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.

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

Trivia

118 is the number of chemical elements in the first 7 periods of the periodic table, meaning that the element names can be used as note names.