
|[[126/125]], [[225/224]], [[121/120]], [[243/242]]
| semicomma
| semicomma
| hydrogen
| hydrogen
| [[126/125]], [[225/224]], [[121/120]], [[243/242]]
|-
|-
| 2
| 2
| 20.34
| 20.34
|  
|  
|[[81/80]], [[531441/524288]]
| comma
| comma
| helium
| helium
| [[81/80]], [[531441/524288]]
|-
|-
| 3
| 3
| 30.51
| 30.51
|  
|  
|[[64/63]], [[49/48]]
|  
|  
| lithium
| lithium
| [[64/63]], [[49/48]]
|-
|-
| 4
| 4
| 40.68
| 40.68
|  
|  
|[[50/49]]
|  
|  
| beryllium
| beryllium
| [[50/49]]
|-
|-
| 5
| 5
| 50.85
| 50.85
|  
|  
|[[36/35]]
|  
|  
| boron
| boron
| [[36/35]]
|-
|-
| 6
| 6
| 61.02
| 61.02
|  
|  
|[[28/27]]
|  
|  
| carbon
| carbon
| [[28/27]]
|-
|-
| 7
| 7
| 71.19
| 71.19
|  
|  
|[[25/24]]
|  
|  
| nitrogen
| nitrogen
| [[25/24]]
|-
|-
| 8
| 8
| 81.36
| 81.36
|  
|  
|[[21/20]], [[22/21]]
|  
|  
| oxygen
| oxygen
| [[21/20]], [[22/21]]
|-
|-
| 9
| 9
| 91.53
| 91.53
| m2
| m2
|[[19/18]], [[20/19]]
| limma, dayavati
| limma, dayavati
| fluorine
| fluorine
| [[19/18]], [[20/19]]
|-
|-
| 10
| 10
| 101.69
| 101.69
|  
|  
|[[17/16]], [[18/17]]
| dodecaic semitone
| dodecaic semitone
| neon
| neon
| [[17/16]], [[18/17]]
|-
|-
| 11
| 11
| 111.86
| 111.86
|  
|  
|[[16/15]]
| apotome, ranjani
| apotome, ranjani
| sodium
| sodium
| [[16/15]]
|-
|-
| 12
| 12
| 122.03
| 122.03
|  
|  
|[[15/14]]
|  
|  
| magnesium
| magnesium
| [[15/14]]
|-
|-
| 13
| 13
| 132.20
| 132.20
|  
|  
|[[27/25]]
|  
|  
| aluminium
| aluminium
| [[27/25]]
|-
|-
| 14
| 14
| 142.37
| 142.37
|  
|  
|[[88/81]]
|  
|  
| silicon
| silicon
| [[88/81]]
|-
|-
| 15
| 15
| 152.54
| 152.54
|  
|  
|[[12/11]]
|  
|  
| phosphorus
| phosphorus
| [[12/11]]
|-
|-
| 16
| 16
| 162.71
| 162.71
|  
|  
|[[11/10]]
|  
|  
| sulphur
| sulphur
| [[11/10]]
|-
|-
| 17
| 17
| 172.88
| 172.88
|  
|  
|[[21/19]]
|  
|  
| chlorine
| chlorine
| [[21/19]]
|-
|-
| 18
| 18
| 183.05
| 183.05
|  
|  
|[[10/9]]
| diminished tone, ratika
| diminished tone, ratika
| argon
| argon
| [[10/9]]
|-
|-
| 19
| 19
| 193.22
| 193.22
|  
|  
|[[28/25]], [[19/17]]
| minor tone
| minor tone
| potassium
| potassium
| [[28/25]], [[19/17]]
|-
|-
| 20
| 20
| 203.39
| 203.39
| M2
| M2
|[[9/8]]
| major tone, raudri
| major tone, raudri
| calcium
| calcium
| [[9/8]]
|-
|-
| 21
| 21
| 213.56
| 213.56
|  
|  
|[[17/15]]
|  
|  
| scandium
| scandium
| [[17/15]]
|-
|-
| 22
| 22
| 223.73
| 223.73
|  
|  
|[[256/225]]
|  
|  
| titanium
| titanium
| [[256/225]]
|-
|-
| 23
| 23
| 233.90
| 233.90
|  
|  
|[[8/7]]
| septimal second, slendric 2
| septimal second, slendric 2
| vanadium
| vanadium
| [[8/7]]
|-
|-
| 24
| 24
| 244.07
| 244.07
|  
|  
|[[144/125]], [[121/105]]
|  
|  
| chromium
| chromium
| [[144/125]], [[121/105]]
|-
|-
| 25
| 25
| 254.24
| 254.24
