118edo: Difference between revisions

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→Intervals: added SKULO notation
Line 29: Line 29:
! Chemical Notation<br>(see below, if base note = 0)
! Chemical Notation<br>(see below, if base note = 0)
! [[Ups and downs notation]]
! [[Ups and downs notation]]
![[SKULO interval names|SKULO]] notation
|-
|-
| 0
| 0
Line 37: Line 38:
| oganesson / neutronium
| oganesson / neutronium
| {{UDnote|step=0}}
| {{UDnote|step=0}}
|D
|-
|-
| 1
| 1
Line 45: Line 47:
| hydrogen
| hydrogen
| {{UDnote|step=1}}
| {{UDnote|step=1}}
|LD
|-
|-
| 2
| 2
Line 53: Line 56:
| helium
| helium
| {{UDnote|step=2}}
| {{UDnote|step=2}}
|KD
|-
|-
| 3
| 3
Line 61: Line 65:
| lithium
| lithium
| {{UDnote|step=3}}
| {{UDnote|step=3}}
|SD
|-
|-
| 4
| 4
Line 69: Line 74:
| beryllium
| beryllium
| {{UDnote|step=4}}
| {{UDnote|step=4}}
|OD, uEb
|-
|-
| 5
| 5
Line 77: Line 83:
| boron
| boron
| {{UDnote|step=5}}
| {{UDnote|step=5}}
|UD, oEb
|-
|-
| 6
| 6
Line 85: Line 92:
| carbon
| carbon
| {{UDnote|step=6}}
| {{UDnote|step=6}}
|sEb, uD#
|-
|-
| 7
| 7
Line 93: Line 101:
| nitrogen
| nitrogen
| {{UDnote|step=7}}
| {{UDnote|step=7}}
|kEb, oD#, (kkD#)
|-
|-
| 8
| 8
Line 101: Line 110:
| oxygen
| oxygen
| {{UDnote|step=8}}
| {{UDnote|step=8}}
|lEb, sD#
|-
|-
| 9
| 9
Line 109: Line 119:
| fluorine
| fluorine
| {{UDnote|step=9}}
| {{UDnote|step=9}}
|Eb, kD#
|-
|-
| 10
| 10
Line 117: Line 128:
| neon
| neon
| {{UDnote|step=10}}
| {{UDnote|step=10}}
|LEb, lD#
|-
|-
| 11
| 11
Line 125: Line 137:
| sodium
| sodium
| {{UDnote|step=11}}
| {{UDnote|step=11}}
|KEb, D#
|-
|-
| 12
| 12
Line 133: Line 146:
| magnesium
| magnesium
| {{UDnote|step=12}}
| {{UDnote|step=12}}
|SEb
|-
|-
| 13
| 13
Line 141: Line 155:
| aluminium
| aluminium
| {{UDnote|step=13}}
| {{UDnote|step=13}}
|OEb
|-
|-
| 14
| 14
Line 149: Line 164:
| silicon
| silicon
| {{UDnote|step=14}}
| {{UDnote|step=14}}
|UEb
|-
|-
| 15
| 15
Line 157: Line 173:
| phosphorus
| phosphorus
| {{UDnote|step=15}}
| {{UDnote|step=15}}
|uE
|-
|-
| 16
| 16
Line 165: Line 182:
| sulphur
| sulphur
| {{UDnote|step=16}}
| {{UDnote|step=16}}
|oE
|-
|-
| 17
| 17
Line 173: Line 191:
| chlorine
| chlorine
| {{UDnote|step=17}}
| {{UDnote|step=17}}
|sE
|-
|-
| 18
| 18
Line 181: Line 200:
| argon
| argon
| {{UDnote|step=18}}
| {{UDnote|step=18}}
|kE
|-
|-
| 19
| 19
Line 189: Line 209:
| potassium
| potassium
| {{UDnote|step=19}}
| {{UDnote|step=19}}
|lE
|-
|-
| 20
| 20
Line 197: Line 218:
| calcium
| calcium
| {{UDnote|step=20}}
| {{UDnote|step=20}}
|E
|-
|-
| 21
| 21
Line 205: Line 227:
| scandium
| scandium
| {{UDnote|step=21}}
| {{UDnote|step=21}}
|LE
|-
|-
| 22
| 22
Line 213: Line 236:
| titanium
| titanium
| {{UDnote|step=22}}
| {{UDnote|step=22}}
|KE
|-
|-
| 23
| 23
