109edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|109}}
{{ED intro}}


== Theory ==
== Theory ==
109edo [[tempering out|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 [[tempering out|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 [[7/1|7th harmonic]], being a denominator of [[semiconvergent]] to log<sub>2</sub>7, 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.  
109edo has an excellent [[7/1|7th harmonic]], being a denominator of [[semiconvergent]] to log<sub>2</sub>7, 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, 2255/2254, 2300/2299, 6860/6859, 10241/10240.  


=== Prime harmonics ===
=== Prime harmonics ===
{{Harmonics in equal|109}}
{{Harmonics in equal|109|columns=16}}


=== Subsets and supersets ===
=== Subsets and supersets ===
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== Intervals ==
== Intervals ==
{{Interval table}}
{{Interval table}}
== Music ==
; [[Francium]]
* "Teenagerges" from ''TOTMC September to December 2024'' (2024) – [https://open.spotify.com/track/4oQglJSEyp6CsL5RNWuiBy Spotify] | [https://francium223.bandcamp.com/track/teenagerges Bandcamp] | [https://www.youtube.com/watch?v=v_J71U392_k YouTube] – in Tetracot[13], 109edo tuning
* "Catbabel" from ''Microtonal Six-Dimensional Cats'' (2025) – [https://open.spotify.com/track/0T7nW3ziEFvjV8c7v1EaMB Spotify] | [https://francium223.bandcamp.com/track/catbabel Bandcamp] | [https://www.youtube.com/watch?v=gtnTdPqiTDQ YouTube]
== See also ==
* [[109-7-comma]]

Latest revision as of 03:56, 19 October 2025

← 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 21/109, 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, 2255/2254, 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 ^D, v6E♭
2 22 ^^D, v5E♭
3 33 ^3D, v4E♭
4 44 38/37, 39/38, 40/39, 41/40 ^4D, v3E♭
5 55 31/30, 32/31, 33/32 ^5D, vvE♭
6 66.1 26/25 ^6D, vE♭
7 77.1 23/22 v5D♯, E♭
8 88.1 20/19, 41/39 v4D♯, ^E♭
9 99.1 18/17 v3D♯, ^^E♭
10 110.1 16/15, 33/31 vvD♯, ^3E♭
11 121.1 15/14, 44/41 vD♯, ^4E♭
12 132.1 41/38 D♯, ^5E♭
13 143.1 25/23, 38/35 ^D♯, v6E
14 154.1 35/32, 47/43 ^^D♯, v5E
15 165.1 11/10 ^3D♯, v4E
16 176.1 31/28, 41/37 ^4D♯, v3E
17 187.2 39/35 ^5D♯, vvE
18 198.2 28/25, 37/33, 46/41 ^6D♯, vE
19 209.2 35/31, 44/39 E
20 220.2 25/22, 42/37 ^E, v6F
21 231.2 8/7 ^^E, v5F
22 242.2 23/20, 38/33 ^3E, v4F
23 253.2 22/19, 37/32 ^4E, v3F
24 264.2 ^5E, vvF
25 275.2 34/29, 41/35 ^6E, vF
26 286.2 33/28, 46/39 F
27 297.2 19/16 ^F, v6G♭
28 308.3 37/31 ^^F, v5G♭
29 319.3 ^3F, v4G♭
30 330.3 23/19 ^4F, v3G♭
31 341.3 28/23, 39/32 ^5F, vvG♭
32 352.3 38/31 ^6F, vG♭
33 363.3 37/30 v5F♯, G♭
34 374.3 31/25, 36/29, 41/33 v4F♯, ^G♭
35 385.3 5/4 v3F♯, ^^G♭
36 396.3 39/31, 44/35 vvF♯, ^3G♭
37 407.3 19/15 vF♯, ^4G♭
38 418.3 14/11 F♯, ^5G♭
39 429.4 32/25, 41/32 ^F♯, v6G
40 440.4 40/31 ^^F♯, v5G
41 451.4 ^3F♯, v4G
42 462.4 ^4F♯, v3G
43 473.4 25/19, 46/35 ^5F♯, vvG
44 484.4 37/28, 41/31, 45/34 ^6F♯, vG
45 495.4 G
46 506.4 ^G, v6A♭
47 517.4 31/23 ^^G, v5A♭
48 528.4 19/14 ^3G, v4A♭
49 539.4 41/30 ^4G, v3A♭
50 550.5 11/8 ^5G, vvA♭
51 561.5 ^6G, vA♭
52 572.5 32/23, 39/28 v5G♯, A♭
53 583.5 7/5 v4G♯, ^A♭
54 594.5 31/22 v3G♯, ^^A♭
55 605.5 44/31 vvG♯, ^3A♭
56 616.5 10/7 vG♯, ^4A♭
57 627.5 23/16 G♯, ^5A♭
58 638.5 ^G♯, v6A
59 649.5 16/11 ^^G♯, v5A
60 660.6 41/28 ^3G♯, v4A
61 671.6 28/19 ^4G♯, v3A
62 682.6 46/31 ^5G♯, vvA
63 693.6 ^6G♯, vA
64 704.6 A
65 715.6 ^A, v6B♭
66 726.6 35/23, 38/25 ^^A, v5B♭
67 737.6 ^3A, v4B♭
68 748.6 37/24 ^4A, v3B♭
69 759.6 31/20, 45/29 ^5A, vvB♭
70 770.6 25/16, 39/25 ^6A, vB♭
71 781.7 11/7 v5A♯, B♭
72 792.7 30/19 v4A♯, ^B♭
73 803.7 35/22 v3A♯, ^^B♭
74 814.7 8/5 vvA♯, ^3B♭
75 825.7 29/18 vA♯, ^4B♭
76 836.7 A♯, ^5B♭
77 847.7 31/19 ^A♯, v6B
78 858.7 23/14 ^^A♯, v5B
79 869.7 38/23, 43/26 ^3A♯, v4B
80 880.7 ^4A♯, v3B
81 891.7 ^5A♯, vvB
82 902.8 32/19 ^6A♯, vB
83 913.8 39/23 B
84 924.8 29/17 ^B, v6C
85 935.8 ^^B, v5C
86 946.8 19/11 ^3B, v4C
87 957.8 33/19, 40/23 ^4B, v3C
88 968.8 7/4 ^5B, vvC
89 979.8 37/21, 44/25 ^6B, vC
90 990.8 39/22 C
91 1001.8 25/14, 41/23 ^C, v6D♭
92 1012.8 ^^C, v5D♭
93 1023.9 47/26 ^3C, v4D♭
94 1034.9 20/11 ^4C, v3D♭
95 1045.9 ^5C, vvD♭
96 1056.9 35/19, 46/25 ^6C, vD♭
97 1067.9 v5C♯, D♭
98 1078.9 28/15, 41/22 v4C♯, ^D♭
99 1089.9 15/8 v3C♯, ^^D♭
100 1100.9 17/9 vvC♯, ^3D♭
101 1111.9 19/10 vC♯, ^4D♭
102 1122.9 44/23 C♯, ^5D♭
103 1133.9 25/13 ^C♯, v6D
104 1145 31/16 ^^C♯, v5D
105 1156 37/19, 39/20 ^3C♯, v4D
106 1167 ^4C♯, v3D
107 1178 ^5C♯, vvD
108 1189 ^6C♯, vD
109 1200 2/1 D

Music

Francium

See also

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