118edo: Difference between revisions

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→Intervals: I appreciate the credit but it's tautological in my opiniion, there is a section devoted to chemical notation explaining it's made by me
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Line 28: Line 28:
! 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 base note = 0)
! [[Ups and downs notation]]
|-
|-
| 0
| 0
Line 35: Line 36:
| unison
| unison
| oganesson / neutronium
| oganesson / neutronium
| {{UDnote|step=0}}
|-
|-
| 1
| 1
Line 42: Line 44:
| semicomma
| semicomma
| hydrogen
| hydrogen
| {{UDnote|step=1}}
|-
|-
| 2
| 2
Line 49: Line 52:
| comma
| comma
| helium
| helium
| {{UDnote|step=2}}
|-
|-
| 3
| 3
Line 56: Line 60:
| augmented comma
| augmented comma
| lithium
| lithium
| {{UDnote|step=3}}
|-
|-
| 4
| 4
Line 63: Line 68:
|  
|  
| beryllium
| beryllium
| {{UDnote|step=4}}
|-
|-
| 5
| 5
Line 70: Line 76:
|  
|  
| boron
| boron
| {{UDnote|step=5}}
|-
|-
| 6
| 6
Line 77: Line 84:
|  
|  
| carbon
| carbon
| {{UDnote|step=6}}
|-
|-
| 7
| 7
Line 84: Line 92:
|  
|  
| nitrogen
| nitrogen
| {{UDnote|step=7}}
|-
|-
| 8
| 8
Line 91: Line 100:
|  
|  
| oxygen
| oxygen
| {{UDnote|step=8}}
|-
|-
| 9
| 9
Line 98: Line 108:
| limma, dayavati
| limma, dayavati
| fluorine
| fluorine
| {{UDnote|step=9}}
|-
|-
| 10
| 10
| 101.69
| 101.69
| {{UDnote|step=10}}
|  
|  
|[[17/16]], [[18/17]]
|[[17/16]], [[18/17]]
| dodecaic semitone
| dodecaic semitone
| neon
| neon
| {{UDnote|step=10}}
|-
|-
| 11
| 11
Line 112: Line 125:
| apotome, ranjani
| apotome, ranjani
| sodium
| sodium
| {{UDnote|step=11}}
|-
|-
| 12
| 12
Line 119: Line 133:
|  
|  
| magnesium
| magnesium
| {{UDnote|step=12}}
|-
|-
| 13
| 13
Line 126: Line 141:
|  
|  
| aluminium
| aluminium
| {{UDnote|step=13}}
|-
|-
| 14
| 14
Line 133: Line 149:
|  
|  
| silicon
| silicon
| {{UDnote|step=14}}
|-
|-
| 15
| 15
Line 140: Line 157:
|  
|  
| phosphorus
| phosphorus
| {{UDnote|step=15}}
|-
|-
| 16
| 16
Line 147: Line 165:
|  
|  
| sulphur
| sulphur
| {{UDnote|step=16}}
|-
|-
| 17
| 17
Line 154: Line 173:
| diminished tone
| diminished tone
| chlorine
| chlorine
| {{UDnote|step=17}}
|-
|-
| 18
| 18
Line 161: Line 181:
| minor tone, ratika
| minor tone, ratika
| argon
| argon
| {{UDnote|step=18}}
|-
|-
| 19
| 19
Line 168: Line 189:
| neutral tone, quasi-meantone
| neutral tone, quasi-meantone
| potassium
| potassium
| {{UDnote|step=19}}
|-
|-
| 20
| 20
Line 175: Line 197:
| major tone, raudri
| major tone, raudri
| calcium
| calcium
| {{UDnote|step=20}}
|-
|-
| 21
| 21
Line 182: Line 205:
| augmented tone
| augmented tone
| scandium
| scandium
| {{UDnote|step=21}}
|-
|-
| 22
| 22
Line 189: Line 213:
| minor slendric second