|  
|  
|[[125/108]], [[81/70]], [[22/19]]
|  
|  
| manganese
| manganese
| [[125/108]], [[81/70]], [[22/19]]
|-
|-
| 26
| 26
| 260.41
| 260.41
|  
|  
|[[7/6]]
| septimal third
| septimal third
| iron
| iron
| [[7/6]]
|-
|-
| 27
| 27
| 274.58
| 274.58
|  
|  
|[[75/64]]
|  
|  
| cobalt
| cobalt
| [[75/64]]
|-
|-
| 28
| 28
| 284.75
| 284.75
|  
|  
|[[33/28]]
|  
|  
| nickel
| nickel
| [[33/28]]
|-
|-
| 29
| 29
| 294.92
| 294.92
| m3
| m3
|[[32/27]], [[19/16]]
| Pythagorean minor 3rd, krodha
| Pythagorean minor 3rd, krodha
| copper
| copper
| [[32/27]], [[19/16]]
|-
|-
| 30
| 30
| 305.08
| 305.08
|  
|  
|[[25/21]]
|  
|  
| zinc
| zinc
| [[25/21]]
|-
|-
| 31
| 31
| 315.25
| 315.25
|  
|  
|[[6/5]]
| Classical minor 3rd, vajrika
| Classical minor 3rd, vajrika
| gallium
| gallium
| [[6/5]]
|-
|-
| 32
| 32
| 325.42
| 325.42
|  
|  
|[[98/81]]
|  
|  
| germanium
| germanium
| [[98/81]]
|-
|-
| 33
| 33
| 335.59
| 335.59
|  
|  
|[[40/33]], [[17/14]]
| Lesser tridecimal third
| Lesser tridecimal third
| arsenic
| arsenic
| [[40/33]], [[17/14]]
|-
|-
| 34
| 34
| 345.76
| 345.76
|  
|  
|[[11/9]]
| Minor-neutral third
| Minor-neutral third
| selenium
| selenium
| [[11/9]]
|-
|-
| 35
| 35
| 355.93
| 355.93
|  
|  
|[[27/22]], 16/13 I**
| Minor tridecimal neurtral third, "major-neutral" third
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
| bromine
| [[27/22]], 16/13 I**
|-
|-
| 36
| 36
| 366.10
| 366.10
|  
|  
|[[99/80]], [[21/17]], 16/13 II**
| Golden ratio 3rd, major-tridecimal neutral third
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
| krypton
| [[99/80]], [[21/17]], 16/13 II**
|-
|-
| 37
| 37
| 376.27
| 376.27
|  
|  
|[[56/45]]
|  
|  
| rubidium
| rubidium
| [[56/45]]
|-
|-
| 38
| 38
| 386.44
| 386.44
|  
|  
|[[5/4]]
| Classical major 3rd, prasarini
| Classical major 3rd, prasarini
| strontium
| strontium
| [[5/4]]
|-
|-
| 39
| 39
| 396.61
| 396.61
|  
|  
|[[63/50]]
|  
|  
| yttrium
| yttrium
| [[63/50]]
|-
|-
| 40
| 40
| 406.78
| 406.78
| M3
| M3
|[[24/19]], [[19/15]]
| Pythagorean major 3rd
| Pythagorean major 3rd
| zirconium
| zirconium
| [[24/19]], [[19/15]]
|-
|-
| 41
| 41
| 416.95
| 416.95
|  
|  
|[[14/11]]
|  
|  
| niobium
| niobium
| [[14/11]]
|-
|-
| 42
| 42
| 427.12
| 427.12
|  
|  
|[[77/60]]
|  
|  
| molybdenum
| molybdenum
| [[77/60]]
|-
|-
| 43
| 43
| 437.29
| 437.29
|  
|  
|[[9/7]]
|  
|  
| technetium
| technetium
| [[9/7]]
|-
|-
| 44
| 44
| 447.46
| 447.46
|  
|  
|[[35/27]], [[22/17]]
|  
|  
| ruthenium
| ruthenium
| [[35/27]], [[22/17]]
|-
|-
| 45
| 45
| 457.63
| 457.63
|  
|  
|[[98/75]]
| Barbados 3rd
| Barbados 3rd
| rhodium
| rhodium
| [[98/75]]
|-
|-
| 46
| 46
| 467.80
| 467.80
|  
|  
|[[21/16]]
| Slendric 3
| Slendric 3
| palladium
