118edo: Difference between revisions

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Line 61: Line 61:
|  
|  
|[[64/63]], [[49/48]]
|[[64/63]], [[49/48]]
|  
| augmented comma
| lithium
| lithium
|-
|-
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| 91.53
| 91.53
| m2
| m2
|[[19/18]], [[20/19]]
|[[19/18]], [[20/19]], [[256/243]]
| limma, dayavati
| limma, dayavati
| fluorine
| fluorine
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| 111.86
| 111.86
|  
|  
|[[16/15]]
|[[16/15]], [[2187/2048]]
| apotome, ranjani
| apotome, ranjani
| sodium
| sodium
Line 159: Line 159:
|  
|  
|[[21/19]]
|[[21/19]]
|  
| diminished tone
| chlorine
| chlorine
|-
|-
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|  
|  
|[[10/9]]
|[[10/9]]
| diminished tone, ratika
| minor tone, ratika
| argon
| argon
|-
|-
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|  
|  
|[[28/25]], [[19/17]]
|[[28/25]], [[19/17]]
| minor tone
| neutral tone, quasi-meantone
| potassium
| potassium
|-
|-
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|  
|  
|[[17/15]]
|[[17/15]]
|  
| augmented tone
| scandium
| scandium
|-
|-
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|  
|  
|[[256/225]]
|[[256/225]]
|  
| minor slendric second
| titanium
| titanium
|-
|-
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|  
|  
|[[144/125]], [[121/105]]
|[[144/125]], [[121/105]]
|  
| major slendric second
| chromium
| chromium
|-
|-
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|  
|  
|[[125/108]], [[81/70]], [[22/19]]
|[[125/108]], [[81/70]], [[22/19]]
|  
| minor septimal third
| manganese
| manganese
|-
|-
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|  
|  
|[[75/64]]
|[[75/64]]
|  
| major septimal third
| cobalt
| cobalt
|-
|-
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| 610.17
| 610.17
| A4
| A4
|[[64/45]]
|[[64/45]], [[729/512]]
| Literal tritone, sandipani
| Literal tritone, sandipani
| neodymium
| neodymium
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|[[10/7]]
|[[10/7]]
|  
|  
|  
| promethium
|-
|-
| 62
| 62
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|[[36/25]]
|[[36/25]]
|  
|  
|  
| samarium
|-
|-
| 63
| 63
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|[[81/56]]
|[[81/56]]
|  
|  
|  
| europium
|-
|-
| 64
| 64
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|[[35/24]], [[16/11]]
|[[35/24]], [[16/11]]
|  
|  
|  
| gadolinium
|-
|-
| 65
| 65
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|[[22/15]]
|[[22/15]]
|  
|  
|  
| terbium
|-
|-
| 66
| 66
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|[[72/49]], [[28/19]]
|[[72/49]], [[28/19]]
|  
|  
|  
| dysprosiu
|-
|-
| 67
| 67
Line 523: Line 523:
| P5
| P5
|[[3/2]]
|[[3/2]]
| perfect 5th
| perfect 5th, slendric 4
| thulium
| thulium
|-
|-
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|  
|  
|[[121/80]], [[128/85]]
|[[121/80]], [[128/85]]
|  
| sheep 5th
| ytterbium
| ytterbium
|-
|-
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|  
|  
|[[243/160]]
|[[243/160]]
|  
| lamb 5th
| lutetium
| lutetium
|-
|-
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|[[28/17]]
|[[28/17]]
|  
|  
|  
| astatine
|-
|-
| 86
| 86
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|[[81/49]]
|[[81/49]]
|  
|  
|  
| radon
|-
|-
| 87
| 87
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|[[42/25]]
|[[42/25]]
|  
|  
|  
| radium
|-
|-
| 89
| 89
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|[[56/33]]
|[[56/33]]
|  
|  
|  
| thorium
|-
|-
| 91
| 91
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|[[128/75]]
|[[128/75]]
|  
|  
|  
| protactinium
|-
|-
| 92
| 92
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|  
|  
|[[12/7]]
|[[12/7]]
