118edo

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← 117edo118edo119edo →
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
Special properties

118 equal divisions of the octave (abbreviated 118edo or 118ed2), also called 118-tone equal temperament (118tet) or 118 equal temperament (118et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 118 equal parts of about 10.2 ¢ each. Each step represents a frequency ratio of 21/118, or the 118th root of 2.

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, v8E♭ LD
2 20.34 81/80, 531441/524288 comma helium ^^D, v7E♭ KD
3 30.51 64/63, 49/48 augmented comma lithium ^3D, v6E♭ 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 ^6D, v3E♭ sEb, uD#
7 71.19 25/24 nitrogen ^7D, vvE♭ kEb, oD#, (kkD#)
8 81.36 21/20, 22/21 oxygen ^8D, vE♭ lEb, sD#
9 91.53 m2 19/18, 20/19, 256/243 limma, dayavati fluorine ^9D, E♭ Eb, kD#
10 101.69 17/16, 18/17 dodecaic semitone neon ^10D, v10E LEb, lD#
11 111.86 16/15, 2187/2048 apotome, ranjani sodium D♯, v9E KEb, D#
12 122.03 15/14 magnesium ^D♯, v8E SEb
13 132.20 27/25 aluminium ^^D♯, v7E OEb
14 142.37 88/81 silicon ^3D♯, v6E 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 ^6D♯, v3E sE
18 183.05 10/9 minor tone, ratika argon ^7D♯, vvE kE
19 193.22 28/25, 19/17 neutral tone, quasi-meantone potassium ^8D♯, vE lE
20 203.39 M2 9/8 major tone, raudri calcium E E
21 213.56 17/15 augmented tone scandium ^E, v8F LE
22 223.73 256/225 minor slendric second titanium ^^E, v7F KE
23 233.90 8/7 septimal second, slendric 2 vanadium ^3E, v6F 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 ^6E, v3F sF
27 274.58 75/64 major septimal third cobalt ^7E, vvF kF
28 284.75 33/28 nickel ^8E, vF lF
29 294.92 m3 32/27, 19/16 Pythagorean minor 3rd, krodha copper F F
30 305.08 25/21 zinc ^F, v8G♭ LF
31 315.25 6/5 Classical minor 3rd, vajrika gallium ^^F, v7G♭ KF
32 325.42 98/81 germanium ^3F, v6G♭ 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 ^6F, v3G♭ uF#
36 366.10 99/80, 21/17, 16/13 II** Golden ratio 3rd, major-tridecimal neutral third krypton ^7F, vvG♭ oF#
37 376.27 56/45 rubidium ^8F, vG♭ sF#
38 386.44 5/4 Classical major 3rd, prasarini strontium ^9F, G♭ kF#
39 396.61 63/50 yttrium ^10F, v10G lF#
40 406.78 M3 24/19, 19/15 Pythagorean major 3rd zirconium F♯, v9G F#
41 416.95 14/11 niobium ^F♯, v8G LF#
42 427.12 77/60 molybdenum ^^F♯, v7G KF#
43 437.29 9/7 technetium ^3F♯, v6G 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 ^6F♯, v3G sG
47 477.97 320/243 silver ^7F♯, vvG kG
48 488.14 160/121, 85/64 cadmium ^8F♯, vG lG
49 498.31 P4 4/3 perfect 4th indium G G
50 508.47 75/56, 51/38 tin ^G, v8A♭ LG
51 518.64 27/20 Kshiti antimony ^^G, v7A♭ KG
52 528.81 49/36, 19/14 tellurium ^3G, v6A♭ 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 ^6G, v3A♭ uG#, sAb
56 569.49 25/18 barium ^7G, vvA♭ oG#, (kkG#), kAb
57 579.66 7/5 lanthanum ^8G, vA♭ sG#, lAb
58 589.83 d5 45/32 Rakta cerium ^9G, A♭ kG#, Ab
59 600.00 99/70, 140/99, 17/12, 24/17 symmetric tritone praseodymium ^10G, v10A lG#, LAb
60 610.17 A4 64/45, 729/512 Literal tritone, sandipani neodymium G♯, v9A G#, KAb
61 620.34 10/7 promethium ^G♯, v8A LG#, SAb
62 630.51 36/25 samarium ^^G♯, v7A KG#, OAb, (KKAb)
63 640.68 81/56 europium ^3G♯, v6A 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 ^6G♯, v3A sA
67 681.36 40/27 wolf 5th holmium ^7G♯, vvA kA
68 691.53 112/75, 76/51 wolf cub 5th erbium ^8G♯, 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, v8B♭ lA
71 722.03 243/160 lamb 5th lutetium ^^A, v7B♭ KA
72 732.20 32/21 hafnium ^3A, v6B♭ 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 ^6A, v3B♭ sBb
76 772.88 120/77 osmium ^7A, vvB♭ kBb
77 783.05 11/7 iridium ^8A, vB♭ lBb
78 793.22 m6 19/12, 30/19 Pythagorean minor 6th platinum ^9A, B♭ Bb
79 803.39 100/63 gold ^10A, v10B LBb
80 813.56 8/5 Classical minor 6th mercury A♯, v9B KBb
81 823.73 45/28 thallium ^A♯, v8B SBb
82 833.90 160/99, 34/21, 13/8 I** Golden ratio sixth, minor-neutral tridecimal sixth lead ^^A♯, v7B OBb
83 844.07 44/27, 13/8 II** Major tridecimal neutral sixth, "minor-neutral" sixth bismuth ^3A♯, v6B 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 ^6A♯, v3B sB
87 884.75 5/3 Classical major 6th francium ^7A♯, vvB kB
88 894.92 42/25 radium ^8A♯, vB lB
89 905.08 M6 27/16, 32/19 Pythagorean major 6th actinium B B
90 915.25 56/33 thorium ^B, v8C LB
91 925.42 128/75 protactinium ^^B, v7C KB
92 935.59 12/7 Septimal supermajor 6th, slendric 5 uranium ^3B, v6C 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 ^6B, v3C sC
96 976.27 225/128 curium ^7B, vvC kC
97 986.44 30/17 berkelium ^8B, vC lC
98 996.61 m7 16/9 Pythagorean minor 7th californium C C
99 1006.78 25/14 einsteinium ^C, v8D♭ LC
100 1016.95 9/5 Tivra fermium ^^C, v7D♭ KC
101 1027.12 38/21 mendelevium ^3C, v6D♭ 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 ^6C, v3D♭ uC#, sDb
105 1067.80 50/27 dubnium ^7C, vvD♭ oC#, kDb
106 1077.97 28/15 seaborgium ^8C, vD♭ sC#, lDb
107 1088.14 15/8 bohrium ^9C, D♭ kC#, Db
108 1098.31 32/17, 17/9 hassium ^10C, v10D lC#, LDb
109 1108.47 M7 36/19, 19/10, 243/128 Pythagorean major 7th meitnerium C♯, v9D C#, KDb
110 1118.64 40/21, 21/11 darmstadtium ^C♯, v8D LC#, SDb
111 1128.81 48/25 roentgenium ^^C♯, v7D KC#, ODb, (kkDb)
112 1138.98 27/14 copernicium ^3C♯, v6D 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 ^6C♯, v3D sD
116 1179.66 160/81 Comma supermajor 7th livermorium ^7C♯, vvD kD
117 1189.83 125/63, 448/225, 240/121, 484/243 Semicomma supermajor 7th tenessine ^8C♯, 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

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* Cents* 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

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct

Music

Mercury Amalgam