| palladium
| [[21/16]]
|-
|-
| 47
| 47
| 477.97
| 477.97
|  
|  
|[[320/243]]
|  
|  
| silver
| silver
| [[320/243]]
|-
|-
| 48
| 48
| 488.14
| 488.14
|  
|  
|[[160/121]], [[85/64]]
|  
|  
| cadmium
| cadmium
| [[160/121]], [[85/64]]
|-
|-
| 49
| 49
| 498.31
| 498.31
| P4
| P4
|[[4/3]]
| perfect 4th
| perfect 4th
| indium
| indium
| [[4/3]]
|-
|-
| 50
| 50
| 508.47
| 508.47
|  
|  
|[[75/56]], [[51/38]]
|  
|  
| tin
| tin
| [[75/56]], [[51/38]]
|-
|-
| 51
| 51
| 518.64
| 518.64
|  
|  
|[[27/20]]
| Kshiti
| Kshiti
| antimony
| antimony
| [[27/20]]
|-
|-
| 52
| 52
| 528.81
| 528.81
|  
|  
|[[49/36]], [[19/14]]
|  
|  
| tellurium
| tellurium
| [[49/36]], [[19/14]]
|-
|-
| 53
| 53
| 538.98
| 538.98
|  
|  
|[[15/11]]
|  
|  
| iodine
| iodine
| [[15/11]]
|-
|-
| 54
| 54
| 549.15
| 549.15
|  
|  
|[[48/35]], [[11/8]]
|  
|  
| xenon
| xenon
| [[48/35]], [[11/8]]
|-
|-
| 55
| 55
| 559.32
| 559.32
|  
|  
|[[112/81]]
|  
|  
| caesium
| caesium
| [[112/81]]
|-
|-
| 56
| 56
| 569.49
| 569.49
|  
|  
|[[25/18]]
|  
|  
| barium
| barium
| [[25/18]]
|-
|-
| 57
| 57
| 579.66
| 579.66
|  
|  
|[[7/5]]
|  
|  
| lanthanum
| lanthanum
| [[7/5]]
|-
|-
| 58
| 58
| 589.83
| 589.83
| d5
| d5
|[[45/32]]
| Rakta
| Rakta
| cerium
| cerium
| [[45/32]]
|-
|-
| 59
| 59
| 600.00
| 600.00
|  
|  
|[[99/70]], [[140/99]], [[17/12]], [[24/17]]
| symmetric tritone
| symmetric tritone
| praseodymium
| praseodymium
| [[99/70]], [[140/99]], [[17/12]], [[24/17]]
|-
|-
| 60
| 60
| 610.17
| 610.17
| A4
| A4
|[[64/45]]
| Literal tritone, sandipani
| Literal tritone, sandipani
| neodymium
| neodymium
| [[64/45]]
|-
|-
| 61
| 61
| 620.34
| 620.34
|  
|  
|[[10/7]]
|  
|  
|  
|  
| [[10/7]]
|-
|-
| 62
| 62
| 630.51
| 630.51
|  
|  
|[[36/25]]
|  
|  
|  
|  
| [[36/25]]
|-
|-
| 63
| 63
| 640.68
| 640.68
|  
|  
|[[81/56]]
|  
|  
|  
|  
| [[81/56]]
|-
|-
| 64
| 64
| 650.85
| 650.85
|  
|  
|[[35/24]], [[16/11]]
|  
|  
|  
|  
| [[35/24]], [[16/11]]
|-
|-
| 65
| 65
| 661.02
| 661.02
|  
|  
|[[22/15]]
|  
|  
|  
|  
| [[22/15]]
|-
|-
| 66
| 66
| 671.19
| 671.19
|  
|  
|[[72/49]], [[28/19]]
|  
|  
|  
|  
| [[72/49]], [[28/19]]
|-
|-
| 67
| 67
| 681.36
| 681.36
|  
|  
|[[40/27]]
| wolf 5th
| wolf 5th
| holmium
| holmium
| [[40/27]]
|-
|-
| 68
| 68
| 691.53
| 691.53
|  
|  
|[[112/75]], [[76/51]]
| wolf cub 5th
| wolf cub 5th
| erbium
| erbium
| [[112/75]], [[76/51]]
|-
|-
| 69
| 69
| 701.69
| 701.69
| P5
| P5
|[[3/2]]
| perfect 5th
| perfect 5th
| thulium
| thulium
| [[3/2]]
|-
|-
| 70
| 70
| 711.86
| 711.86
|  
|  
|[[121/80]], [[128/85]]
|  
|  
| ytterbium
| ytterbium
| [[121/80]], [[128/85]]
|-
|-
| 71
| 71
| 722.03
| 722.03
|  
|  
|[[243/160]]
|  
|  
| lutetium
| lutetium
| [[243/160]]
|-
|-
| 72
| 72
| 732.20
| 732.20
|  
|  
|[[32/21]]
|  
|  
| hafnium
| hafnium
| [[32/21]]
|-
|-
| 73
| 73
| 742.37
| 742.37
|  
|  
|[[75/49]]
|  
|  
| tantalum
| tantalum
| [[75/49]]
|-
|-
| 74
| 74
| 752.54
| 752.54
|  
|  
|[[54/35]], [[17/11]]
|  
|  
| tungsten
| tungsten
| [[54/35]], [[17/11]]
|-
|-
| 75
| 75
| 762.71
| 762.71
|  
|  
|[[14/9]]
|  
|  
| rhenium
| rhenium
| [[14/9]]
|-
|-
| 76
| 76
| 772.88
| 772.88
|  
|  
|[[120/77]]
|  
|  
| osmium
| osmium
| [[120/77]]
|-
|-
| 77
| 77
| 783.05
| 783.05
|  
|  
|[[11/7]]
|  
|  
| iridium
| iridium
| [[11/7]]
|-
|-
| 78
| 78
| 793.22
| 793.22
| m6
| m6
|[[19/12]], [[30/19]]
| Pythagorean minor 6th
| Pythagorean minor 6th
| platinum
| platinum
| [[19/12]], [[30/19]]
|-
|-
| 79
| 79
| 803.39
| 803.39
|  
|  
|[[100/63]]
|  
|  
| gold
| gold
| [[100/63]]
|-
|-
| 80
| 80
| 813.56
| 813.56
|  
|  
|[[8/5]]
| Classical minor 6th
| Classical minor 6th
| mercury
| mercury
| [[8/5]]
|-
|-
| 81
| 81
| 823.73
| 823.73
|  
|  
|[[45/28]]
|  
|  
| thallium
| thallium
| [[45/28]]
|-
|-
| 82
| 82
| 833.90
| 833.90
|  
|  
|[[160/99]], [[34/21]], [[13/8]] I**
| Golden ratio sixth, minor-neutral tridecimal sixth
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
| lead
| [[160/99]], [[34/21]], [[13/8]] I**
|-
|-
| 83
| 83
| 844.07
| 844.07
|  
|  
|[[44/27]], 13/8 II**
| Major tridecimal neutral sixth, "minor-neutral" sixth
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
| bismuth
| [[44/27]], 13/8 II**
|-
|-
| 84
| 84
| 854.24
| 854.24
|  
|  
|[[18/11]]
| Major-neutral sixth
| Major-neutral sixth
| polonium
| polonium
| [[18/11]]
|-
|-
| 85
| 85
| 864.41
| 864.41
|  
|  
|[[28/17]]
|  
|  
|  
|  
| [[28/17]]
|-
|-
| 86
| 86
| 874.58
| 874.58
|  
|  
|[[81/49]]
|  
|  
|  
|  
| [[81/49]]
|-
|-
| 87
| 87
| 884.75
| 884.75
|  
|  
|[[5/3]]
| Classical major 6th
| Classical major 6th
| francium
| francium
| [[5/3]]
|-
|-
| 88
| 88
| 894.92
| 894.92
|  
|  
|[[42/25]]
|  
|  
|  
|  
| [[42/25]]
|-
|-
| 89
| 89
| 905.08
| 905.08
| M6
| M6
|[[27/16]], [[32/19]]
| Pythagorean major 6th
| Pythagorean major 6th
| actinium
| actinium
| [[27/16]], [[32/19]]
|-
|-
| 90
| 90
| 915.25
| 915.25
|  
|  
|[[56/33]]
|  
|  
|  
|  
| [[56/33]]
|-
|-
| 91
| 91
| 925.42
| 925.42
|  
|  
|[[128/75]]
|  
|  
|  
|  
| [[128/75]]
|-
|-
| 92
| 92
| 935.59
| 935.59
|  
|  
|[[12/7]]
| Septimal supermajor 6th, slendro 5
| Septimal supermajor 6th, slendro 5
| uranium
| uranium
| [[12/7]]
|-
|-
| 93
| 93
| 945.76
| 945.76
|  
|  
|[[216/125]], [[140/81]], [[121/70]], [[19/11]]
|  
|  
| neptunium
| neptunium
| [[216/125]], [[140/81]], [[121/70]], [[19/11]]
|-
|-
| 94
| 94
| 955.93
| 955.93
|  
|  
|[[125/72]]
|  
|  
| plutonium
| plutonium
| [[125/72]]
|-
