109edo

← 108edo

109edo

110edo →

Prime factorization	109 (prime)
Step size	11.0092¢
Fifth	64\109 (704.587¢)
Semitones (A1:m2)	12:7 (132.1¢ : 77.06¢)
Consistency limit	7
Distinct consistency limit	7

109 equal divisions of the octave (abbreviated 109edo or 109ed2), also called 109-tone equal temperament (109tet) or 109 equal temperament (109et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 109 equal parts of about 11 ¢ each. Each step represents a frequency ratio of 2^1/109, or the 109th root of 2.

Theory

109edo tempers out 20000/19683 (tetracot comma) in the 5-limit; 245/243, 2401/2400 and 65625/65536 in the 7-limit; 385/384, 1375/1372, and 4000/3993 in the 11-limit. It provides the optimal patent val for 7-limit octacot temperament, and 11- and 13-limit leapweek; plus 109ef provides an excellent tuning for 11- and 13-limit octacot.

109edo has an excellent 7th harmonic, being a denominator of semiconvergent to log₂7, and it is overall a strong 2.5.7.11.19.23.31.41 subgroup tuning, with errors of less than 10% on all harmonics. Some commas it tempers out in this subgroup are 575/574, 1331/1330, 1375/1372, 2255/2244, 2300/2299, 6860/6859, 10241/10240.

Prime harmonics

Approximation of prime harmonics in 109edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53
Error	Absolute (¢)	+0.00	+2.63	-0.99	-0.02	-0.86	-3.83	+5.14	-0.27	-0.75	+5.29	-0.08	+1.87	+0.30	-5.10	-4.96	-3.78
	Relative (%)	+0.0	+23.9	-9.0	-0.2	-7.8	-34.8	+46.7	-2.4	-6.8	+48.0	-0.7	+17.0	+2.7	-46.3	-45.0	-34.3
Steps (reduced)		109 (0)	173 (64)	253 (35)	306 (88)	377 (50)	403 (76)	446 (10)	463 (27)	493 (57)	530 (94)	540 (104)	568 (23)	584 (39)	591 (46)	605 (60)	624 (79)

Subsets and supersets

109edo is the 29th prime edo, following 107edo and before 113edo. 436edo, which slices each step of 109edo in four, provides correction for the approximation to harmonic 3.

Nonoctave temperaments

Taking every 8 degree of 109edo produces a scale extremely close to 88cET.