| minor slendric second
| titanium
| titanium
| {{UDnote|step=22}}
|-
|-
| 23
| 23
Line 196: Line 221:
| septimal second, slendric 2
| septimal second, slendric 2
| vanadium
| vanadium
| {{UDnote|step=23}}
|-
|-
| 24
| 24
Line 203: Line 229:
| major slendric second
| major slendric second
| chromium
| chromium
| {{UDnote|step=24}}
|-
|-
| 25
| 25
Line 210: Line 237:
| minor septimal third
| minor septimal third
| manganese
| manganese
| {{UDnote|step=25}}
|-
|-
| 26
| 26
Line 217: Line 245:
| septimal third
| septimal third
| iron
| iron
| {{UDnote|step=26}}
|-
|-
| 27
| 27
Line 224: Line 253:
| major septimal third
| major septimal third
| cobalt
| cobalt
| {{UDnote|step=27}}
|-
|-
| 28
| 28
Line 231: Line 261:
|  
|  
| nickel
| nickel
| {{UDnote|step=28}}
|-
|-
| 29
| 29
Line 238: Line 269:
| Pythagorean minor 3rd, krodha
| Pythagorean minor 3rd, krodha
| copper
| copper
| {{UDnote|step=29}}
|-
|-
| 30
| 30
Line 245: Line 277:
|  
|  
| zinc
| zinc
| {{UDnote|step=30}}
|-
|-
| 31
| 31
Line 252: Line 285:
| Classical minor 3rd, vajrika
| Classical minor 3rd, vajrika
| gallium
| gallium
| {{UDnote|step=31}}
|-
|-
| 32
| 32
Line 259: Line 293:
|  
|  
| germanium
| germanium
| {{UDnote|step=32}}
|-
|-
| 33
| 33
Line 266: Line 301:
| Lesser tridecimal third
| Lesser tridecimal third
| arsenic
| arsenic
| {{UDnote|step=33}}
|-
|-
| 34
| 34
Line 273: Line 309:
| Minor-neutral third
| Minor-neutral third
| selenium
| selenium
| {{UDnote|step=34}}
|-
|-
| 35
| 35
Line 280: Line 317:
| Minor tridecimal neurtral third, "major-neutral" third
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
| bromine
| {{UDnote|step=35}}
|-
|-
| 36
| 36
Line 287: Line 325:
| Golden ratio 3rd, major-tridecimal neutral third
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
| krypton
| {{UDnote|step=36}}
|-
|-
| 37
| 37
Line 294: Line 333:
|  
|  
| rubidium
| rubidium
| {{UDnote|step=37}}
|-
|-
| 38
| 38
Line 301: Line 341:
| Classical major 3rd, prasarini
| Classical major 3rd, prasarini
| strontium
| strontium
| {{UDnote|step=38}}
|-
|-
| 39
| 39
Line 308: Line 349:
|  
|  
| yttrium
| yttrium
| {{UDnote|step=39}}
|-
|-
| 40
| 40
Line 315: Line 357:
| Pythagorean major 3rd
| Pythagorean major 3rd
| zirconium
| zirconium
| {{UDnote|step=40}}
|-
|-
| 41
| 41
Line 322: Line 365:
|  
|  
| niobium
| niobium
| {{UDnote|step=41}}
|-
|-
| 42
| 42
Line 329: Line 373:
|  
|  
| molybdenum
| molybdenum
| {{UDnote|step=42}}
|-
|-
| 43
| 43
Line 336: Line 381:
|  
|  
| technetium
| technetium
| {{UDnote|step=43}}
|-
|-
| 44
| 44
Line 343: Line 389:
|  
|  
| ruthenium
| ruthenium
| {{UDnote|step=44}}
|-
|-
| 45
| 45
Line 350: Line 397:
| Barbados 3rd
| Barbados 3rd
| rhodium
| rhodium
