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 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. In addition, one step of 118edo is close to the 2097152/2083725 (the bronzisma), 169/168, and 170/169.

118edo is the 17th zeta peak edo.

Intervals

Table of intervals in 118edo

Step	Cents	Marks	Approximate Ratios *	Eliora's Naming System (+Shruti 22 correspondence)	Chemical Notation (see below, if base note = 0)	Ups and downs notation
0	0.00	P1	1/1	unison	oganesson / neutronium	D
1	10.17		126/125, 225/224, 121/120, 243/242	semicomma	hydrogen	^D, ^3E♭♭
2	20.34		81/80, 531441/524288	comma	helium	^^D, ^4E♭♭
3	30.51		64/63, 49/48	augmented comma	lithium	^3D, ^5E♭♭
4	40.68		50/49		beryllium	^4D, v5E♭
5	50.85		36/35		boron	^5D, v4E♭
6	61.02		28/27		carbon	v5D♯, v3E♭
7	71.19		25/24		nitrogen	v4D♯, vvE♭
8	81.36		21/20, 22/21		oxygen	v3D♯, vE♭
9	91.53	m2	19/18, 20/19, 256/243	limma, dayavati	fluorine	vvD♯, E♭
10	101.69	vD♯, ^E♭		17/16, 18/17	dodecaic semitone	neon	vD♯, ^E♭
11	111.86		16/15, 2187/2048	apotome, ranjani	sodium	D♯, ^^E♭
12	122.03		15/14		magnesium	^D♯, ^3E♭
13	132.20		27/25		aluminium	^^D♯, ^4E♭
14	142.37		88/81		silicon	^3D♯, ^5E♭
15	152.54		12/11		phosphorus	^4D♯, v5E
16	162.71		11/10		sulphur	^5D♯, v4E
17	172.88		21/19	diminished tone	chlorine	v5D𝄪, v3E
18	183.05		10/9	minor tone, ratika	argon	v4D𝄪, vvE
19	193.22		28/25, 19/17	neutral tone, quasi-meantone	potassium	v3D𝄪, vE
20	203.39	M2	9/8	major tone, raudri	calcium	E
21	213.56		17/15	augmented tone	scandium	^E, ^3F♭
22	223.73		256/225	minor slendric second	titanium	^^E, ^4F♭
23	233.90		8/7	septimal second, slendric 2	vanadium	^3E, ^5F♭
24	244.07		144/125, 121/105	major slendric second	chromium	^4E, v5F
25	254.24		125/108, 81/70, 22/19	minor septimal third	manganese	^5E, v4F
26	260.41		7/6	septimal third	iron	v5E♯, v3F
27	274.58		75/64	major septimal third	cobalt	v4E♯, vvF
28	284.75		33/28		nickel	v3E♯, vF
29	294.92	m3	32/27, 19/16	Pythagorean minor 3rd, krodha	copper	F
30	305.08		25/21		zinc	^F, ^3G♭♭
31	315.25		6/5	Classical minor 3rd, vajrika	gallium	^^F, ^4G♭♭
32	325.42		98/81		germanium	^3F, ^5G♭♭
33	335.59		40/33, 17/14	Lesser tridecimal third	arsenic	^4F, v5G♭
34	345.76		11/9	Minor-neutral third	selenium	^5F, v4G♭
35	355.93		27/22, 16/13 I**	Minor tridecimal neurtral third, "major-neutral" third	bromine	v5F♯, v3G♭
36	366.10		99/80, 21/17, 16/13 II**	Golden ratio 3rd, major-tridecimal neutral third	krypton	v4F♯, vvG♭
37	376.27		56/45		rubidium	v3F♯, vG♭
38	386.44		5/4	Classical major 3rd, prasarini	strontium	vvF♯, G♭
39	396.61		63/50		yttrium	vF♯, ^G♭
40	406.78	M3	24/19, 19/15	Pythagorean major 3rd	zirconium	F♯, ^^G♭
41	416.95		14/11		niobium	^F♯, ^3G♭
42	427.12		77/60		molybdenum	^^F♯, ^4G♭
43	437.29		9/7		technetium	^3F♯, ^5G♭
44	447.46		35/27, 22/17		ruthenium	^4F♯, v5G
45	457.63		98/75	Barbados 3rd	rhodium	^5F♯, v4G
46	467.80		21/16	Slendric 3	palladium	v5F𝄪, v3G
47	477.97		320/243		silver	v4F𝄪, vvG
48	488.14		160/121, 85/64		cadmium	v3F𝄪, vG
49	498.31	P4	4/3	perfect 4th	indium	G
50	508.47		75/56, 51/38		tin	^G, ^3A♭♭
51	518.64		27/20	Kshiti	antimony	^^G, ^4A♭♭
52	528.81		49/36, 19/14		tellurium	^3G, ^5A♭♭
53	538.98		15/11	^4G, v5A♭		iodine
54	549.15		48/35, 11/8	^5G, v4A♭		xenon
55	559.32		112/81		caesium	v5G♯, v3A♭
56	569.49		25/18		barium	v4G♯, vvA♭