| Septimal supermajor 6th, slendro 5
| Septimal supermajor 6th, slendric 5
| uranium
| uranium
|-
|-
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| m7
| m7
|[[16/9]]
|[[16/9]]
|  
| Pythagorean minor 7th
|  
| californium
|-
|-
| 99
| 99
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|[[25/14]]
|[[25/14]]
|  
|  
|  
| einsteinium
|-
|-
| 100
| 100
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|[[38/21]]
|[[38/21]]
|  
|  
|  
| mendelevium
|-
|-
| 102
| 102
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|[[20/11]]
|[[20/11]]
|  
|  
|  
| nobelium
|-
|-
| 103
| 103
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|[[11/6]]
|[[11/6]]
|  
|  
|  
| lawrencium
|-
|-
| 104
| 104
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|[[81/44]]
|[[81/44]]
|  
|  
|  
| rutherfordium
|-
|-
| 105
| 105
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|[[50/27]]
|[[50/27]]
|  
|  
|  
| dubnium
|-
|-
| 106
| 106
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|[[28/15]]
|[[28/15]]
|  
|  
|  
| seaborgium
|-
|-
| 107
| 107
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|[[15/8]]
|[[15/8]]
|  
|  
|  
| bohrium
|-
|-
| 108
| 108
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|[[32/17]], [[17/9]]
|[[32/17]], [[17/9]]
|  
|  
|  
| hassium
|-
|-
| 109
| 109
| 1108.47
| 1108.47
| M7
| M7
|[[36/19]], [[19/10]]
|[[36/19]], [[19/10]], [[243/128]]
| Pythagorean major 7th
| Pythagorean major 7th
| meitnerium
| meitnerium
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|  
|  
|[[160/81]]
|[[160/81]]
| Comma 7th
| Comma supermajor 7th
| livermorium
| livermorium
|-
|-

Revision as of 23:17, 4 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 augmented comma 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, 256/243 limma, dayavati fluorine
10 101.69 17/16, 18/17 dodecaic semitone neon
11 111.86 16/15, 2187/2048 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 diminished tone chlorine
18 183.05 10/9 minor tone, ratika argon
19 193.22 28/25, 19/17 neutral tone, quasi-meantone potassium
20 203.39 M2 9/8 major tone, raudri calcium
21 213.56 17/15 augmented tone scandium
22 223.73 256/225 minor slendric second titanium
23 233.90 8/7 septimal second, slendric 2 vanadium
24 244.07 144/125, 121/105 major slendric second chromium
25 254.24 125/108, 81/70, 22/19 minor septimal third manganese
26 260.41 7/6 septimal third iron
27 274.58 75/64 major septimal third 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, 729/512 Literal tritone, sandipani neodymium
61 620.34 10/7 promethium
62 630.51 36/25 samarium
63 640.68 81/56 europium
64 650.85 35/24, 16/11 gadolinium
65 661.02 22/15 terbium
66 671.19 72/49, 28/19 dysprosiu
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, slendric 4 thulium
70 711.86 121/80, 128/85 sheep 5th ytterbium
71 722.03 243/160 lamb 5th 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 astatine
86 874.58 81/49 radon
87 884.75 5/3 Classical major 6th francium
88 894.92 42/25 radium
89 905.08 M6 27/16, 32/19 Pythagorean major 6th actinium
90 915.25 56/33 thorium
91 925.42 128/75 protactinium
92 935.59 12/7 Septimal supermajor 6th, slendric 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 Pythagorean minor 7th californium
99 1006.78 25/14 einsteinium
100 1016.95 9/5 Tivra fermium
101 1027.12 38/21 mendelevium
102 1037.29 20/11 nobelium
103 1047.46 11/6 lawrencium
104 1057.63 81/44 rutherfordium
105 1067.80 50/27 dubnium
106 1077.97 28/15 seaborgium
107 1088.14 15/8 bohrium
108 1098.31 32/17, 17/9 hassium
109 1108.47 M7 36/19, 19/10, 243/128 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 supermajor 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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