118edo

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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 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 stack of 10 12edos minus the said comma.

118edo is the 17th zeta peak edo.

Proposed chemical notation

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.

Interval names can be made using the suffix "-ide", as to how a 2-atom molecule would be named in chemistry. For example, fluoride for limma, chloride for 17 steps, strontide for classical major third. These are appended to the initial note. For example, an interval of 17 steps on the 1st note would be called hydrogen chloride, a limma on the 11th note would be called sodium fluoride. Classical minor 6th on 29th note would be called copper mercuride, etc.

It should be noted that vast majority of these intervals do not make chemical sense, such as apotome on the 9th note, which would be chlorine sodiide. In real life, this is impossible, as flow of electrons is always to chlorine, not from it.

Table of intervals

Selected 118 EDO intervals

Step	Name	Chemical notation if base note = 0	Associated ratio
0	unison	oganesson / neutronium	1/1 exact
2	comma	helium	531441/524288, 81/80
9	limma	fluorine	256/243
11	apotome	sodium	2187/2048
20	whole tone	calcium	9/8
23	septimal second	vanadium	8/7
26	septimal third	iron	7/6
29	Pythagorean minor 3rd	copper	32/27
31	Classical minor 3rd	gallium	6/5
38	Classical major 3rd	strontium	5/4
40	Pythagorean major 3rd	zirconium	81/64
49	perfect 4th	indium	4/3
59	symmetric tritone	praseodymium
69	perfect 5th	thulium	3/2
78	Pythagorean minor 6th	platinum	128/81
80	Classical minor 6th	mercury	8/5
87	Classical major 6th	francium	5/3
89	Pythagorean major 6th	actinium	27/16
118	perfect 8ve	oganesson / neutronium	2/1 exact

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

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