|-
| 95
| 95
| 966.10
| 966.10
|  
|  
|[[7/4]]
| Harmonic 7th
| Harmonic 7th
| americium
| americium
| [[7/4]]
|-
|-
| 96
| 96
| 976.27
| 976.27
|  
|  
|[[225/128]]
|  
|  
| curium
| curium
| [[225/128]]
|-
|-
| 97
| 97
| 986.44
| 986.44
|  
|  
|[[30/17]]
|  
|  
| berkelium
| berkelium
| [[30/17]]
|-
|-
| 98
| 98
| 996.61
| 996.61
| m7
| m7
|[[16/9]]
|  
|  
|  
|  
| [[16/9]]
|-
|-
| 99
| 99
| 1006.78
| 1006.78
|  
|  
|[[25/14]]
|  
|  
|  
|  
| [[25/14]]
|-
|-
| 100
| 100
| 1016.95
| 1016.95
|  
|  
|[[9/5]]
| Tivra
| Tivra
| fermium
| fermium
| [[9/5]]
|-
|-
| 101
| 101
| 1027.12
| 1027.12
|  
|  
|[[38/21]]
|  
|  
|  
|  
| [[38/21]]
|-
|-
| 102
| 102
| 1037.29
| 1037.29
|  
|  
|[[20/11]]
|  
|  
|  
|  
| [[20/11]]
|-
|-
| 103
| 103
| 1047.46
| 1047.46
|  
|  
|[[11/6]]
|  
|  
|  
|  
| [[11/6]]
|-
|-
| 104
| 104
| 1057.63
| 1057.63
|  
|  
|[[81/44]]
|  
|  
|  
|  
| [[81/44]]
|-
|-
| 105
| 105
| 1067.80
| 1067.80
|  
|  
|[[50/27]]
|  
|  
|  
|  
| [[50/27]]
|-
|-
| 106
| 106
| 1077.97
| 1077.97
|  
|  
|[[28/15]]
|  
|  
|  
|  
| [[28/15]]
|-
|-
| 107
| 107
| 1088.14
| 1088.14
|  
|  
|[[15/8]]
|  
|  
|  
|  
| [[15/8]]
|-
|-
| 108
| 108
| 1098.31
| 1098.31
|  
|  
|[[32/17]], [[17/9]]
|  
|  
|  
|  
| [[32/17]], [[17/9]]
|-
|-
| 109
| 109
| 1108.47
| 1108.47
| M7
| M7
|[[36/19]], [[19/10]]
| Pythagorean major 7th
| Pythagorean major 7th
| meitnerium
| meitnerium
| [[36/19]], [[19/10]]
|-
|-
| 110
| 110
| 1118.64
| 1118.64
|  
|  
|[[40/21]], [[21/11]]
|  
|  
| darmstadtium
| darmstadtium
| [[40/21]], [[21/11]]
|-
|-
| 111
| 111
| 1128.81
| 1128.81
|  
|  
|[[48/25]]
|  
|  
| roentgenium
| roentgenium
| [[48/25]]
|-
|-
| 112
| 112
| 1138.98
| 1138.98
|  
|  
|[[27/14]]
|  
|  
| copernicium
| copernicium
| [[27/14]]
|-
|-
| 113
| 113
| 1149.15
| 1149.15
|  
|  
|[[35/18]], [[64/33]]
|  
|  
| nihonium
| nihonium
| [[35/18]], [[64/33]]
|-
|-
| 114
| 114
| 1159.32
| 1159.32
|  
|  
|[[49/25]]
|  
|  
| flerovium
| flerovium
| [[49/25]]
|-
|-
| 115
| 115
| 1169.49
| 1169.49
|  
|  
|[[63/32]], [[96/49]]
|  
|  
| moscovium
| moscovium
| [[63/32]], [[96/49]]
|-
|-
| 116
| 116
| 1179.66
| 1179.66
|  
|  
|[[160/81]]
| Comma 7th
| Comma 7th
| livermorium
| livermorium
| [[160/81]]
|-
|-
| 117
| 117
| 1189.83
| 1189.83
|  
|  
|[[125/63]], [[448/225]], [[240/121]], [[484/243]]
| Semicomma supermajor 7th
| Semicomma supermajor 7th
| tenessine
| tenessine
| [[125/63]], [[448/225]], [[240/121]], [[484/243]]
|-
|-
| 118
| 118
| 1200.00
| 1200.00
| P8
| P8
|[[2/1]]
| perfect 8ve
| perfect 8ve
| oganesson / neutronium
| oganesson / neutronium
| [[2/1]]
|}
|}
<nowiki>*</nowiki> treated as a 2.3.5.7.11.17.19 system
<nowiki>*</nowiki> treated as a 2.3.5.7.11.17.19 system

