118edo: Difference between revisions

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== 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 118edo
|+ style=white-space:nowrap | Table of intervals in 118edo
! Step
! Step
! Cents
! Marks
! Eliora's Naming System
! Eliora's Naming System
! Eliora's Chemical Notation<br>(if base note = 0)
! Eliora's Chemical Notation<br>(if base note = 0)
! Approximate Ratios
! Approximate Ratios *
|-
|-
| 0
| 0
| 0.00
| P1
| unison
| unison
| oganesson / neutronium
| oganesson / neutronium
Line 40: Line 44:
|-
|-
| 1
| 1
| 10.17
|
| semicomma
| semicomma
| hydrogen
| hydrogen
Line 45: Line 51:
|-
|-
| 2
| 2
| 20.34
|
| comma
| comma
| helium
| helium
Line 50: Line 58:
|-
|-
| 3
| 3
| 30.51
|
|  
|  
|  
|  
Line 55: Line 65:
|-
|-
| 4
| 4
| 40.68
|
|  
|  
|  
|  
Line 60: Line 72:
|-
|-
| 5
| 5
| 50.85
|
|  
|  
|  
|  
Line 65: Line 79:
|-
|-
| 6
| 6
| 61.02
|
|  
|  
|  
|  
Line 70: Line 86:
|-
|-
| 7
| 7
| 71.19
|
|  
|  
|  
|  
Line 75: Line 93:
|-
|-
| 8
| 8
| 81.36
|
|  
|  
|  
|  
Line 80: Line 100:
|-
|-
| 9
| 9
| 91.53
| m2
| limma, dayavati
| limma, dayavati
| fluorine
| fluorine
Line 85: Line 107:
|-
|-
| 10
| 10
| 101.69
|
| dodecaic semitone
| dodecaic semitone
| neon
| neon
Line 90: Line 114:
|-
|-
| 11
| 11
| 111.86
|
| apotome, ranjani
| apotome, ranjani
| sodium
| sodium
Line 95: Line 121:
|-
|-
| 12
| 12
| 122.03
|
|  
|  
|  
|  
Line 100: Line 128:
|-
|-
| 13
| 13
| 132.20
|
|  
|  
|  
|  
Line 105: Line 135:
|-
|-
| 14
| 14
| 142.37
|
|  
|  
|  
|  
Line 110: Line 142:
|-
|-
| 15
| 15
| 152.54
|
|  
|  
|  
|  
Line 115: Line 149:
|-
|-
| 16
| 16
| 162.71
|
|  
|  
|  
|  
Line 120: Line 156:
|-
|-
| 17
| 17
| 172.88
|
|  
|  
|  
|  
| [[21/19]]
| [[21/19]]
|-
|-
| 18
| 18
| 183.05
|
| diminished tone, ratika
| diminished tone, ratika
| argon
| argon
Line 131: Line 170:
|-
|-
| 19
| 19
| 193.22
|
| minor tone
| minor tone
| potassium
| potassium
Line 136: Line 177:
|-
|-
| 20
| 20
| 203.39
| M2
| major tone, raudri
| major tone, raudri
| calcium
| calcium
Line 141: Line 184:
|-
|-
| 21
| 21
| 213.56
|
|  
|  
|  
|  
Line 146: Line 191:
|-
|-
| 22
| 22
| 223.73
|
|  
|  
|  
|  
Line 151: Line 198:
|-
|-
| 23
| 23
| 233.90
|
| septimal second, slendric 2
| septimal second, slendric 2
| vanadium
| vanadium
Line 156: Line 205:
|-
|-
| 24
| 24
| 244.07
|
|  
|  
|  
|  
Line 161: Line 212:
|-
|-
| 25
| 25
| 254.24
|
|  
|  
|  
|  
Line 166: Line 219:
|-
|-
| 26
| 26
| 260.41
|
| septimal third
| septimal third
| iron
| iron
Line 171: Line 226:
|-
|-
| 27
| 27
| 274.58
|
|  
|  
|  
|  
Line 176: Line 233:
|-
|-
| 28
| 28
| 284.75
|
|  
|  
|  
|  
Line 181: Line 240:
|-
|-
| 29
| 29
| 294.92
| m3
| Pythagorean minor 3rd, krodha
| Pythagorean minor 3rd, krodha
| copper
| copper