| {{UDnote|step=45}}
|-
|-
| 46
| 46
Line 357: Line 405:
| Slendric 3
| Slendric 3
| palladium
| palladium
| {{UDnote|step=46}}
|-
|-
| 47
| 47
Line 364: Line 413:
|  
|  
| silver
| silver
| {{UDnote|step=47}}
|-
|-
| 48
| 48
Line 371: Line 421:
|  
|  
| cadmium
| cadmium
| {{UDnote|step=48}}
|-
|-
| 49
| 49
Line 378: Line 429:
| perfect 4th
| perfect 4th
| indium
| indium
| {{UDnote|step=49}}
|-
|-
| 50
| 50
Line 385: Line 437:
|  
|  
| tin
| tin
| {{UDnote|step=50}}
|-
|-
| 51
| 51
Line 392: Line 445:
| Kshiti
| Kshiti
| antimony
| antimony
| {{UDnote|step=51}}
|-
|-
| 52
| 52
Line 399: Line 453:
|  
|  
| tellurium
| tellurium
| {{UDnote|step=52}}
|-
|-
| 53
| 53
Line 404: Line 459:
|  
|  
|[[15/11]]
|[[15/11]]
| {{UDnote|step=53}}
|  
|  
| iodine
| iodine
Line 411: Line 467:
|  
|  
|[[48/35]], [[11/8]]
|[[48/35]], [[11/8]]
| {{UDnote|step=54}}
|  
|  
| xenon
| xenon
Line 420: Line 477:
|  
|  
| caesium
| caesium
| {{UDnote|step=55}}
|-
|-
| 56
| 56
Line 427: Line 485:
|  
|  
| barium
| barium
| {{UDnote|step=56}}
|-
|-
| 57
| 57
Line 434: Line 493:
|  
|  
| lanthanum
| lanthanum
| {{UDnote|step=57}}
|-
|-
| 58
| 58
Line 441: Line 501:
| Rakta
| Rakta
| cerium
| cerium
| {{UDnote|step=58}}
|-
|-
| 59
| 59
Line 448: Line 509:
| symmetric tritone
| symmetric tritone
| praseodymium
| praseodymium
| {{UDnote|step=59}}
|-
|-
| 60
| 60
Line 455: Line 517:
| Literal tritone, sandipani
| Literal tritone, sandipani
| neodymium
| neodymium
| {{UDnote|step=60}}
|-
|-
| 61
| 61
Line 462: Line 525:
|  
|  
| promethium
| promethium
| {{UDnote|step=61}}
|-
|-
| 62
| 62
Line 469: Line 533:
|  
|  
| samarium
| samarium
| {{UDnote|step=62}}
|-
|-
| 63
| 63
Line 476: Line 541:
|  
|  
| europium
| europium
| {{UDnote|step=63}}
|-
|-
| 64
| 64
Line 483: Line 549:
|  
|  
| gadolinium
| gadolinium
| {{UDnote|step=64}}
|-
|-
| 65
| 65
Line 490: Line 557:
|  
|  
| terbium
| terbium
| {{UDnote|step=65}}
|-
|-
| 66
| 66
Line 496: Line 564:
|[[72/49]], [[28/19]]
|[[72/49]], [[28/19]]
|  
|  
| dysprosiu
| dysprosium
| {{UDnote|step=66}}
|-
|-
| 67
| 67
Line 504: Line 573:
| wolf 5th
| wolf 5th
| holmium
| holmium
| {{UDnote|step=67}}
|-
|-
| 68
| 68
Line 511: Line 581:
| wolf cub 5th
| wolf cub 5th
| erbium
| erbium
| {{UDnote|step=68}}
|-
|-
| 69
| 69
Line 518: Line 589:
| perfect 5th, slendric 4
| perfect 5th, slendric 4
| thulium
| thulium
| {{UDnote|step=69}}
|-
|-
| 70
| 70
Line 525: Line 597:
| sheep 5th
| sheep 5th
| ytterbium
| ytterbium
| {{UDnote|step=70}}
|-
|-
| 71
| 71
Line 532: Line 605:
| lamb 5th
| lamb 5th
| lutetium
| lutetium
| {{UDnote|step=71}}
|-
|-
| 72
| 72
Line 539: Line 613:
|  
|  
| hafnium
| hafnium
| {{UDnote|step=72}}
|-
|-
| 73
| 73
Line 546: Line 621:
|  
|  
| tantalum
| tantalum
| {{UDnote|step=73}}
|-
|-
| 74
| 74
Line 553: Line 629:
|  
|  
| tungsten
| tungsten
| {{UDnote|step=74}}
|-
|-
| 75
| 75
Line 560: Line 637:
|  
|  
| rhenium
| rhenium
| {{UDnote|step=75}}
|-
|-
| 76
| 76
Line 567: Line 645:
|  
|  
| osmium
| osmium
| {{UDnote|step=76}}
|-
|-
| 77
| 77
Line 574: Line 653:
|  
|  
| iridium
| iridium
| {{UDnote|step=77}}
|-
|-
| 78
| 78
Line 581: Line 661:
| Pythagorean minor 6th
| Pythagorean minor 6th
| platinum
| platinum
| {{UDnote|step=78}}
|-
|-
| 79
| 79
Line 588: Line 669:
|  
|  
| gold
| gold
| {{UDnote|step=79}}
|-
|-
| 80
| 80
Line 595: Line 677:
| Classical minor 6th
| Classical minor 6th
| mercury
| mercury
| {{UDnote|step=80}}
|-
|-
| 81
| 81
Line 602: Line 685:
|  
|  
| thallium
| thallium
| {{UDnote|step=81}}
|-
|-
| 82
| 82
Line 609: Line 693:
| Golden ratio sixth, minor-neutral tridecimal sixth
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
| lead
| {{UDnote|step=82}}
|-
|-
| 83
| 83
Line 616: Line 701:
| Major tridecimal neutral sixth, "minor-neutral" sixth
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
| bismuth
| {{UDnote|step=83}}
|-
|-
| 84
| 84
Line 623: Line 709:
| Major-neutral sixth
| Major-neutral sixth
| polonium
| polonium
| {{UDnote|step=84}}
|-
|-
| 85
| 85
Line 630: Line 717:
|  
|  
| astatine
| astatine
| {{UDnote|step=85}}
|-
|-
| 86
| 86
Line 637: Line 725:
|  
|  
| radon
| radon
| {{UDnote|step=86}}
|-
|-
| 87
| 87
Line 644: Line 733:
| Classical major 6th
| Classical major 6th
| francium
| francium
| {{UDnote|step=87}}
|-
|-
| 88
| 88
Line 651: Line 741:
|  
|  
| radium
| radium
| {{UDnote|step=88}}
|-
|-
| 89
| 89
Line 658: Line 749:
| Pythagorean major 6th
| Pythagorean major 6th
| actinium
| actinium
| {{UDnote|step=89}}
|-
|-
| 90
| 90
Line 665: Line 757:
|  
|  
| thorium
| thorium
| {{UDnote|step=90}}
|-
|-
| 91
| 91
Line 672: Line 765:
|  
|  
| protactinium
| protactinium
| {{UDnote|step=91}}
|-
|-
| 92
| 92
Line 679: Line 773:
| Septimal supermajor 6th, slendric 5
| Septimal supermajor 6th, slendric 5
| uranium
| uranium
| {{UDnote|step=92}}
|-
|-
| 93
| 93
Line 686: Line 781:
|  
|  
| neptunium
| neptunium
| {{UDnote|step=93}}
|-
|-
| 94
| 94
Line 693: Line 789:
|  
|  
| plutonium
| plutonium
| {{UDnote|step=94}}
|-
|-
| 95
| 95
Line 700: Line 797:
| Harmonic 7th
| Harmonic 7th
| americium
| americium
| {{UDnote|step=95}}
|-
|-
| 96
| 96
Line 707: Line 805:
|  
|  
| curium
| curium
| {{UDnote|step=96}}
|-
|-
| 97
| 97
Line 714: Line 813:
|  
|  
| berkelium
| berkelium
| {{UDnote|step=97}}
|-
|-
| 98
| 98
Line 721: Line 821:
| Pythagorean minor 7th
| Pythagorean minor 7th
| californium
| californium
| {{UDnote|step=98}}
|-
|-
| 99
| 99
Line 728: Line 829:
|  
|  
| einsteinium
| einsteinium
| {{UDnote|step=99}}
|-
|-
| 100
| 100
Line 735: Line 837:
| Tivra
| Tivra
| fermium
| fermium
| {{UDnote|step=100}}
|-
|-
| 101
| 101
Line 742: Line 845:
|  
|  
| mendelevium
| mendelevium
| {{UDnote|step=101}}
|-
|-
| 102
| 102
Line 749: Line 853:
|  
|  
| nobelium
| nobelium
| {{UDnote|step=102}}
|-
|-
| 103
| 103
Line 756: Line 861:
|  
|  
| lawrencium
| lawrencium
| {{UDnote|step=103}}
|-
|-
| 104
| 104
Line 763: Line 869:
|  
|  
| rutherfordium
| rutherfordium
| {{UDnote|step=104}}
|-
|-
| 105
| 105
Line 770: Line 877:
|  
|  
| dubnium
| dubnium
| {{UDnote|step=105}}
|-
|-
| 106
| 106
Line 777: Line 885:
|  
|  
| seaborgium
| seaborgium
| {{UDnote|step=106}}
|-
|-
| 107
| 107
Line 784: Line 893:
|  
|  
| bohrium
| bohrium
| {{UDnote|step=107}}
|-
|-
| 108
| 108
Line 791: Line 901:
|  
|  
| hassium
| hassium
| {{UDnote|step=108}}
|-
|-
| 109
| 109
Line 798: Line 909:
| Pythagorean major 7th
| Pythagorean major 7th
| meitnerium
| meitnerium
| {{UDnote|step=109}}
|-
|-
| 110
| 110
Line 805: Line 917:
|  
|  
| darmstadtium
| darmstadtium
| {{UDnote|step=110}}
|-
|-
| 111
| 111
Line 812: Line 925:
|  
|  
| roentgenium
| roentgenium
| {{UDnote|step=111}}
|-
|-
| 112
| 112
Line 819: Line 933:
|  
|  
| copernicium
| copernicium
| {{UDnote|step=112}}
|-
|-
| 113
| 113
Line 826: Line 941:
|  
|  
| nihonium
| nihonium
| {{UDnote|step=113}}
|-
|-
| 114
| 114
Line 833: Line 949:
|  
|  
| flerovium
| flerovium
| {{UDnote|step=114}}
|-
|-
| 115
| 115
Line 840: Line 957:
|  
|  
| moscovium
| moscovium
| {{UDnote|step=115}}
|-
|-
| 116
| 116
Line 847: Line 965:
| Comma supermajor 7th
| Comma supermajor 7th
| livermorium
| livermorium
| {{UDnote|step=116}}
|-
|-
| 117
| 117
Line 854: Line 973:
| Semicomma supermajor 7th
| Semicomma supermajor 7th
| tenessine
| tenessine
| {{UDnote|step=117}}
|-
|-
| 118
| 118
Line 861: Line 981:
| perfect 8ve
| perfect 8ve
| oganesson / neutronium
| oganesson / neutronium
| {{UDnote|step=118}}
|}
|}
<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 08:19, 16 June 2023

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

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) 		Chemical Notation
(see below, if base note = 0) 		Ups and downs notation
0 0.00 P1 1/1 unison oganesson / neutronium D
1 10.17 126/125, 225/224, 121/120, 243/242 semicomma hydrogen ^D, ^3E♭♭
2 20.34 81/80, 531441/524288 comma helium ^^D, ^4E♭♭
3 30.51 64/63, 49/48 augmented comma lithium ^3D, ^5E♭♭
4 40.68 50/49 beryllium ^4D, v5E♭
5 50.85 36/35 boron ^5D, v4E♭
6 61.02 28/27 carbon v5D♯, v3E♭
7 71.19 25/24 nitrogen v4D♯, vvE♭
8 81.36 21/20, 22/21 oxygen v3D♯, vE♭
9 91.53 m2 19/18, 20/19, 256/243 limma, dayavati fluorine vvD♯, E♭
10 101.69 vD♯, ^E♭ 17/16, 18/17 dodecaic semitone neon vD♯, ^E♭
11 111.86 16/15, 2187/2048 apotome, ranjani sodium D♯, ^^E♭
12 122.03 15/14 magnesium ^D♯, ^3E♭
13 132.20 27/25 aluminium ^^D♯, ^4E♭
14 142.37 88/81 silicon ^3D♯, ^5E♭
15 152.54 12/11 phosphorus ^4D♯, v5E
16 162.71 11/10 sulphur ^5D♯, v4E
17 172.88 21/19 diminished tone chlorine v5D𝄪, v3E
18 183.05 10/9 minor tone, ratika argon v4D𝄪, vvE
19 193.22 28/25, 19/17 neutral tone, quasi-meantone potassium v3D𝄪, vE
20 203.39 M2 9/8 major tone, raudri calcium E
21 213.56 17/15 augmented tone scandium ^E, ^3F♭
22 223.73 256/225 minor slendric second titanium ^^E, ^4F♭
23 233.90 8/7 septimal second, slendric 2 vanadium ^3E, ^5F♭
24 244.07 144/125, 121/105 major slendric second chromium ^4E, v5F
25 254.24 125/108, 81/70, 22/19 minor septimal third manganese ^5E, v4F
26 260.41 7/6 septimal third iron v5E♯, v3F
27 274.58 75/64 major septimal third cobalt v4E♯, vvF
28 284.75 33/28 nickel v3E♯, vF
29 294.92 m3 32/27, 19/16 Pythagorean minor 3rd, krodha copper F
30 305.08 25/21 zinc ^F, ^3G♭♭
31 315.25 6/5 Classical minor 3rd, vajrika gallium ^^F, ^4G♭♭
32 325.42 98/81 germanium ^3F, ^5G♭♭
33 335.59 40/33, 17/14 Lesser tridecimal third arsenic ^4F, v5G♭
34 345.76 11/9 Minor-neutral third selenium ^5F, v4G♭
35 355.93 27/22, 16/13 I** Minor tridecimal neurtral third, "major-neutral" third bromine v5F♯, v3G♭
36 366.10 99/80, 21/17, 16/13 II** Golden ratio 3rd, major-tridecimal neutral third krypton v4F♯, vvG♭
37 376.27 56/45 rubidium v3F♯, vG♭
38 386.44 5/4 Classical major 3rd, prasarini strontium vvF♯, G♭
39 396.61 63/50 yttrium vF♯, ^G♭
40 406.78 M3 24/19, 19/15 Pythagorean major 3rd zirconium F♯, ^^G♭
41 416.95 14/11 niobium ^F♯, ^3G♭
42 427.12 77/60 molybdenum ^^F♯, ^4G♭
43 437.29 9/7 technetium ^3F♯, ^5G♭
44 447.46 35/27, 22/17 ruthenium ^4F♯, v5G
45 457.63 98/75 Barbados 3rd rhodium ^5F♯, v4G
46 467.80 21/16 Slendric 3 palladium v5F𝄪, v3G
47 477.97 320/243 silver v4F𝄪, vvG
48 488.14 160/121, 85/64 cadmium v3F𝄪, vG
49 498.31 P4 4/3 perfect 4th indium G
50 508.47 75/56, 51/38 tin ^G, ^3A♭♭
51 518.64 27/20 Kshiti antimony ^^G, ^4A♭♭
52 528.81 49/36, 19/14 tellurium ^3G, ^5A♭♭
53 538.98 15/11 ^4G, v5A♭ iodine
54 549.15 48/35, 11/8 ^5G, v4A♭ xenon
55 559.32 112/81 caesium v5G♯, v3A♭
56 569.49 25/18 barium v4G♯, vvA♭
57 579.66 7/5 lanthanum v3G♯, vA♭
58 589.83 d5 45/32 Rakta cerium vvG♯, A♭
59 600.00 99/70, 140/99, 17/12, 24/17 symmetric tritone praseodymium vG♯, ^A♭
60 610.17 A4 64/45, 729/512 Literal tritone, sandipani neodymium G♯, ^^A♭
61 620.34 10/7 promethium ^G♯, ^3A♭
62 630.51 36/25 samarium ^^G♯, ^4A♭
63 640.68 81/56 europium ^3G♯, ^5A♭
64 650.85 35/24, 16/11 gadolinium ^4G♯, v5A
65 661.02 22/15 terbium ^5G♯, v4A