Revision as of 21:52, 3 January 2022

← 117edo 118edo 119edo →
Prime factorization 2 × 59
Step size 10.1695 ¢ 
Fifth 69\118 (701.695 ¢)
Semitones (A1:m2) 11:9 (111.9 ¢ : 91.53 ¢)
Consistency limit 11
Distinct consistency limit 11

The 118 equal divisions of the octave (118edo), or the 118(-tone) equal temperament (118tet, 118et) when viewed from a regular temperament perspective, is the equal division of the octave into 118 parts of about 10.2 cents each.

Theory

118edo represents the intersection of the 5-limit schismatic and parakleismic temperaments, tempering out both the schisma, [-15 8 1⟩ and the parakleisma, [8 14 -13⟩, as well as the vishnuzma, [23 6 -14⟩, the hemithirds comma, [38 -2 -15⟩, and the kwazy, [-53 10 16⟩. It is the first 5-limit equal division which clearly gives microtempering, with errors well under half a cent. In addition, 118edo excellently approximates the 22 Shruti scale.

In the 7-limit, it is particularly notable for tempering out the gamelisma, 1029/1024, and is an excellent tuning for the rank three gamelan temperament, and for guiron, the rank two temperament also tempering out the schisma, 32805/32768. It also tempers out 3136/3125, the hemimean comma, but 99edo does better with that.

In the 11-limit, it tempers out 385/384 and 441/440, and is an excellent tuning for portent, the temperament tempering out both, and for the 11-limit version of guiron, which does also.

It has two reasonable mappings for 13. The patent val tempers out 196/195, 352/351, 625/624, 729/728, 1001/1000, 1575/1573 and 4096/4095. The 118f val tempers out 169/168, 325/324, 351/350, 364/363, 1573/1568, 1716/1715 and 2080/2079. It is, however, better viewed as a no-13 19-limit temperament, on which subgroup it is consistent through the 21-odd-limit.

Since the Pythagorean comma maps to 2 steps of 118edo, it can be interpreted as a series of ten segments of twelve Pythagorean fifths minus the said comma.

118edo is the 17th zeta peak edo.

Prime harmonics

Script error: No such module "primes_in_edo".

