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.

118edo is the 17th zeta peak edo.

Intervals

Table of intervals in 118edo

Step	Cents	FJS Name	Eliora's Naming System (+Shruti correspondence)	Eliora's Chemical Notation (if base note = 0)	Approximate Ratios *
0	0.00	P1	unison	oganesson / neutronium	1/1
1	10.17		semicomma	hydrogen	126/125, 225/224, 121/120, 243/242
2	20.34		comma	helium	81/80, 531441/524288
3	30.51				64/63, 49/48
4	40.68				50/49
5	50.85				36/35
6	61.02				28/27
7	71.19				25/24
8	81.36				21/20, 22/21
9	91.53	m2	limma, dayavati	fluorine	19/18, 20/19
10	101.69		dodecaic semitone	neon	17/16, 18/17
11	111.86		apotome, ranjani	sodium	16/15
12	122.03				15/14
13	132.20				27/25
14	142.37				88/81
15	152.54				12/11
16	162.71				11/10
17	172.88				21/19
18	183.05		diminished tone, ratika	argon	10/9
19	193.22		minor tone	potassium	28/25, 19/17
20	203.39	M2	major tone, raudri	calcium	9/8
21	213.56				17/15
22	223.73				256/225
23	233.90		septimal second, slendric 2	vanadium	8/7
24	244.07				144/125, 121/105
25	254.24				125/108, 81/70, 22/19
26	260.41		septimal third	iron	7/6
27	274.58				75/64
28	284.75				33/28
29	294.92	m3	Pythagorean minor 3rd, krodha	copper	32/27, 19/16
30	305.08				25/21
31	315.25		Classical minor 3rd, vajrika	gallium	6/5
32	325.42				98/81
33	335.59		Lesser tridecimal third	germanium	40/33, 17/14
34	345.76		Minor-neutral third	selenium	11/9
35	355.93		Minor tridecimal neurtral third, "major-neutral" third	bromine	27/22, 16/13 I**
36	366.10		Golden ratio 3rd, major-tridecimal neutral third	krypton	99/80, 21/17, 16/13 II**
37	376.27				56/45
38	386.44		Classical major 3rd, prasarini	strontium	5/4
39	396.61				63/50
40	406.78	M3	Pythagorean major 3rd	zirconium	24/19, 19/15
41	416.95				14/11
42	427.12				77/60
43	437.29				9/7
44	447.46				35/27, 22/17
45	457.63		Barbados 3rd	rhodium	98/75
46	467.80		Slendric 3	palladium	21/16
47	477.97				320/243
48	488.14				160/121, 85/64
49	498.31	P4	perfect 4th	indium	4/3
50	508.47				75/56, 51/38
51	518.64		Kshiti	antimony	27/20
52	528.81				49/36, 19/14
53	538.98				15/11
54	549.15				48/35, 11/8
55	559.32				112/81
56	569.49				25/18
57	579.66				7/5
58	589.83	d5	Rakta	cerium	45/32
59	600.00		symmetric tritone	praseodymium	99/70, 140/99, 17/12, 24/17
60	610.17	A4	Literal tritone, sandipani	neodymium	64/45
61	620.34				10/7
62	630.51				36/25
63	640.68				81/56
64	650.85				35/24, 16/11
65	661.02				22/15
66	671.19				72/49, 28/19
67	681.36				40/27
68	691.53				112/75, 76/51
69	701.69	P5	perfect 5th	thulium	3/2
70	711.86				121/80, 128/85
71	722.03				243/160
72	732.20				32/21
73	742.37				75/49
74	752.54				54/35, 17/11
75	762.71				14/9
76	772.88				120/77
77	783.05				11/7
78	793.22	m6	Pythagorean minor 6th	platinum	19/12, 30/19
79	803.39				100/63
80	813.56		Classical minor 6th	mercury	8/5
81	823.73				45/28
82	833.90		Golden ratio sixth, minor-neutral tridecimal sixth	lead	160/99, 34/21, 13/8 I**
83	844.07		Major tridecimal neutral sixth, "minor-neutral" sixth	bismuth	44/27, 13/8 II**
84	854.24		Major-neutral sixth	polonium	18/11
85	864.41				28/17
86	874.58				81/49
87	884.75		Classical major 6th	francium	5/3
88	894.92				42/25
89	905.08	M6	Pythagorean major 6th	actinium	27/16, 32/19
90	915.25				56/33
91	925.42				128/75
92	935.59		Septimal supermajor 6th, slendro 5	uranium	12/7
93	945.76				216/125, 140/81, 121/70, 19/11
94	955.93				125/72
95	966.10		Harmonic 7th	americium	7/4
96	976.27				225/128
97	986.44				30/17
98	996.61	m7			16/9
99	1006.78				25/14
100	1016.95		Tivra	fermium	9/5
101	1027.12				38/21
102	1037.29				20/11
103	1047.46				11/6
104	1057.63				81/44
105	1067.80				50/27
106	1077.97				28/15
107	1088.14				15/8
108	1098.31				32/17, 17/9
109	1108.47	M7	Pythagorean major 7th	meitnerium	36/19, 19/10
110	1118.64				40/21, 21/11
111	1128.81				48/25
112	1138.98				27/14
113	1149.15				35/18, 64/33
114	1159.32				49/25
115	1169.49				63/32, 96/49
116	1179.66		Comma 7th	livermorium	160/81
117	1189.83		Semicomma supermajor 7th	tenessine	125/63, 448/225, 240/121, 484/243
118	1200.00	P8	perfect 8ve	oganesson / neutronium	2/1

* 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. In addition, uniqueness of pitch class is a disadvantage as well - since all the notes are separately named, it does not reflect the harmonic structure of 118edo.

This being said, there are a few correspondences - 2\118 is the width of the s-block, and is also the size of the Pythagorean and syntonic commas in 118edo. In addition, 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 (magnesium), the 2s metal.

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 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