118edo

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

Special properties

zeta peak
zeta peak integer

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 2^1/118, 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

Approximation of prime harmonics in 118edo
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
Error	relative (%)	+0	-3	+1	-27	-21	+35	-32	-26	+22	-24	+40
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

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	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, ↓⁸E♭	LD
2	20.34		81/80, 531441/524288	comma	helium	↑↑D, ↓⁷E♭	KD
3	30.51		64/63, 49/48	augmented comma	lithium	↑³D, ↓⁶E♭	SD
4	40.68		50/49		beryllium	↑⁴D, ↓⁵E♭	OD, uEb
5	50.85		36/35		boron	↑⁵D, ↓⁴E♭	UD, oEb
6	61.02		28/27		carbon	↑⁶D, ↓³E♭	sEb, uD#
7	71.19		25/24		nitrogen	↑⁷D, ↓↓E♭	kEb, oD#, (kkD#)
8	81.36		21/20, 22/21		oxygen	↑⁸D, ↓E♭	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, ↓¹⁰E	LEb, lD#
11	111.86		16/15, 2187/2048	apotome, ranjani	sodium	D♯, ↓⁹E	KEb, D#
12	122.03		15/14		magnesium	↑D♯, ↓⁸E	SEb
13	132.20		27/25		aluminium	↑↑D♯, ↓⁷E	OEb
14	142.37		88/81		silicon	↑³D♯, ↓⁶E	UEb
15	152.54		12/11		phosphorus	↑⁴D♯, ↓⁵E	uE
16	162.71		11/10		sulphur	↑⁵D♯, ↓⁴E	oE
17	172.88		21/19	diminished tone	chlorine	↑⁶D♯, ↓³E	sE
18	183.05		10/9	minor tone, ratika	argon	↑⁷D♯, ↓↓E	kE
19	193.22		28/25, 19/17	neutral tone, quasi-meantone	potassium	↑⁸D♯, ↓E	lE
20	203.39	M2	9/8	major tone, raudri	calcium	E	E
21	213.56		17/15	augmented tone	scandium	↑E, ↓⁸F	LE
22	223.73		256/225	minor slendric second	titanium	↑↑E, ↓⁷F	KE
23	233.90		8/7	septimal second, slendric 2	vanadium	↑³E, ↓⁶F	SE
24	244.07		144/125, 121/105	major slendric second	chromium	↑⁴E, ↓⁵F	OE, uF
25	254.24		125/108, 81/70, 22/19	minor septimal third	manganese	↑⁵E, ↓⁴F	UE, oF
26	260.41		7/6	septimal third	iron	↑⁶E, ↓³F	sF
27	274.58		75/64	major septimal third	cobalt	↑⁷E, ↓↓F	kF
28	284.75		33/28		nickel	↑⁸E, ↓F	lF
29	294.92	m3	32/27, 19/16	Pythagorean minor 3rd, krodha	copper	F	F
30	305.08		25/21		zinc	↑F, ↓⁸G♭	LF
31	315.25		6/5	Classical minor 3rd, vajrika	gallium	↑↑F, ↓⁷G♭	KF
32	325.42		98/81		germanium	↑³F, ↓⁶G♭	SF
33	335.59		40/33, 17/14	Lesser tridecimal third	arsenic	↑⁴F, ↓⁵G♭	OF
34	345.76		11/9	Minor-neutral third	selenium	↑⁵F, ↓⁴G♭	UF
35	355.93		27/22, 16/13 I**	Minor tridecimal neurtral third, "major-neutral" third	bromine	↑⁶F, ↓³G♭	uF#
36	366.10		99/80, 21/17, 16/13 II**	Golden ratio 3rd, major-tridecimal neutral third	krypton	↑⁷F, ↓↓G♭	oF#
37	376.27		56/45		rubidium	↑⁸F, ↓G♭	sF#
38	386.44		5/4	Classical major 3rd, prasarini	strontium	↑⁹F, G♭	kF#
39	396.61		63/50		yttrium	↑¹⁰F, ↓¹⁰G	lF#
40	406.78	M3	24/19, 19/15	Pythagorean major 3rd	zirconium	F♯, ↓⁹G	F#
41	416.95		14/11		niobium	↑F♯, ↓⁸G	LF#
42	427.12		77/60		molybdenum	↑↑F♯, ↓⁷G	KF#
43	437.29		9/7		technetium	↑³F♯, ↓⁶G	SF#
44	447.46		35/27, 22/17		ruthenium	↑⁴F♯, ↓⁵G	OF#, uG
45	457.63		98/75	Barbados 3rd	rhodium	↑⁵F♯, ↓⁴G	UF#, oG
46	467.80		21/16	Slendric 3	palladium	↑⁶F♯, ↓³G	sG
47	477.97		320/243		silver	↑⁷F♯, ↓↓G	kG
48	488.14		160/121, 85/64		cadmium	↑⁸F♯, ↓G	lG
49	498.31	P4	4/3	perfect 4th	indium	G	G
50	508.47		75/56, 51/38		tin	↑G, ↓⁸A♭	LG
51	518.64		27/20	Kshiti	antimony	↑↑G, ↓⁷A♭	KG
52	528.81		49/36, 19/14		tellurium	↑³G, ↓⁶A♭	SG
53	538.98		15/11	↑⁴G, ↓⁵A♭		iodine	OG, uGb
54	549.15		48/35, 11/8	↑⁵G, ↓⁴A♭		xenon	UG, oAb
55	559.32		112/81		caesium	↑⁶G, ↓³A♭	uG#, sAb
56	569.49		25/18		barium	↑⁷G, ↓↓A♭	oG#, (kkG#), kAb
57	579.66		7/5		lanthanum	↑⁸G, ↓A♭	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, ↓¹⁰A	lG#, LAb
60	610.17	A4	64/45, 729/512	Literal tritone, sandipani	neodymium	G♯, ↓⁹A	G#, KAb
61	620.34		10/7		promethium	↑G♯, ↓⁸A	LG#, SAb
62	630.51		36/25		samarium	↑↑G♯, ↓⁷A	KG#, OAb, (KKAb)
63	640.68		81/56		europium	↑³G♯, ↓⁶A	SG#, UAb
64	650.85		35/24, 16/11		gadolinium	↑⁴G♯, ↓⁵A	OG#, uA
65	661.02		22/15		terbium	↑⁵G♯, ↓⁴A	oA
66	671.19		72/49, 28/19		dysprosium	↑⁶G♯, ↓³A	sA
67	681.36		40/27	wolf 5th	holmium	↑⁷G♯, ↓↓A	kA
68	691.53		112/75, 76/51	wolf cub 5th	erbium	↑⁸G♯, ↓A	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, ↓⁸B♭	lA
71	722.03		243/160	lamb 5th	lutetium	↑↑A, ↓⁷B♭	KA
72	732.20		32/21		hafnium	↑³A, ↓⁶B♭	SA
73	742.37		75/49		tantalum	↑⁴A, ↓⁵B♭	OA, uBb
74	752.54		54/35, 17/11		tungsten	↑⁵A, ↓⁴B♭	UA, oBb
75	762.71		14/9		rhenium	↑⁶A, ↓³B♭	sBb
76	772.88		120/77		osmium	↑⁷A, ↓↓B♭	kBb
77	783.05		11/7		iridium	↑⁸A, ↓B♭	lBb
78	793.22	m6	19/12, 30/19	Pythagorean minor 6th	platinum	↑⁹A, B♭	Bb
79	803.39		100/63		gold	↑¹⁰A, ↓¹⁰B	LBb
80	813.56		8/5	Classical minor 6th	mercury	A♯, ↓⁹B	KBb
81	823.73		45/28		thallium	↑A♯, ↓⁸B	SBb
82	833.90		160/99, 34/21, 13/8 I**	Golden ratio sixth, minor-neutral tridecimal sixth	lead	↑↑A♯, ↓⁷B	OBb
83	844.07		44/27, 13/8 II**	Major tridecimal neutral sixth, "minor-neutral" sixth	bismuth	↑³A♯, ↓⁶B	UBb
84	854.24		18/11	Major-neutral sixth	polonium	↑⁴A♯, ↓⁵B	uB
85	864.41		28/17		astatine	↑⁵A♯, ↓⁴B	oB
86	874.58		81/49		radon	↑⁶A♯, ↓³B	sB
87	884.75		5/3	Classical major 6th	francium	↑⁷A♯, ↓↓B	kB
88	894.92		42/25		radium	↑⁸A♯, ↓B	lB
89	905.08	M6	27/16, 32/19	Pythagorean major 6th	actinium	B	B
90	915.25		56/33		thorium	↑B, ↓⁸C	LB
91	925.42		128/75		protactinium	↑↑B, ↓⁷C	KB
92	935.59		12/7	Septimal supermajor 6th, slendric 5	uranium	↑³B, ↓⁶C	SB
93	945.76		216/125, 140/81, 121/70, 19/11		neptunium	↑⁴B, ↓⁵C	OB, uC
94	955.93		125/72		plutonium	↑⁵B, ↓⁴C	UB, oC
95	966.10		7/4	Harmonic 7th	americium	↑⁶B, ↓³C	sC
96	976.27		225/128		curium	↑⁷B, ↓↓C	kC
97	986.44		30/17		berkelium	↑⁸B, ↓C	lC
98	996.61	m7	16/9	Pythagorean minor 7th	californium	C	C
99	1006.78		25/14		einsteinium	↑C, ↓⁸D♭	LC
100	1016.95		9/5	Tivra	fermium	↑↑C, ↓⁷D♭	KC
101	1027.12		38/21		mendelevium	↑³C, ↓⁶D♭	SC
102	1037.29		20/11		nobelium	↑⁴C, ↓⁵D♭	OC, uDb
103	1047.46		11/6		lawrencium	↑⁵C, ↓⁴D♭	UC, oDb
104	1057.63		81/44		rutherfordium	↑⁶C, ↓³D♭	uC#, sDb
105	1067.80		50/27		dubnium	↑⁷C, ↓↓D♭	oC#, kDb
106	1077.97		28/15		seaborgium	↑⁸C, ↓D♭	sC#, lDb
107	1088.14		15/8		bohrium	↑⁹C, D♭	kC#, Db
108	1098.31		32/17, 17/9		hassium	↑¹⁰C, ↓¹⁰D	lC#, LDb
109	1108.47	M7	36/19, 19/10, 243/128	Pythagorean major 7th	meitnerium	C♯, ↓⁹D	C#, KDb
110	1118.64		40/21, 21/11		darmstadtium	↑C♯, ↓⁸D	LC#, SDb
111	1128.81		48/25		roentgenium	↑↑C♯, ↓⁷D	KC#, ODb, (kkDb)
112	1138.98		27/14		copernicium	↑³C♯, ↓⁶D	SC#, UDb
113	1149.15		35/18, 64/33		nihonium	↑⁴C♯, ↓⁵D	OC#, uD
114	1159.32		49/25		flerovium	↑⁵C♯, ↓⁴D	UC#, oD
115	1169.49		63/32, 96/49		moscovium	↑⁶C♯, ↓³D	sD
116	1179.66		160/81	Comma supermajor 7th	livermorium	↑⁷C♯, ↓↓D	kD
117	1189.83		125/63, 448/225, 240/121, 484/243	Semicomma supermajor 7th	tenessine	↑⁸C♯, ↓D	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

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
Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	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

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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