Line 221: Line 245:
| vanadium
| vanadium
| {{UDnote|step=23}}
| {{UDnote|step=23}}
|SE
|-
|-
| 24
| 24
Line 229: Line 254:
| chromium
| chromium
| {{UDnote|step=24}}
| {{UDnote|step=24}}
|OE, uF
|-
|-
| 25
| 25
Line 237: Line 263:
| manganese
| manganese
| {{UDnote|step=25}}
| {{UDnote|step=25}}
|UE, oF
|-
|-
| 26
| 26
Line 245: Line 272:
| iron
| iron
| {{UDnote|step=26}}
| {{UDnote|step=26}}
|sF
|-
|-
| 27
| 27
Line 253: Line 281:
| cobalt
| cobalt
| {{UDnote|step=27}}
| {{UDnote|step=27}}
|kF
|-
|-
| 28
| 28
Line 261: Line 290:
| nickel
| nickel
| {{UDnote|step=28}}
| {{UDnote|step=28}}
|lF
|-
|-
| 29
| 29
Line 269: Line 299:
| copper
| copper
| {{UDnote|step=29}}
| {{UDnote|step=29}}
|F
|-
|-
| 30
| 30
Line 277: Line 308:
| zinc
| zinc
| {{UDnote|step=30}}
| {{UDnote|step=30}}
|LF
|-
|-
| 31
| 31
Line 285: Line 317:
| gallium
| gallium
| {{UDnote|step=31}}
| {{UDnote|step=31}}
|KF
|-
|-
| 32
| 32
Line 293: Line 326:
| germanium
| germanium
| {{UDnote|step=32}}
| {{UDnote|step=32}}
|SF
|-
|-
| 33
| 33
Line 301: Line 335:
| arsenic
| arsenic
| {{UDnote|step=33}}
| {{UDnote|step=33}}
|OF
|-
|-
| 34
| 34
Line 309: Line 344:
| selenium
| selenium
| {{UDnote|step=34}}
| {{UDnote|step=34}}
|UF
|-
|-
| 35
| 35
Line 317: Line 353:
| bromine
| bromine
| {{UDnote|step=35}}
| {{UDnote|step=35}}
|uF#
|-
|-
| 36
| 36
Line 325: Line 362:
| krypton
| krypton
| {{UDnote|step=36}}
| {{UDnote|step=36}}
|oF#
|-
|-
| 37
| 37
Line 333: Line 371:
| rubidium
| rubidium
| {{UDnote|step=37}}
| {{UDnote|step=37}}
|sF#
|-
|-
| 38
| 38
Line 341: Line 380:
| strontium
| strontium
| {{UDnote|step=38}}
| {{UDnote|step=38}}
|kF#
|-
|-
| 39
| 39
Line 349: Line 389:
| yttrium
| yttrium
| {{UDnote|step=39}}
| {{UDnote|step=39}}
|lF#
|-
|-
| 40
| 40
Line 357: Line 398:
| zirconium
| zirconium
| {{UDnote|step=40}}
| {{UDnote|step=40}}
|F#
|-
|-
| 41
| 41
Line 365: Line 407:
| niobium
| niobium
| {{UDnote|step=41}}
| {{UDnote|step=41}}
|LF#
|-
|-
| 42
| 42
Line 373: Line 416:
| molybdenum
| molybdenum
| {{UDnote|step=42}}
| {{UDnote|step=42}}
|KF#
|-
|-
| 43
| 43
Line 381: Line 425:
| technetium
| technetium
| {{UDnote|step=43}}
| {{UDnote|step=43}}
|SF#
|-
|-
| 44
| 44
Line 389: Line 434:
| ruthenium
| ruthenium
| {{UDnote|step=44}}
| {{UDnote|step=44}}
|OF#, uG
|-
|-
| 45
| 45
Line 397: Line 443:
| rhodium
| rhodium
| {{UDnote|step=45}}
| {{UDnote|step=45}}
|UF#, oG
|-
|-
| 46
| 46
Line 405: Line 452:
| palladium
| palladium
| {{UDnote|step=46}}
| {{UDnote|step=46}}
|sG
|-
|-
| 47
| 47
Line 413: Line 461:
| silver
| silver
| {{UDnote|step=47}}
| {{UDnote|step=47}}
|kG
|-
|-
| 48
| 48
Line 421: Line 470:
| cadmium
| cadmium
| {{UDnote|step=48}}
| {{UDnote|step=48}}
|lG
|-
|-
| 49
| 49
Line 429: Line 479:
| indium
| indium