Intervals

Table of intervals in 118edo
Step Cents Marks Approximate Ratios * Eliora's Naming System
(+Shruti 22 correspondence) 		Eliora's Chemical Notation
(if base note = 0)
0 0.00 P1 1/1 unison oganesson / neutronium
1 10.17 126/125, 225/224, 121/120, 243/242 semicomma hydrogen
2 20.34 81/80, 531441/524288 comma helium
3 30.51 64/63, 49/48 lithium
4 40.68 50/49 beryllium
5 50.85 36/35 boron
6 61.02 28/27 carbon
7 71.19 25/24 nitrogen
8 81.36 21/20, 22/21 oxygen
9 91.53 m2 19/18, 20/19 limma, dayavati fluorine
10 101.69 17/16, 18/17 dodecaic semitone neon
11 111.86 16/15 apotome, ranjani sodium
12 122.03 15/14 magnesium
13 132.20 27/25 aluminium
14 142.37 88/81 silicon
15 152.54 12/11 phosphorus
16 162.71 11/10 sulphur
17 172.88 21/19 chlorine
18 183.05 10/9 diminished tone, ratika argon
19 193.22 28/25, 19/17 minor tone potassium
20 203.39 M2 9/8 major tone, raudri calcium
21 213.56 17/15 scandium
22 223.73 256/225 titanium
23 233.90 8/7 septimal second, slendric 2 vanadium
24 244.07 144/125, 121/105 chromium
25 254.24 125/108, 81/70, 22/19 manganese
26 260.41 7/6 septimal third iron
27 274.58 75/64 cobalt
28 284.75 33/28 nickel
29 294.92 m3 32/27, 19/16 Pythagorean minor 3rd, krodha copper
30 305.08 25/21 zinc
31 315.25 6/5 Classical minor 3rd, vajrika gallium
32 325.42 98/81 germanium
33 335.59 40/33, 17/14 Lesser tridecimal third arsenic
34 345.76 11/9 Minor-neutral third selenium
35 355.93 27/22, 16/13 I** Minor tridecimal neurtral third, "major-neutral" third bromine
36 366.10 99/80, 21/17, 16/13 II** Golden ratio 3rd, major-tridecimal neutral third krypton
37 376.27 56/45 rubidium
38 386.44 5/4 Classical major 3rd, prasarini strontium
39 396.61 63/50 yttrium
40 406.78 M3 24/19, 19/15 Pythagorean major 3rd zirconium
41 416.95 14/11 niobium
42 427.12 77/60 molybdenum
43 437.29 9/7 technetium
44 447.46 35/27, 22/17 ruthenium
45 457.63 98/75 Barbados 3rd rhodium
46 467.80 21/16 Slendric 3 palladium
47 477.97 320/243 silver
48 488.14 160/121, 85/64 cadmium
49 498.31 P4 4/3 perfect 4th indium
50 508.47 75/56, 51/38 tin
51 518.64 27/20 Kshiti antimony
52 528.81 49/36, 19/14 tellurium
53 538.98 15/11 iodine
54 549.15 48/35, 11/8 xenon
55 559.32 112/81 caesium
56 569.49 25/18 barium
57 579.66 7/5 lanthanum
58 589.83 d5 45/32 Rakta cerium
59 600.00 99/70, 140/99, 17/12, 24/17 symmetric tritone praseodymium
60 610.17 A4 64/45 Literal tritone, sandipani neodymium
61 620.34 10/7
62 630.51 36/25
63 640.68 81/56
64 650.85 35/24, 16/11
65 661.02 22/15
66 671.19 72/49, 28/19
67 681.36 40/27 wolf 5th holmium
68 691.53 112/75, 76/51 wolf cub 5th erbium
69 701.69 P5 3/2 perfect 5th thulium
70 711.86 121/80, 128/85 ytterbium
71 722.03 243/160 lutetium
72 732.20 32/21 hafnium
73 742.37 75/49 tantalum
74 752.54 54/35, 17/11 tungsten
75 762.71 14/9 rhenium
76 772.88 120/77 osmium
77 783.05 11/7 iridium
78 793.22 m6 19/12, 30/19 Pythagorean minor 6th platinum
79 803.39 100/63 gold
80 813.56 8/5 Classical minor 6th mercury
81 823.73 45/28 thallium
82 833.90 160/99, 34/21, 13/8 I** Golden ratio sixth, minor-neutral tridecimal sixth lead
83 844.07 44/27, 13/8 II** Major tridecimal neutral sixth, "minor-neutral" sixth bismuth
84 854.24 18/11 Major-neutral sixth polonium
85 864.41 28/17
86 874.58 81/49
87 884.75 5/3 Classical major 6th francium
88 894.92 42/25
89 905.08 M6 27/16, 32/19 Pythagorean major 6th actinium
90 915.25 56/33
91 925.42 128/75
92 935.59 12/7 Septimal supermajor 6th, slendro 5 uranium
93 945.76 216/125, 140/81, 121/70, 19/11 neptunium
94 955.93 125/72 plutonium
95 966.10 7/4 Harmonic 7th americium
96 976.27 225/128 curium
97 986.44 30/17 berkelium
98 996.61 m7 16/9
99 1006.78 25/14
100 1016.95 9/5 Tivra fermium
101 1027.12 38/21
102 1037.29 20/11
103 1047.46 11/6
104 1057.63 81/44
105 1067.80 50/27
106 1077.97 28/15
107 1088.14 15/8
108 1098.31 32/17, 17/9
109 1108.47 M7 36/19, 19/10 Pythagorean major 7th meitnerium
110 1118.64 40/21, 21/11 darmstadtium
111 1128.81 48/25 roentgenium
112 1138.98 27/14 copernicium
113 1149.15 35/18, 64/33 nihonium
114 1159.32 49/25 flerovium
115 1169.49 63/32, 96/49 moscovium
116 1179.66 160/81 Comma 7th livermorium
117 1189.83 125/63, 448/225, 240/121, 484/243 Semicomma supermajor 7th tenessine
118 1200.00 P8 2/1 perfect 8ve oganesson / neutronium

