118edo
| ← 117edo | 118edo | 119edo → |
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, a size close to 169/168.
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
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Intervals
| 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.
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
| 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 |