Intervals

Steps	Cents	Approximate ratios	Ups and downs notation
0	0	1/1	D
1	11.009		^D, v6E♭
2	22.018		^^D, v5E♭
3	33.028		^3D, v4E♭
4	44.037	38/37, 39/38, 40/39, 41/40	^4D, v3E♭
5	55.046	31/30, 32/31, 33/32	^5D, vvE♭
6	66.055	26/25	^6D, vE♭
7	77.064	23/22	^7D, E♭
8	88.073	20/19, 41/39	^8D, v11E
9	99.083	18/17	^9D, v10E
10	110.092	16/15, 33/31	^10D, v9E
11	121.101	15/14, 44/41	^11D, v8E
12	132.11	41/38	D♯, v7E
13	143.119	25/23, 38/35	^D♯, v6E
14	154.128	35/32, 47/43	^^D♯, v5E
15	165.138	11/10	^3D♯, v4E
16	176.147	31/28, 41/37	^4D♯, v3E
17	187.156	39/35	^5D♯, vvE
18	198.165	28/25, 37/33, 46/41	^6D♯, vE
19	209.174	35/31, 44/39	E
20	220.183	25/22, 42/37	^E, v6F
21	231.193	8/7	^^E, v5F
22	242.202	23/20, 38/33	^3E, v4F
23	253.211	22/19, 37/32	^4E, v3F
24	264.22		^5E, vvF
25	275.229	34/29, 41/35	^6E, vF
26	286.239	33/28, 46/39	F
27	297.248	19/16	^F, v6G♭
28	308.257	37/31	^^F, v5G♭
29	319.266		^3F, v4G♭
30	330.275	23/19	^4F, v3G♭
31	341.284	28/23, 39/32	^5F, vvG♭
32	352.294	38/31	^6F, vG♭
33	363.303	37/30	^7F, G♭
34	374.312	31/25, 36/29, 41/33	^8F, v11G
35	385.321	5/4	^9F, v10G
36	396.33	39/31, 44/35	^10F, v9G
37	407.339	19/15	^11F, v8G
38	418.349	14/11	F♯, v7G
39	429.358	32/25, 41/32	^F♯, v6G
40	440.367	40/31	^^F♯, v5G
41	451.376		^3F♯, v4G
42	462.385		^4F♯, v3G
43	473.394	25/19, 46/35	^5F♯, vvG
44	484.404	37/28, 41/31, 45/34	^6F♯, vG
45	495.413		G
46	506.422		^G, v6A♭
47	517.431	31/23	^^G, v5A♭
48	528.44	19/14	^3G, v4A♭
49	539.45	41/30	^4G, v3A♭
50	550.459	11/8	^5G, vvA♭
51	561.468		^6G, vA♭
52	572.477	32/23, 39/28	^7G, A♭
53	583.486	7/5	^8G, v11A
54	594.495	31/22	^9G, v10A
55	605.505	44/31	^10G, v9A
56	616.514	10/7	^11G, v8A
57	627.523	23/16	G♯, v7A
58	638.532		^G♯, v6A
59	649.541	16/11	^^G♯, v5A
60	660.55	41/28	^3G♯, v4A
61	671.56	28/19	^4G♯, v3A
62	682.569	46/31	^5G♯, vvA
63	693.578		^6G♯, vA
64	704.587		A
65	715.596		^A, v6B♭
66	726.606	35/23, 38/25	^^A, v5B♭
67	737.615		^3A, v4B♭
68	748.624	37/24	^4A, v3B♭
69	759.633	31/20, 45/29	^5A, vvB♭
70	770.642	25/16, 39/25	^6A, vB♭
71	781.651	11/7	^7A, B♭
72	792.661	30/19	^8A, v11B
73	803.67	35/22	^9A, v10B
74	814.679	8/5	^10A, v9B
75	825.688	29/18	^11A, v8B
76	836.697		A♯, v7B
77	847.706	31/19	^A♯, v6B
78	858.716	23/14	^^A♯, v5B
79	869.725	38/23, 43/26	^3A♯, v4B
80	880.734		^4A♯, v3B
81	891.743		^5A♯, vvB
82	902.752	32/19	^6A♯, vB
83	913.761	39/23	B
84	924.771	29/17	^B, v6C
85	935.78		^^B, v5C
86	946.789	19/11	^3B, v4C
87	957.798	33/19, 40/23	^4B, v3C
88	968.807	7/4	^5B, vvC
89	979.817	37/21, 44/25	^6B, vC
90	990.826	39/22	C
91	1001.835	25/14, 41/23	^C, v6D♭
92	1012.844		^^C, v5D♭
93	1023.853	47/26	^3C, v4D♭
94	1034.862	20/11	^4C, v3D♭
95	1045.872		^5C, vvD♭
96	1056.881	35/19, 46/25	^6C, vD♭
97	1067.89		^7C, D♭
98	1078.899	28/15, 41/22	^8C, v11D
99	1089.908	15/8	^9C, v10D
100	1100.917	17/9	^10C, v9D
101	1111.927	19/10	^11C, v8D
102	1122.936	44/23	C♯, v7D
103	1133.945	25/13	^C♯, v6D
104	1144.954	31/16	^^C♯, v5D
105	1155.963	37/19, 39/20	^3C♯, v4D
106	1166.972		^4C♯, v3D
107	1177.982		^5C♯, vvD
108	1188.991		^6C♯, vD
109	1200	2/1	D

Music

Francium

• Teenagerges (2024) – tetracot[13] in 109edo tuning

See also

• 109-7-comma