Line 186: Line 247:
|-
|-
| 30
| 30
| 305.08
|
|  
|  
|  
|  
Line 191: Line 254:
|-
|-
| 31
| 31
| 315.25
|
| Classical minor 3rd, vajrika
| Classical minor 3rd, vajrika
| gallium
| gallium
Line 196: Line 261:
|-
|-
| 32
| 32
| 325.42
|
|  
|  
|  
|  
Line 201: Line 268:
|-
|-
| 33
| 33
| 335.59
|
| Lesser tridecimal third
| Lesser tridecimal third
| germanium
| germanium
Line 206: Line 275:
|-
|-
| 34
| 34
| 345.76
|
| Minor-neutral third
| Minor-neutral third
| selenium
| selenium
Line 211: Line 282:
|-
|-
| 35
| 35
| 355.93
|
| Minor tridecimal neurtral third, "major-neutral" third
| Minor tridecimal neurtral third, "major-neutral" third
| bromine
| bromine
Line 216: Line 289:
|-
|-
| 36
| 36
| 366.10
|
| Golden ratio 3rd, major-tridecimal neutral third
| Golden ratio 3rd, major-tridecimal neutral third
| krypton
| krypton
Line 221: Line 296:
|-
|-
| 37
| 37
| 376.27
|
|  
|  
|  
|  
Line 226: Line 303:
|-
|-
| 38
| 38
| 386.44
|
| Classical major 3rd, prasarini
| Classical major 3rd, prasarini
| strontium
| strontium
Line 231: Line 310:
|-
|-
| 39
| 39
| 396.61
|
|  
|  
|  
|  
Line 236: Line 317:
|-
|-
| 40
| 40
| 406.78
| M3
| Pythagorean major 3rd
| Pythagorean major 3rd
| zirconium
| zirconium
Line 241: Line 324:
|-
|-
| 41
| 41
| 416.95
|
|  
|  
|  
|  
Line 246: Line 331:
|-
|-
| 42
| 42
| 427.12
|
|  
|  
|  
|  
Line 251: Line 338:
|-
|-
| 43
| 43
| 437.29
|
|  
|  
|  
|  
Line 256: Line 345:
|-
|-
| 44
| 44
| 447.46
|
|  
|  
|  
|  
Line 261: Line 352:
|-
|-
| 45
| 45
| 457.63
|
| Barbados 3rd
| Barbados 3rd
| rhodium
| rhodium
Line 266: Line 359:
|-
|-
| 46
| 46
| 467.80
|
| Slendric 3
| Slendric 3
| palladium
| palladium
Line 271: Line 366:
|-
|-
| 47
| 47
| 477.97
|
|  
|  
|  
|  
Line 276: Line 373:
|-
|-
| 48
| 48
| 488.14
|
|  
|  
|  
|  
Line 281: Line 380:
|-
|-
| 49
| 49
| 498.31
| P4
| perfect 4th
| perfect 4th
| indium
| indium
Line 286: Line 387:
|-
|-
| 50
| 50
| 508.47
|
|  
|  
|  
|  
Line 291: Line 394:
|-
|-
| 51
| 51
| 518.64
|
| Kshiti
| Kshiti
| antimony
| antimony
Line 296: Line 401:
|-
|-
| 52
| 52
| 528.81
|
|  
|  
|  
|  
Line 301: Line 408:
|-
|-
| 53
| 53
| 538.98
|
|  
|  
|  
|  
Line 306: Line 415:
|-
|-
| 54
| 54
| 549.15
|
|  
|  
|  
|  
Line 311: Line 422:
|-
|-
| 55
| 55
| 559.32
|
|  
|  
|  
|  
Line 316: Line 429:
|-
|-
| 56
| 56
| 569.49
|
|  
|  
|  
|  
Line 321: Line 436:
|-
|-
| 57
| 57
| 579.66
|
|  
|  
|  
|  
Line 326: Line 443:
|-
|-
| 58
| 58
| 589.83
| d5
| Rakta
| Rakta
| cerium
| cerium
Line 331: Line 450:
|-
|-
| 59
| 59
| 600.00
|
| symmetric tritone
| symmetric tritone
| praseodymium
| praseodymium
Line 336: Line 457:
|-
|-
| 60
| 60
| 610.17
| A4
| Literal tritone, sandipani
| Literal tritone, sandipani
| neodymium
| neodymium
Line 341: Line 464:
|-
|-
| 61
| 61
| 620.34
|
|  
|  
|  
|  
Line 346: Line 471:
|-
|-
| 62
| 62
| 630.51
|
|  
|  
|  
|  
Line 351: Line 478:
|-
|-
| 63
| 63
| 640.68
|
|  
|  
|  
|  
Line 356: Line 485:
|-
|-
| 64
| 64
| 650.85
|
|  
|  
|  
|  
Line 361: Line 492:
|-
|-
| 65
| 65
| 661.02
|
|  
|  
|  
|  
Line 366: Line 499:
|-
|-
| 66
| 66
| 671.19
|
|  
|  
|  
|  
Line 371: Line 506:
|-
|-
| 67
| 67
| 681.36
|
|  
|  
|  
|  
Line 376: Line 513:
|-
|-
| 68
| 68
| 691.53
|
|  
|  
|  
|  
Line 381: Line 520:
|-
|-
| 69
| 69
| 701.69
| P5
| perfect 5th
| perfect 5th
| thulium
| thulium
Line 386: Line 527:
|-
|-
| 70
| 70
| 711.86
|
|  
|  
|  
|  
Line 391: Line 534:
|-
|-
| 71
| 71
| 722.03
|
|  
|  
|  
|  
Line 396: Line 541:
|-
|-
| 72
| 72
| 732.20
|
|  
|  
|  
|  
Line 401: Line 548:
|-
|-
| 73
| 73
| 742.37
|
|  
|  
|  
|  
Line 406: Line 555:
|-
|-
| 74
| 74
| 752.54
|
|  
|  
|  
|  
Line 411: Line 562:
|-
|-
| 75
| 75
| 762.71
|
|  
|  
|  
|  
Line 416: Line 569:
|-
|-
| 76
| 76
| 772.88
|
|  
|  
|  
|  
Line 421: Line 576:
|-
|-
| 77
| 77
| 783.05
|
|  
|  
|  
|  
Line 426: Line 583:
|-
|-
| 78
| 78
| 793.22
| m6
| Pythagorean minor 6th
| Pythagorean minor 6th
| platinum
| platinum
Line 431: Line 590:
|-
|-
| 79
| 79
| 803.39
|
|  
|  
|  
|  
Line 436: Line 597:
|-
|-
| 80
| 80
| 813.56
|
| Classical minor 6th
| Classical minor 6th
| mercury
| mercury
Line 441: Line 604:
|-
|-
| 81
| 81
| 823.73
|
|  
|  
|  
|  
Line 446: Line 611:
|-
|-
| 82
| 82
| 833.90
|
| Golden ratio sixth, minor-neutral tridecimal sixth
| Golden ratio sixth, minor-neutral tridecimal sixth
| lead
| lead
Line 451: Line 618:
|-
|-
| 83
| 83
| 844.07
|
| Major tridecimal neutral sixth, "minor-neutral" sixth
| Major tridecimal neutral sixth, "minor-neutral" sixth
| bismuth
| bismuth
Line 456: Line 625:
|-
|-
| 84
| 84
| 854.24
|
| Major-neutral sixth
| Major-neutral sixth
| polonium
| polonium
Line 461: Line 632:
|-
|-
| 85
| 85
| 864.41
|
|  
|  
|  
|  
Line 466: Line 639:
|-
|-
| 86
| 86
| 874.58
|
|  
|  
|  
|  
Line 471: Line 646:
|-
|-
| 87
| 87
| 884.75
|
| Classical major 6th
| Classical major 6th
| francium
| francium
Line 476: Line 653:
|-
|-
| 88
| 88
| 894.92
|
|  
|  
|  
|  
Line 481: Line 660:
|-
|-
| 89
| 89
| 905.08
| M6
| Pythagorean major 6th
| Pythagorean major 6th
| actinium
| actinium
Line 486: Line 667:
|-
|-
| 90
| 90
| 915.25
|
|  
|  
|  
|  
Line 491: Line 674:
|-
|-
| 91
| 91
| 925.42
|
|  
|  
|  
|  
Line 496: Line 681:
|-
|-
| 92
| 92
| 935.59
|
| Septimal supermajor 6th, slendro 5
| Septimal supermajor 6th, slendro 5
| uranium
| uranium
Line 501: Line 688:
|-
|-
| 93
| 93
| 945.76
|
|  
|  
|  
|  
Line 506: Line 695:
|-
|-
| 94
| 94
| 955.93
|
|  
|  
|  
|  
Line 511: Line 702:
|-
|-
| 95
| 95
| 966.10
|
| Harmonic 7th
| Harmonic 7th
| americium
| americium
Line 516: Line 709:
|-
|-
| 96
| 96
| 976.27
|
|  
|  
|  
|  
Line 521: Line 716:
|-
|-
| 97
| 97
| 986.44
|
|  
|  
|  
|  
Line 526: Line 723:
|-
|-
| 98
| 98
| 996.61
| m7
|  
|  
|  
|  
Line 531: Line 730:
|-
|-
| 99
| 99
| 1006.78
|
|  
|  
|  
|  
Line 536: Line 737:
|-
|-
| 100
| 100
| 1016.95
|
| Tivra
| Tivra
| fermium
| fermium
Line 541: Line 744:
|-
|-
| 101
| 101
| 1027.12
|
|  
|  
|  
|  
Line 546: Line 751:
|-
|-
| 102
| 102
| 1037.29
|
|  
|  
|  
|  
Line 551: Line 758:
|-
|-
| 103
| 103
| 1047.46
|
|  
|  
|  
|  
Line 556: Line 765:
|-
|-
| 104
| 104
| 1057.63
|
|  
|  
|  
|  
Line 561: Line 772:
|-
|-
| 105
| 105
| 1067.80
|
|  
|  
|  
|  
Line 566: Line 779:
|-
|-
| 106
| 106
| 1077.97
|
|  
|  
|  
|  
Line 571: Line 786:
|-
|-
| 107
| 107
| 1088.14
|
|  
|  
|  
|  
Line 576: Line 793:
|-
|-
| 108
| 108
| 1098.31
|
|  
|  
|  
|  
Line 581: Line 800:
|-
|-
| 109
| 109
| 1108.47
| M7
| Pythagorean major 7th
| Pythagorean major 7th
| meitnerium
| meitnerium
Line 586: Line 807:
|-
|-
| 110
| 110
| 1118.64
|
|  
|  
|  
|  
Line 591: Line 814:
|-
|-
| 111
| 111
| 1128.81
|
|  
|  
|  
|  
Line 596: Line 821:
|-
|-
| 112
| 112
| 1138.98
|
|  
|  
|  
|  
Line 601: Line 828:
|-
|-
| 113
| 113
| 1149.15
|
|  
|  
|  
|  
Line 606: Line 835:
|-
|-
| 114
| 114
| 1159.32
|
|  
|  
|  
|  
Line 611: Line 842:
|-
|-
| 115
| 115
| 1169.49
|
|  
|  
|  
|  
Line 616: Line 849:
|-
|-
| 116
| 116
| 1179.66
|
| Comma 7th
| Comma 7th
| livermorium
| livermorium
Line 621: Line 856:
|-
|-
| 117
| 117
| 1189.83
|
| Semicomma supermajor 7th
| Semicomma supermajor 7th
| tenessine
| tenessine
Line 626: Line 863:
|-
|-
| 118
| 118
| 1200.00
| P8
| perfect 8ve
| perfect 8ve
| oganesson / neutronium
| oganesson / neutronium
| [[2/1]]
| [[2/1]]
|}
|}
<nowiki>*</nowiki> treated as a 2.3.5.7.11.17.19 system


== Notation ==
== Notation ==

Revision as of 19:45, 4 December 2021

← 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 Eliora's Naming System Eliora's Chemical Notation
(if base note = 0) 		Approximate Ratios *
0 0.00 P1 unison oganesson / neutronium 1/1
1 10.17 semicomma hydrogen 126/125, 225/224, 121/120, 243/242
2 20.34 comma helium 81/80
3 30.51 64/63, 49/48
4 40.68 50/49
5 50.85 36/35
6 61.02 28/27
7 71.19 25/24
8 81.36 21/20, 22/21
9 91.53 m2 limma, dayavati fluorine 19/18, 20/19
10 101.69 dodecaic semitone neon 17/16, 18/17
11 111.86 apotome, ranjani sodium 16/15
12 122.03 15/14
13 132.20 27/25
14 142.37 88/81
15 152.54 12/11
16 162.71 11/10
17 172.88 21/19
18 183.05 diminished tone, ratika argon 10/9
19 193.22 minor tone potassium 28/25, 19/17
20 203.39 M2 major tone, raudri calcium 9/8
21 213.56 17/15
22 223.73 256/225
23 233.90 septimal second, slendric 2 vanadium 8/7
24 244.07 144/125, 121/105
25 254.24 125/108, 81/70, 22/19
26 260.41 septimal third iron 7/6
27 274.58 75/64
28 284.75 33/28
29 294.92 m3 Pythagorean minor 3rd, krodha copper 32/27, 19/16
30 305.08 25/21
31 315.25 Classical minor 3rd, vajrika gallium 6/5
32 325.42 135/112
33 335.59 Lesser tridecimal third germanium 40/33, 17/14
34 345.76 Minor-neutral third selenium 11/9
35 355.93 Minor tridecimal neurtral third, "major-neutral" third bromine 27/22
36 366.10 Golden ratio 3rd, major-tridecimal neutral third krypton 99/80, 21/17
37 376.27 56/45
38 386.44 Classical major 3rd, prasarini strontium 5/4
39 396.61 63/50
40 406.78 M3 Pythagorean major 3rd zirconium 24/19, 19/15
41 416.95 14/11
42 427.12 77/60
43 437.29 9/7
44 447.46 35/27, 22/17
45 457.63 Barbados 3rd rhodium 98/75
46 467.80 Slendric 3 palladium 21/16
47 477.97 320/243
48 488.14 160/121, 85/64
49 498.31 P4 perfect 4th indium 4/3
50 508.47 75/56, 51/38
51 518.64 Kshiti antimony 27/20
52 528.81 49/36
53 538.98 15/11
54 549.15 48/35, 11/8
55 559.32 112/81
56 569.49 25/18
57 579.66 7/5
58 589.83 d5 Rakta cerium 45/32
59 600.00 symmetric tritone praseodymium 99/70, 140/99, 17/12, 24/17
60 610.17 A4 Literal tritone, sandipani neodymium 64/45
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
67 681.36 40/27
68 691.53 112/75, 76/51
69 701.69 P5 perfect 5th thulium 3/2
70 711.86 121/80, 128/85
71 722.03 243/160
72 732.20 32/21
73 742.37 75/49
74 752.54 54/35, 17/11
75 762.71 14/9
76 772.88 120/77
77 783.05 11/7
78 793.22 m6 Pythagorean minor 6th platinum 19/12, 30/19
79 803.39 100/63
80 813.56 Classical minor 6th mercury 8/5
81 823.73 45/28
82 833.90 Golden ratio sixth, minor-neutral tridecimal sixth lead 160/99, 34/21
83 844.07 Major tridecimal neutral sixth, "minor-neutral" sixth bismuth 44/27
84 854.24 Major-neutral sixth polonium 18/11
85 864.41 28/17
86 874.58 224/135
87 884.75 Classical major 6th francium 5/3
88 894.92 42/25
89 905.08 M6 Pythagorean major 6th actinium 27/16, 32/19
90 915.25 56/33
91 925.42 128/75
92 935.59 Septimal supermajor 6th, slendro 5 uranium 12/7
93 945.76 216/125, 140/81, 121/70, 19/11
94 955.93 125/72
95 966.10 Harmonic 7th americium 7/4
96 976.27 225/128
97 986.44 30/17
98 996.61 m7 16/9
99 1006.78 25/14
100 1016.95 Tivra fermium 9/5
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 Pythagorean major 7th meitnerium 36/19, 19/10
110 1118.64 40/21, 21/11
111 1128.81 48/25
112 1138.98 27/14
113 1149.15 35/18, 64/33
114 1159.32 49/25
115 1169.49 63/32, 96/49
116 1179.66 Comma 7th livermorium 160/81
117 1189.83 Semicomma supermajor 7th tenessine 125/63, 448/225, 240/121, 484/243
118 1200.00 P8 perfect 8ve oganesson / neutronium 2/1

* treated as a 2.3.5.7.11.17.19 system

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.

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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