140edo

← 139edo

140edo

141edo →

Prime factorization	22 × 5 × 7
Step size	8.57143 ¢
Fifth	82\140 (702.857 ¢) (→ 41\70)
Semitones (A1:m2)	14:10 (120 ¢ : 85.71 ¢)
Consistency limit	9
Distinct consistency limit	9

Steps	Cents	Approximate ratios	Ups and downs notation
0	0	1/1	D
1	8.6		^D, ^5E♭♭
2	17.1		^^D, ^6E♭♭
3	25.7		^3D, v7E♭
4	34.3	49/48, 50/49, 51/50, 52/51	^4D, v6E♭
5	42.9	40/39, 41/40, 42/41	^5D, v5E♭
6	51.4	34/33, 35/34	^6D, v4E♭
7	60	29/28, 30/29	^7D, v3E♭
8	68.6	26/25, 51/49	v6D♯, vvE♭
9	77.1	23/22, 45/43	v5D♯, vE♭
10	85.7	21/20, 41/39	v4D♯, E♭
11	94.3	19/18	v3D♯, ^E♭
12	102.9	35/33, 52/49	vvD♯, ^^E♭
13	111.4	16/15	vD♯, ^3E♭
14	120	15/14	D♯, ^4E♭
15	128.6	14/13	^D♯, ^5E♭
16	137.1	13/12	^^D♯, ^6E♭
17	145.7	25/23, 37/34	^3D♯, v7E
18	154.3	35/32, 47/43	^4D♯, v6E
19	162.9		^5D♯, v5E
20	171.4	32/29	^6D♯, v4E
21	180	51/46	^7D♯, v3E
22	188.6	29/26, 39/35	v6D𝄪, vvE
23	197.1	28/25, 37/33	v5D𝄪, vE
24	205.7		E
25	214.3	43/38	^E, ^5F♭
26	222.9	25/22, 33/29	^^E, ^6F♭
27	231.4	8/7	^3E, v7F
28	240	31/27, 54/47	^4E, v6F
29	248.6	15/13	^5E, v5F
30	257.1	29/25, 51/44	^6E, v4F
31	265.7	7/6	^7E, v3F
32	274.3	34/29, 41/35, 48/41	v6E♯, vvF
33	282.9	20/17, 33/28	v5E♯, vF
34	291.4	45/38	F
35	300	44/37	^F, ^5G♭♭
36	308.6	43/36, 49/41, 55/46	^^F, ^6G♭♭
37	317.1	6/5	^3F, v7G♭
38	325.7	35/29, 41/34	^4F, v6G♭
39	334.3	40/33	^5F, v5G♭
40	342.9	39/32, 50/41	^6F, v4G♭
41	351.4	38/31, 49/40	^7F, v3G♭
42	360	16/13	v6F♯, vvG♭
43	368.6	26/21, 47/38	v5F♯, vG♭
44	377.1	41/33, 46/37, 51/41	v4F♯, G♭
45	385.7	5/4	v3F♯, ^G♭
46	394.3	49/39, 54/43	vvF♯, ^^G♭
47	402.9	24/19, 29/23	vF♯, ^3G♭
48	411.4	33/26, 52/41	F♯, ^4G♭
49	420	51/40	^F♯, ^5G♭
50	428.6	32/25, 41/32, 50/39	^^F♯, ^6G♭
51	437.1		^3F♯, v7G
52	445.7	22/17	^4F♯, v6G
53	454.3	13/10	^5F♯, v5G
54	462.9	17/13, 47/36	^6F♯, v4G
55	471.4	21/16	^7F♯, v3G
56	480	33/25	v6F𝄪, vvG
57	488.6		v5F𝄪, vG
58	497.1	4/3	G
59	505.7		^G, ^5A♭♭
60	514.3	35/26, 39/29	^^G, ^6A♭♭
61	522.9	23/17, 50/37	^3G, v7A♭
62	531.4	34/25	^4G, v6A♭
63	540	41/30	^5G, v5A♭
64	548.6		^6G, v4A♭
65	557.1	29/21, 40/29, 51/37	^7G, v3A♭
66	565.7	43/31	v6G♯, vvA♭
67	574.3	39/28, 46/33	v5G♯, vA♭
68	582.9	7/5	v4G♯, A♭
69	591.4	38/27, 45/32	v3G♯, ^A♭
70	600	41/29	vvG♯, ^^A♭
71	608.6	27/19	vG♯, ^3A♭
72	617.1	10/7	G♯, ^4A♭
73	625.7	33/23	^G♯, ^5A♭
74	634.3	49/34	^^G♯, ^6A♭
75	642.9	29/20, 42/29	^3G♯, v7A
76	651.4	51/35	^4G♯, v6A