* treated as a 2.3.5.7.11.17.19 system

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

Notation

Possible chemical notation

This notation was proposed by Eliora in November 2021.

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

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

This being said, there are a few correspondences - 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo. In addition, 87\118 (francium, start of period 7) and 89\118 (actinium, start of the 7f-block), form 5/3 and 27/16 respectively. Mercury, ending the 6d-block, corresponds to 8/5. The minor tone 10/9 corresponds to 18 (argon), a noble gas, ending 3 periods, while 9/8 corresponds to 20 (calcium), the 2s metal.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-187 118⟩ [⟨118 187]] -0.119 0.082 0.81
2.3.5 32805/32768, [8 14 -13⟩ [⟨118 187 274]] +0.036 0.093 0.91
2.3.5.7 1029/1024, 3136/3125, 4375/4374 [⟨118 187 274 331]] +0.270 0.412 4.05
2.3.5.7.11 385/384, 441/440, 3136/3125, 4375/4374 [⟨118 187 274 331 408]] +0.341 0.370 3.89
2.3.5.7.11.13 196/195, 352/351, 384/384, 625/624, 729/728 [⟨118 187 274 331 408 437]] (118) +0.125 0.604 5.93
2.3.5.7.11.13 169/168, 325/324, 364/363, 385/384, 3136/3125 [⟨118 187 274 331 408 436]] (118f) +0.583 0.650 6.39
2.3.5.7.11.17 289/288, 385/384, 441/440, 561/560, 3136/3125 [⟨118 187 274 331 408 482]] +0.417 0.399 3.92
2.3.5.7.11.17.19 289/288, 361/360, 385/384, 441/440, 476/475, 513/512, 969/968 [⟨118 187 274 331 408 482 501]] +0.445 0.376 3.69
  • 118et is lower in relative error than any previous ETs in the 5-limit. Not until 171 do we find a better ET in terms of absolute error, and not until 441 do we find one in terms of relative error.

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per octave 		Generator
(reduced) 		Cents
(reduced) 		Associated
ratio 		Temperaments
1 11\118 111.86 16/15 Vavoom
1 19\118 193.22 28/25 Luna / hemithirds / lunatic
1 23\118 233.90 8/7 Slendric / guiron
1 31\118 315.25 6/5 Parakleismic / paralytic
1 39\118 396.61 44/35 Squarschmidt
1 49\118 498.31 4/3 Helmholtz / pontiac / helenoid / pontic
1 55\118 559.32 242/175 Tritriple
2 2\118 20.34 81/80 Commatic
2 5\118 50.85 33/32~36/35 Kleischismic
2 7\118 71.19 25/24 Vishnu / ananta (118) / acyuta (118f)
2 10\118 101.69 35/33 Bischismic / bipont (118) / counterbipont (118f)
2 16\118 162.71 11/10 Kwazy / bisupermajor
2 18\118 183.05 10/9 Unidec / ekadash (118) / hendec (118f)
2 19\118 193.22 121/108 Semiluna
2 31\118
(28\118) 		315.25
(284.75) 		6/5
(33/28) 		Semiparakleismic
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