| {{UDnote|step=49}}
| {{UDnote|step=49}}
|G
|-
|-
| 50
| 50
Line 437: Line 488:
| tin
| tin
| {{UDnote|step=50}}
| {{UDnote|step=50}}
|LG
|-
|-
| 51
| 51
Line 445: Line 497:
| antimony
| antimony
| {{UDnote|step=51}}
| {{UDnote|step=51}}
|KG
|-
|-
| 52
| 52
Line 453: Line 506:
| tellurium
| tellurium
| {{UDnote|step=52}}
| {{UDnote|step=52}}
|SG
|-
|-
| 53
| 53
Line 461: Line 515:
|  
|  
| iodine
| iodine
|OG, uGb
|-
|-
| 54
| 54
Line 469: Line 524:
|  
|  
| xenon
| xenon
|UG, oAb
|-
|-
| 55
| 55
Line 477: Line 533:
| caesium
| caesium
| {{UDnote|step=55}}
| {{UDnote|step=55}}
|uG#, sAb
|-
|-
| 56
| 56
Line 485: Line 542:
| barium
| barium
| {{UDnote|step=56}}
| {{UDnote|step=56}}
|oG#, (kkG#), kAb
|-
|-
| 57
| 57
Line 493: Line 551:
| lanthanum
| lanthanum
| {{UDnote|step=57}}
| {{UDnote|step=57}}
|sG#, lAb
|-
|-
| 58
| 58
Line 501: Line 560:
| cerium
| cerium
| {{UDnote|step=58}}
| {{UDnote|step=58}}
|kG#, Ab
|-
|-
| 59
| 59
Line 509: Line 569:
| praseodymium
| praseodymium
| {{UDnote|step=59}}
| {{UDnote|step=59}}
|lG#, LAb
|-
|-
| 60
| 60
Line 517: Line 578:
| neodymium
| neodymium
| {{UDnote|step=60}}
| {{UDnote|step=60}}
|G#, KAb
|-
|-
| 61
| 61
Line 525: Line 587:
| promethium
| promethium
| {{UDnote|step=61}}
| {{UDnote|step=61}}
|LG#, SAb
|-
|-
| 62
| 62
Line 533: Line 596:
| samarium
| samarium
| {{UDnote|step=62}}
| {{UDnote|step=62}}
|KG#, OAb, (KKAb)
|-
|-
| 63
| 63
Line 541: Line 605:
| europium
| europium
| {{UDnote|step=63}}
| {{UDnote|step=63}}
|SG#, UAb
|-
|-
| 64
| 64
Line 549: Line 614:
| gadolinium
| gadolinium
| {{UDnote|step=64}}
| {{UDnote|step=64}}
|OG#, uA
|-
|-
| 65
| 65
Line 557: Line 623:
| terbium
| terbium
| {{UDnote|step=65}}
| {{UDnote|step=65}}
|oA
|-
|-
| 66
| 66
Line 565: Line 632:
| dysprosium
| dysprosium
| {{UDnote|step=66}}
| {{UDnote|step=66}}
|sA
|-
|-
| 67
| 67
Line 573: Line 641:
| holmium
| holmium
| {{UDnote|step=67}}
| {{UDnote|step=67}}
|kA
|-
|-
| 68
| 68
Line 581: Line 650:
| erbium
| erbium
| {{UDnote|step=68}}
| {{UDnote|step=68}}
|lA
|-
|-
| 69
| 69
Line 589: Line 659:
| thulium
| thulium
| {{UDnote|step=69}}
| {{UDnote|step=69}}
|A
|-
|-
| 70
| 70
Line 597: Line 668:
| ytterbium
| ytterbium
| {{UDnote|step=70}}
| {{UDnote|step=70}}
|lA
|-
|-
| 71
| 71
Line 605: Line 677:
| lutetium
| lutetium
| {{UDnote|step=71}}
| {{UDnote|step=71}}
|KA
|-
|-
| 72
| 72
Line 613: Line 686:
| hafnium
| hafnium
| {{UDnote|step=72}}
| {{UDnote|step=72}}
|SA
|-
|-
| 73
| 73
Line 621: Line 695:
| tantalum
| tantalum
| {{UDnote|step=73}}
| {{UDnote|step=73}}
|OA, uBb
|-
|-
| 74
| 74
Line 629: Line 704:
| tungsten
| tungsten
| {{UDnote|step=74}}
| {{UDnote|step=74}}
|UA, oBb
|-
|-
| 75
| 75
Line 637: Line 713:
| rhenium
| rhenium
| {{UDnote|step=75}}
| {{UDnote|step=75}}
|sBb
|-
|-
| 76
| 76
Line 645: Line 722:
| osmium
| osmium
| {{UDnote|step=76}}
| {{UDnote|step=76}}
|kBb
|-
|-
| 77
| 77
Line 653: Line 731:
| iridium
| iridium
| {{UDnote|step=77}}
| {{UDnote|step=77}}
|lBb
|-
|-
| 78
| 78
Line 661: Line 740:
| platinum
| platinum
| {{UDnote|step=78}}
| {{UDnote|step=78}}
|Bb
|-
|-
| 79
| 79
Line 669: Line 749:
| gold
| gold
| {{UDnote|step=79}}
| {{UDnote|step=79}}
|LBb
|-
|-
| 80
| 80
Line 677: Line 758:
| mercury
| mercury
| {{UDnote|step=80}}
| {{UDnote|step=80}}
|KBb
|-
|-
| 81
| 81
Line 685: Line 767:
| thallium
| thallium
| {{UDnote|step=81}}
| {{UDnote|step=81}}
|SBb
|-
|-
| 82
| 82
Line 693: Line 776:
| lead
| lead
| {{UDnote|step=82}}
| {{UDnote|step=82}}
|OBb
|-
|-
| 83
| 83
Line 701: Line 785:
| bismuth
| bismuth
| {{UDnote|step=83}}
| {{UDnote|step=83}}
|UBb
|-
|-
| 84
| 84
Line 709: Line 794:
| polonium
| polonium
| {{UDnote|step=84}}
| {{UDnote|step=84}}
|uB
|-
|-
| 85
| 85
Line 717: Line 803:
| astatine
| astatine
| {{UDnote|step=85}}
| {{UDnote|step=85}}
|oB
|-
|-
| 86
| 86
Line 725: Line 812:
| radon
| radon
| {{UDnote|step=86}}
| {{UDnote|step=86}}
|sB
|-
|-
| 87
| 87
Line 733: Line 821:
| francium
| francium
| {{UDnote|step=87}}
| {{UDnote|step=87}}
|kB
|-
|-
| 88
| 88
Line 741: Line 830:
| radium
| radium
| {{UDnote|step=88}}
| {{UDnote|step=88}}
|lB
|-
|-
| 89
| 89
Line 749: Line 839:
| actinium
| actinium
| {{UDnote|step=89}}
| {{UDnote|step=89}}
|B
|-
|-
| 90
| 90
Line 757: Line 848:
| thorium
| thorium
| {{UDnote|step=90}}
| {{UDnote|step=90}}
|LB
|-
|-
| 91
| 91
Line 765: Line 857:
| protactinium
| protactinium
| {{UDnote|step=91}}
| {{UDnote|step=91}}
|KB
|-
|-
| 92
| 92
Line 773: Line 866:
| uranium
| uranium
| {{UDnote|step=92}}
| {{UDnote|step=92}}
|SB
|-
|-
| 93
| 93
Line 781: Line 875:
| neptunium
| neptunium
| {{UDnote|step=93}}
| {{UDnote|step=93}}
|OB, uC
|-
|-
| 94
| 94
Line 789: Line 884:
| plutonium
| plutonium
| {{UDnote|step=94}}
| {{UDnote|step=94}}
|UB, oC
|-
|-
| 95
| 95
Line 797: Line 893:
| americium
| americium
| {{UDnote|step=95}}
| {{UDnote|step=95}}
|sC
|-
|-
| 96
| 96
Line 805: Line 902:
| curium
| curium
| {{UDnote|step=96}}
| {{UDnote|step=96}}
|kC
|-
|-
| 97
| 97
Line 813: Line 911:
| berkelium
| berkelium
| {{UDnote|step=97}}
| {{UDnote|step=97}}
|lC
|-
|-
| 98
| 98
Line 821: Line 920:
| californium
| californium
| {{UDnote|step=98}}
| {{UDnote|step=98}}
|C
|-
|-
| 99
| 99
Line 829: Line 929:
| einsteinium
| einsteinium
| {{UDnote|step=99}}
| {{UDnote|step=99}}
|LC
|-
|-
| 100
| 100
Line 837: Line 938:
| fermium
| fermium
| {{UDnote|step=100}}
| {{UDnote|step=100}}
|KC
|-
|-
| 101
| 101
Line 845: Line 947:
| mendelevium
| mendelevium
| {{UDnote|step=101}}
| {{UDnote|step=101}}
|SC
|-
|-
| 102
| 102
Line 853: Line 956:
| nobelium
| nobelium
| {{UDnote|step=102}}
| {{UDnote|step=102}}
|OC, uDb
|-
|-
| 103
| 103
Line 861: Line 965:
| lawrencium
| lawrencium
| {{UDnote|step=103}}
| {{UDnote|step=103}}
|UC, oDb
|-
|-
| 104
| 104
Line 869: Line 974:
| rutherfordium
| rutherfordium
| {{UDnote|step=104}}
| {{UDnote|step=104}}
|uC#, sDb
|-
|-
| 105
| 105
Line 877: Line 983:
| dubnium
| dubnium
| {{UDnote|step=105}}
| {{UDnote|step=105}}
|oC#, kDb
|-
|-
| 106
| 106
Line 885: Line 992:
| seaborgium
| seaborgium
| {{UDnote|step=106}}
| {{UDnote|step=106}}
|sC#, lDb
|-
|-
| 107
| 107
Line 893: Line 1,001:
| bohrium
| bohrium
| {{UDnote|step=107}}
| {{UDnote|step=107}}
|kC#, Db
|-
|-
| 108
| 108
Line 901: Line 1,010:
| hassium
| hassium
| {{UDnote|step=108}}
| {{UDnote|step=108}}
|lC#, LDb
|-
|-
| 109
| 109
Line 909: Line 1,019:
| meitnerium
| meitnerium
| {{UDnote|step=109}}
| {{UDnote|step=109}}
|C#, KDb
|-
|-
| 110
| 110
Line 917: Line 1,028:
| darmstadtium
| darmstadtium
| {{UDnote|step=110}}
| {{UDnote|step=110}}
|LC#, SDb
|-
|-
| 111
| 111
Line 925: Line 1,037:
| roentgenium
| roentgenium
| {{UDnote|step=111}}
| {{UDnote|step=111}}
|KC#, ODb, (kkDb)
|-
|-
| 112
| 112
Line 933: Line 1,046:
| copernicium
| copernicium
| {{UDnote|step=112}}
| {{UDnote|step=112}}
|SC#, UDb
|-
|-
| 113
| 113
Line 941: Line 1,055:
| nihonium
| nihonium
| {{UDnote|step=113}}
| {{UDnote|step=113}}
|OC#, uD
|-
|-
| 114
| 114
Line 949: Line 1,064:
| flerovium
| flerovium
| {{UDnote|step=114}}
| {{UDnote|step=114}}
|UC#, oD
|-
|-
| 115
| 115
Line 957: Line 1,073:
| moscovium
| moscovium
| {{UDnote|step=115}}
| {{UDnote|step=115}}
|sD
|-
|-
| 116
| 116
Line 965: Line 1,082:
| livermorium
| livermorium
| {{UDnote|step=116}}
| {{UDnote|step=116}}
|kD
|-
|-
| 117
| 117
Line 973: Line 1,091:
| tenessine
| tenessine
| {{UDnote|step=117}}
| {{UDnote|step=117}}
|lD
|-
|-
| 118
| 118
Line 981: Line 1,100:
| 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>*</nowiki> treated as a 2.3.5.7.11.17.19 system

Revision as of 05:44, 19 March 2024

← 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

Template:EDO intro

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. As a result, 118edo also excellently approximates the 22 Shruti scale.

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)

Subsets and supersets

118edo contains 2edo and 59edo as subsets. 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

* 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. 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. I
  • 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.

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
(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

Music

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