77	660	41/28	^5G♯, v5A
78	668.6	25/17	^6G♯, v4A
79	677.1	34/23, 37/25	^7G♯, v3A
80	685.7	49/33, 52/35, 55/37	v6G𝄪, vvA
81	694.3		v5G𝄪, vA
82	702.9	3/2	A
83	711.4		^A, ^5B♭♭
84	720	47/31, 50/33	^^A, ^6B♭♭
85	728.6	32/21	^3A, v7B♭
86	737.1	26/17, 49/32	^4A, v6B♭
87	745.7	20/13	^5A, v5B♭
88	754.3	17/11	^6A, v4B♭
89	762.9		^7A, v3B♭
90	771.4	25/16, 39/25	v6A♯, vvB♭
91	780		v5A♯, vB♭
92	788.6	41/26, 52/33	v4A♯, B♭
93	797.1	19/12, 46/29	v3A♯, ^B♭
94	805.7	43/27, 51/32	vvA♯, ^^B♭
95	814.3	8/5	vA♯, ^3B♭
96	822.9	37/23, 45/28	A♯, ^4B♭
97	831.4	21/13, 55/34	^A♯, ^5B♭
98	840	13/8	^^A♯, ^6B♭
99	848.6	31/19, 49/30	^3A♯, v7B
100	857.1	41/25	^4A♯, v6B
101	865.7	33/20	^5A♯, v5B
102	874.3		^6A♯, v4B
103	882.9	5/3	^7A♯, v3B
104	891.4		v6A𝄪, vvB
105	900	37/22	v5A𝄪, vB
106	908.6	49/29	B
107	917.1	17/10	^B, ^5C♭
108	925.7	29/17, 41/24	^^B, ^6C♭
109	934.3	12/7	^3B, v7C
110	942.9	50/29	^4B, v6C
111	951.4	26/15	^5B, v5C
112	960	47/27, 54/31	^6B, v4C
113	968.6	7/4	^7B, v3C
114	977.1	44/25, 51/29	v6B♯, vvC
115	985.7		v5B♯, vC
116	994.3		C
117	1002.9	25/14	^C, ^5D♭♭
118	1011.4	52/29	^^C, ^6D♭♭
119	1020		^3C, v7D♭
120	1028.6	29/16	^4C, v6D♭
121	1037.1	51/28	^5C, v5D♭
122	1045.7		^6C, v4D♭
123	1054.3	46/25	^7C, v3D♭
124	1062.9	24/13	v6C♯, vvD♭
125	1071.4	13/7	v5C♯, vD♭
126	1080	28/15	v4C♯, D♭
127	1088.6	15/8	v3C♯, ^D♭
128	1097.1	49/26	vvC♯, ^^D♭
129	1105.7	36/19	vC♯, ^3D♭
130	1114.3	40/21	C♯, ^4D♭
131	1122.9	44/23	^C♯, ^5D♭
132	1131.4	25/13	^^C♯, ^6D♭
133	1140	29/15	^3C♯, v7D
134	1148.6	33/17	^4C♯, v6D
135	1157.1	39/20, 41/21	^5C♯, v5D
136	1165.7	49/25, 51/26	^6C♯, v4D
137	1174.3		^7C♯, v3D
138	1182.9		v6C𝄪, vvD
139	1191.4		v5C𝄪, vD
140	1200	2/1	D

Theory

In the 5-limit, 140et tempers out 15625/15552, making it a kleismic system, and the kwazy comma, [-53 10 16⟩. It is most notable, however, in the 7-limit, where it tempers out 2401/2400, 5120/5103, 10976/10935 and 65625/65536. It supports the 7-limit rank-2 temperaments tertiaseptal, hemififths, countercata and bisupermajor, and is a good tuning recommendation for countercata, the 53 & 87 temperament tempering out 15625/15552 and 5120/5103, and provides the optimal patent val for 13-limit countercata. In the 11-limit it tempers out 385/384, 1331/1323, 1375/1372, 5632/5625, 6250/6237 and 9801/9800, and in the 13-limit 325/324, 352/351, 625/624, 676/675, 847/845, 1001/1000, 1716/1715 and 2080/2079.