66 671.19 72/49, 28/19 dysprosium v5G𝄪, v3A
67 681.36 40/27 wolf 5th holmium v4G𝄪, vvA
68 691.53 112/75, 76/51 wolf cub 5th erbium v3G𝄪, vA
69 701.69 P5 3/2 perfect 5th, slendric 4 thulium A
70 711.86 121/80, 128/85 sheep 5th ytterbium ^A, ^3B♭♭
71 722.03 243/160 lamb 5th lutetium ^^A, ^4B♭♭
72 732.20 32/21 hafnium ^3A, ^5B♭♭
73 742.37 75/49 tantalum ^4A, v5B♭
74 752.54 54/35, 17/11 tungsten ^5A, v4B♭
75 762.71 14/9 rhenium v5A♯, v3B♭
76 772.88 120/77 osmium v4A♯, vvB♭
77 783.05 11/7 iridium v3A♯, vB♭
78 793.22 m6 19/12, 30/19 Pythagorean minor 6th platinum vvA♯, B♭
79 803.39 100/63 gold vA♯, ^B♭
80 813.56 8/5 Classical minor 6th mercury A♯, ^^B♭
81 823.73 45/28 thallium ^A♯, ^3B♭
82 833.90 160/99, 34/21, 13/8 I** Golden ratio sixth, minor-neutral tridecimal sixth lead ^^A♯, ^4B♭
83 844.07 44/27, 13/8 II** Major tridecimal neutral sixth, "minor-neutral" sixth bismuth ^3A♯, ^5B♭
84 854.24 18/11 Major-neutral sixth polonium ^4A♯, v5B
85 864.41 28/17 astatine ^5A♯, v4B
86 874.58 81/49 radon v5A𝄪, v3B
87 884.75 5/3 Classical major 6th francium v4A𝄪, vvB
88 894.92 42/25 radium v3A𝄪, vB
89 905.08 M6 27/16, 32/19 Pythagorean major 6th actinium B
90 915.25 56/33 thorium ^B, ^3C♭
91 925.42 128/75 protactinium ^^B, ^4C♭
92 935.59 12/7 Septimal supermajor 6th, slendric 5 uranium ^3B, ^5C♭
93 945.76 216/125, 140/81, 121/70, 19/11 neptunium ^4B, v5C
94 955.93 125/72 plutonium ^5B, v4C
95 966.10 7/4 Harmonic 7th americium v5B♯, v3C
96 976.27 225/128 curium v4B♯, vvC
97 986.44 30/17 berkelium v3B♯, vC
98 996.61 m7 16/9 Pythagorean minor 7th californium C
99 1006.78 25/14 einsteinium ^C, ^3D♭♭
100 1016.95 9/5 Tivra fermium ^^C, ^4D♭♭
101 1027.12 38/21 mendelevium ^3C, ^5D♭♭
102 1037.29 20/11 nobelium ^4C, v5D♭
103 1047.46 11/6 lawrencium ^5C, v4D♭
104 1057.63 81/44 rutherfordium v5C♯, v3D♭
105 1067.80 50/27 dubnium v4C♯, vvD♭
106 1077.97 28/15 seaborgium v3C♯, vD♭
107 1088.14 15/8 bohrium vvC♯, D♭
108 1098.31 32/17, 17/9 hassium vC♯, ^D♭
109 1108.47 M7 36/19, 19/10, 243/128 Pythagorean major 7th meitnerium C♯, ^^D♭
110 1118.64 40/21, 21/11 darmstadtium ^C♯, ^3D♭
111 1128.81 48/25 roentgenium ^^C♯, ^4D♭
112 1138.98 27/14 copernicium ^3C♯, ^5D♭
113 1149.15 35/18, 64/33 nihonium ^4C♯, v5D
114 1159.32 49/25 flerovium ^5C♯, v4D
115 1169.49 63/32, 96/49 moscovium v5C𝄪, v3D
116 1179.66 160/81 Comma supermajor 7th livermorium v4C𝄪, vvD
117 1189.83 125/63, 448/225, 240/121, 484/243 Semicomma supermajor 7th tenessine v3C𝄪, vD
118 1200.00 P8 2/1 perfect 8ve oganesson / neutronium 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 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 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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