138ed22

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← 137ed22 138ed22 139ed22 →
Prime factorization 2 × 3 × 23
Step size 38.7777 ¢ 
Octave 31\138ed22 (1202.11 ¢)
Twelfth 49\138ed22 (1900.11 ¢)
Consistency limit 12
Distinct consistency limit 9

138 equal divisions of the 22nd harmonic (abbreviated 138ed22) is a nonoctave tuning system that divides the interval of 22/1 into 138 equal parts of about 38.8 ¢ each. Each step represents a frequency ratio of 221/138, or the 138th root of 22.

Theory

138ed22 is related to 31edo, but with the 22/1 rather than the 2/1 being just, which stretches the octave by 2.11 ¢. Like 31edo, 138ed22 is consistent through the 12-integer-limit, and unlike 107ed11 it does not have a discrepancy for the 13th harmonic.

Harmonics

Approximation of harmonics in 138ed22
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +2.1 -1.8 +4.2 +5.7 +0.3 +4.8 +6.3 -3.7 +7.8 -2.1 +2.4
Relative (%) +5.4 -4.8 +10.9 +14.6 +0.7 +12.5 +16.3 -9.5 +20.1 -5.4 +6.1
Steps
(reduced) 		31
(31) 		49
(49) 		62
(62) 		72
(72) 		80
(80) 		87
(87) 		93
(93) 		98
(98) 		103
(103) 		107
(107) 		111
(111)
Approximation of harmonics in 138ed22 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +18.9 +6.9 +3.8 +8.4 -19.0 -1.6 -17.6 +9.9 +3.0 +0.0 +0.6 +4.5
Relative (%) +48.7 +17.9 +9.9 +21.7 -48.9 -4.1 -45.5 +25.5 +7.7 +0.0 +1.5 +11.5
Steps
(reduced) 		115
(115) 		118
(118) 		121
(121) 		124
(124) 		126
(126) 		129
(129) 		131
(131) 		134
(134) 		136
(136) 		138
(0) 		140
(2) 		142
(4)

Intervals

Steps Cents Approximate ratios
0 0 1/1
1 38.8 44/43, 45/44, 46/45, 47/46
2 77.6 23/22, 45/43
3 116.3 31/29, 46/43
4 155.1 35/32, 47/43
5 193.9 19/17, 47/42
6 232.7 8/7
7 271.4 48/41
8 310.2 49/41
9 349 11/9
10 387.8 5/4
11 426.6 32/25
12 465.3
13 504.1
14 542.9
15 581.7 7/5
16 620.4
17 659.2 41/28
18 698
19 736.8 49/32
20 775.6 36/23, 47/30
21 814.3 8/5
22 853.1 18/11
23 891.9
24 930.7
25 969.4 7/4
26 1008.2 34/19, 43/24
27 1047
28 1085.8
29 1124.6 44/23
30 1163.3 45/23, 47/24, 49/25
31 1202.1
32 1240.9 43/21
33 1279.7 44/21
34 1318.4 15/7
35 1357.2 46/21
36 1396 47/21
37 1434.8
38 1473.6
39 1512.3
40 1551.1 49/20
41 1589.9
42 1628.7 41/16
43 1667.4
44 1706.2
45 1745
46 1783.8 14/5
47 1822.6 43/15
48 1861.3 41/14
49 1900.1
50 1938.9 46/15, 49/16
51 1977.7 47/15
52 2016.4
53 2055.2
54 2094
55 2132.8 24/7
56 2171.5
57 2210.3 43/12
58 2249.1 11/3
59 2287.9 15/4
60 2326.7 23/6
61 2365.4
62 2404.2
63 2443 41/10
64 2481.8
65 2520.5 30/7
66 2559.3
67 2598.1
68 2636.9
69 2675.7
70 2714.4 24/5
71 2753.2 54/11
72 2792
73 2830.8
74 2869.5 21/4
75 2908.3
76 2947.1
77 2985.9
78 3024.7
79 3063.4
80 3102.2 6/1
81 3141 43/7
82 3179.8
83 3218.5
84 3257.3
85 3296.1 47/7
86 3334.9
87 3373.7
88 3412.4
89 3451.2
90 3490
91 3528.8
92 3567.5
93 3606.3
94 3645.1
95 3683.9 42/5
96 3722.7
97 3761.4
98 3800.2
99 3839
100 3877.8 47/5
101 3916.5 48/5
102 3955.3
103 3994.1
104 4032.9
105 4071.7 21/2
106 4110.4 43/4
107 4149.2
108 4188
109 4226.8 23/2
110 4265.5 47/4
111 4304.3
112 4343.1
113 4381.9
114 4420.7
115 4459.4
116 4498.2
117 4537
118 4575.8
119 4614.5
120 4653.3
121 4692.1
122 4730.9
123 4769.7
124 4808.4
125 4847.2
126 4886
127 4924.8
128 4963.5
129 5002.3
130 5041.1
131 5079.9
132 5118.7
133 5157.4
134 5196.2
135 5235
136 5273.8
137 5312.5
138 5351.3

See also

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