If we use the val ⟨140 223 325 394] (140bbd) we obtain a tuning for porcupine temperament; the generator 19\140 is 0.023 cents flat of the POTE generator.

Prime harmonics

Approximation of prime harmonics in 140edo

Harmonic		2	3	5	7	11	13	17	19	23	29
Error	Absolute (¢)	+0.00	+0.90	-0.60	-0.25	-2.75	-0.53	-2.10	+2.49	-2.56	-1.01
	Relative (%)	+0.0	+10.5	-7.0	-3.0	-32.0	-6.2	-24.5	+29.0	-29.9	-11.7
Steps (reduced)		140 (0)	222 (82)	325 (45)	393 (113)	484 (64)	518 (98)	572 (12)	595 (35)	633 (73)	680 (120)

Approximation of prime harmonics in 140edo (continued)

Harmonic		31	37	41	43	47	53	59	61	67	71
Error	Absolute (¢)	+3.54	-2.77	-0.49	+2.77	+3.06	+0.78	+3.69	-2.60	-2.16	+0.30
	Relative (%)	+41.3	-32.3	-5.7	+32.3	+35.8	+9.1	+43.0	-30.3	-25.2	+3.5
Steps (reduced)		694 (134)	729 (29)	750 (50)	760 (60)	778 (78)	802 (102)	824 (124)	830 (130)	849 (9)	861 (21)

Subsets and supersets

Since 140 factors into 2² × 5 × 7, 140edo has subset edos 2, 4, 5, 7, 10, 14, 20, 28, 35, and 70.

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3.5	15625/15552, [35 -25 2⟩	[⟨140 222 325]]	−0.104	0.346	4.03
2.3.5.7	2401/2400, 5120/5103, 15625/15552	[⟨140 222 325 393]]	−0.055	0.311	3.63
2.3.5.7.11	385/384, 1331/1323, 1375/1372, 2200/2187	[⟨140 222 325 393 484]]	+0.115	0.439	5.12
2.3.5.7.11.13	325/324, 352/351, 385/384, 625/624, 1331/1323	[⟨140 222 325 393 484 518]]	+0.119	0.401	4.68
2.3.5.7.11.13.17	289/288, 325/324, 352/351, 385/384, 442/441, 625/624	[⟨140 222 325 393 484 518 572]]	+0.176	0.396	4.62

• 140et has lower absolute errors than any previous equal temperaments in the 17-, 19-, and 23-limit, and perhaps beyond. In the 17-limit it is the first to beat 121 and is superseded by 171. In the 19- and 23-limit it is the first to beat 130 and is superseded by 152fg.

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperaments
1	9\140	77.14	22/21	Tertiaseptal / tertia
1	13\140	111.43	16/15	Stockhausenic
1	37\140	317.14	6/5	Countercata
1	41\140	351.43	49/40	Hemififths
1	53\140	454.29	13/10	Fibo
1	59\140	505.71	75/56	Marfifths
2	3\140	25.71	64/63	Ketchup
2	19\140	162.86	11/10	Bisupermajor
2	41\140 (29\140)	351.43 (248.57)	49/40 (15/13)	Semihemi
4	37\140 (2\140)	317.14 (17.14)	6/5 (126/125)	Quadritikleismic
4	58\140 (12\140)	497.14 (102.86)	4/3 (35/33)	Undim
5	43\140 (13\140)	368.57 (111.43)	1024/891 (16/15)	Quintosec
10	29\140 (1\140)	248.57 (8.57)	15/13 (176/175)	Decoid
20	54\140 (2\140)	497.14 (17.14)	4/3 (126/125)	Degrees
28	54\140 (2\140)	497.14 (17.14)	4/3 (126/125)	Oquatonic

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct