118edo
| ← 117edo | 118edo | 119edo → |
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
| 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
| 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
| 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
- Pops (2022)