57	579.66		7/5		lanthanum	v3G♯, vA♭
58	589.83	d5	45/32	Rakta	cerium	vvG♯, A♭
59	600.00		99/70, 140/99, 17/12, 24/17	symmetric tritone	praseodymium	vG♯, ^A♭
60	610.17	A4	64/45, 729/512	Literal tritone, sandipani	neodymium	G♯, ^^A♭
61	620.34		10/7		promethium	^G♯, ^3A♭
62	630.51		36/25		samarium	^^G♯, ^4A♭
63	640.68		81/56		europium	^3G♯, ^5A♭
64	650.85		35/24, 16/11		gadolinium	^4G♯, v5A
65	661.02		22/15		terbium	^5G♯, v4A
66	671.19		72/49, 28/19		dysprosium	v5G𝄪, v3A
67	681.36		40/27	wolf 5th	holmium	v4G𝄪, vvA
68	691.53		112/75, 76/51	wolf cub 5th	erbium	v3G𝄪, vA
69	701.69	P5	3/2	perfect 5th, slendric 4	thulium	A
70	711.86		121/80, 128/85	sheep 5th	ytterbium	^A, ^3B♭♭
71	722.03		243/160	lamb 5th	lutetium	^^A, ^4B♭♭
72	732.20		32/21		hafnium	^3A, ^5B♭♭
73	742.37		75/49		tantalum	^4A, v5B♭
74	752.54		54/35, 17/11		tungsten	^5A, v4B♭
75	762.71		14/9		rhenium	v5A♯, v3B♭
76	772.88		120/77		osmium	v4A♯, vvB♭
77	783.05		11/7		iridium	v3A♯, vB♭
78	793.22	m6	19/12, 30/19	Pythagorean minor 6th	platinum	vvA♯, B♭
79	803.39		100/63		gold	vA♯, ^B♭
80	813.56		8/5	Classical minor 6th	mercury	A♯, ^^B♭
81	823.73		45/28		thallium	^A♯, ^3B♭
82	833.90		160/99, 34/21, 13/8 I**	Golden ratio sixth, minor-neutral tridecimal sixth	lead	^^A♯, ^4B♭
83	844.07		44/27, 13/8 II**	Major tridecimal neutral sixth, "minor-neutral" sixth	bismuth	^3A♯, ^5B♭
84	854.24		18/11	Major-neutral sixth	polonium	^4A♯, v5B
85	864.41		28/17		astatine	^5A♯, v4B
86	874.58		81/49		radon	v5A𝄪, v3B
87	884.75		5/3	Classical major 6th	francium	v4A𝄪, vvB
88	894.92		42/25		radium	v3A𝄪, vB
89	905.08	M6	27/16, 32/19	Pythagorean major 6th	actinium	B
90	915.25		56/33		thorium	^B, ^3C♭
91	925.42		128/75		protactinium	^^B, ^4C♭
92	935.59		12/7	Septimal supermajor 6th, slendric 5	uranium	^3B, ^5C♭
93	945.76		216/125, 140/81, 121/70, 19/11		neptunium	^4B, v5C
94	955.93		125/72		plutonium	^5B, v4C
95	966.10		7/4	Harmonic 7th	americium	v5B♯, v3C
96	976.27		225/128		curium	v4B♯, vvC
97	986.44		30/17		berkelium	v3B♯, vC
98	996.61	m7	16/9	Pythagorean minor 7th	californium	C
99	1006.78		25/14		einsteinium	^C, ^3D♭♭
100	1016.95		9/5	Tivra	fermium	^^C, ^4D♭♭
101	1027.12		38/21		mendelevium	^3C, ^5D♭♭
102	1037.29		20/11		nobelium	^4C, v5D♭
103	1047.46		11/6		lawrencium	^5C, v4D♭
104	1057.63		81/44		rutherfordium	v5C♯, v3D♭
105	1067.80		50/27		dubnium	v4C♯, vvD♭
106	1077.97		28/15		seaborgium	v3C♯, vD♭
107	1088.14		15/8		bohrium	vvC♯, D♭
108	1098.31		32/17, 17/9		hassium	vC♯, ^D♭
109	1108.47	M7	36/19, 19/10, 243/128	Pythagorean major 7th	meitnerium	C♯, ^^D♭
110	1118.64		40/21, 21/11		darmstadtium	^C♯, ^3D♭
111	1128.81		48/25		roentgenium	^^C♯, ^4D♭
112	1138.98		27/14		copernicium	^3C♯, ^5D♭
113	1149.15		35/18, 64/33		nihonium	^4C♯, v5D
114	1159.32		49/25		flerovium	^5C♯, v4D
115	1169.49		63/32, 96/49		moscovium	v5C𝄪, v3D
116	1179.66		160/81	Comma supermajor 7th	livermorium	v4C𝄪, vvD
117	1189.83		125/63, 448/225, 240/121, 484/243	Semicomma supermajor 7th	tenessine	v3C𝄪, vD
118	1200.00	P8	2/1	perfect 8ve	oganesson / neutronium	D

* 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. 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 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 8ve	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

Music

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