201ed12: Difference between revisions

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== Theory ==
== Theory ==
201ed12 acts as a compressed version of [[56edo]].
201ed12 acts as a compressed version of [[56edo]]. It improves primes [[3/1|3]] and [[11/1|11]], but at the cost of [[2/1|2]], [[5/1|5]], [[7/1|7]], [[13/1|13]], [[17/1|17]], and [[19/1|19]]. Thus, most would consider this amount of compression as too much.


{{Harmonics in equal|201|12}}
The main reason one would want to use this much compression is if they are using the 56d [[val]], which most notably supports [[pajara]]. This significantly improves the 3rd, 7th, and 11th harmonics, while not damaging prime 5 too much.
 
{{Harmonics in equal|201|12|columns=11}}
{{Harmonics in equal|201|12|columns=11|start=12}}


== Intervals ==
== Intervals ==
{{Interval table}}
{{Interval table}}


{{Stub}}
{{Todo|expand}}

Latest revision as of 06:21, 18 January 2026

← 200ed12 201ed12 202ed12 →
Prime factorization 3 × 67
Step size 21.4028 ¢ 
Octave 56\201ed12 (1198.55 ¢)
Twelfth 89\201ed12 (1904.85 ¢)
Consistency limit 6
Distinct consistency limit 6

201 equal divisions of the 12th harmonic (abbreviated 201ed12) is a nonoctave tuning system that divides the interval of 12/1 into 201 equal parts of about 21.4 ¢ each. Each step represents a frequency ratio of 121/201, or the 201st root of 12.

Theory

201ed12 acts as a compressed version of 56edo. It improves primes 3 and 11, but at the cost of 2, 5, 7, 13, 17, and 19. Thus, most would consider this amount of compression as too much.

The main reason one would want to use this much compression is if they are using the 56d val, which most notably supports pajara. This significantly improves the 3rd, 7th, and 11th harmonics, while not damaging prime 5 too much.


Approximation of harmonics in 201ed12
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -1.45 +2.89 -2.89 -3.95 +1.45 -8.59 -4.34 +5.78 -5.40 +0.82 +0.00
Relative (%) -6.8 +13.5 -13.5 -18.5 +6.8 -40.1 -20.3 +27.0 -25.2 +3.8 +0.0
Steps
(reduced) 		56
(56) 		89
(89) 		112
(112) 		130
(130) 		145
(145) 		157
(157) 		168
(168) 		178
(178) 		186
(186) 		194
(194) 		201
(0)
Approximation of harmonics in 201ed12
Harmonic 13 14 15 16 17 18 19 20 21 22 23
Error Absolute (¢) -10.16 -10.04 -1.06 -5.78 -3.72 +4.34 -3.66 -6.85 -5.70 -0.63 +8.03
Relative (%) -47.5 -46.9 -5.0 -27.0 -17.4 +20.3 -17.1 -32.0 -26.6 -2.9 +37.5
Steps
(reduced) 		207
(6) 		213
(12) 		219
(18) 		224
(23) 		229
(28) 		234
(33) 		238
(37) 		242
(41) 		246
(45) 		250
(49) 		254
(53)

Intervals

Steps Cents Approximate ratios
0 0 1/1
1 21.4
2 42.81 40/39, 41/40, 42/41
3 64.21
4 85.61 41/39
5 107.01 50/47
6 128.42 14/13
7 149.82 12/11
8 171.22 32/29
9 192.62 19/17
10 214.03 43/38
11 235.43 47/41, 55/48, 63/55
12 256.83 29/25
13 278.24 27/23, 47/40
14 299.64 44/37
15 321.04
16 342.44 39/32
17 363.85 37/30, 58/47
18 385.25 5/4
19 406.65 43/34
20 428.06 32/25, 41/32
21 449.46
22 470.86 21/16
23 492.26
24 513.67 35/26, 39/29
25 535.07 64/47
26 556.47 40/29, 51/37
27 577.87
28 599.28 41/29
29 620.68 63/44
30 642.08 42/29
31 663.49 22/15
32 684.89 52/35
33 706.29
34 727.69
35 749.1 37/24, 57/37
36 770.5 39/25, 64/41
37 791.9
38 813.3 8/5
39 834.71 34/21
40 856.11 41/25
41 877.51
42 898.92 42/25
43 920.32 63/37
44 941.72 31/18
45 963.12 68/39
46 984.53
47 1005.93
48 1027.33 38/21
49 1048.74 11/6
50 1070.14
51 1091.54 62/33
52 1112.94
53 1134.35
54 1155.75 39/20
55 1177.15
56 1198.55
57 1219.96
58 1241.36 43/21
59 1262.76
60 1284.17 21/10
61 1305.57 17/8
62 1326.97
63 1348.37
64 1369.78 64/29
65 1391.18
66 1412.58
67 1433.99
68 1455.39 51/22
69 1476.79 54/23
70 1498.19 19/8
71 1519.6
72 1541
73 1562.4 37/15
74 1583.8
75 1605.21 48/19
76 1626.61 64/25
77 1648.01 57/22
78 1669.42
79 1690.82
80 1712.22 43/16
81 1733.62
82 1755.03
83 1776.43
84 1797.83 48/17
85 1819.23
86 1840.64 55/19
87 1862.04 44/15
88 1883.44
89 1904.85
90 1926.25
91 1947.65
92 1969.05
93 1990.46 60/19
94 2011.86
95 2033.26 55/17, 68/21
96 2054.67
97 2076.07 63/19
98 2097.47 47/14
99 2118.87 17/5
100 2140.28 31/9
101 2161.68
102 2183.08 60/17
103 2204.48 25/7
104 2225.89 47/13
105 2247.29
106 2268.69 63/17
107 2290.1
108 2311.5 19/5
109 2332.9 50/13
110 2354.3
111 2375.71
112 2397.11
113 2418.51
114 2439.91 45/11
115 2461.32 29/7
116 2482.72
117 2504.12 17/4
118 2525.53 43/10
119 2546.93
120 2568.33
121 2589.73 58/13
122 2611.14
123 2632.54
124 2653.94
125 2675.35
126 2696.75 19/4
127 2718.15
128 2739.55
129 2760.96
130 2782.36
131 2803.76
132 2825.16 46/9
133 2846.57
134 2867.97
135 2889.37
136 2910.78 43/8
137 2932.18
138 2953.58
139 2974.98
140 2996.39
141 3017.79 40/7
142 3039.19
143 3060.59 41/7
144 3082
145 3103.4
146 3124.8
147 3146.21
148 3167.61
149 3189.01
150 3210.41
151 3231.82
152 3253.22
153 3274.62
154 3296.03 47/7
155 3317.43
156 3338.83
157 3360.23
158 3381.64
159 3403.04 50/7
160 3424.44
161 3445.84
162 3467.25
163 3488.65 15/2
164 3510.05
165 3531.46
166 3552.86
167 3574.26
168 3595.66
169 3617.07
170 3638.47
171 3659.87 58/7
172 3681.27
173 3702.68
174 3724.08
175 3745.48
176 3766.89
177 3788.29
178 3809.69
179 3831.09 64/7
180 3852.5
181 3873.9
182 3895.3
183 3916.71 48/5
184 3938.11
185 3959.51
186 3980.91
187 4002.32
188 4023.72
189 4045.12
190 4066.52
191 4087.93
192 4109.33
193 4130.73
194 4152.14 11/1
195 4173.54
196 4194.94
197 4216.34
198 4237.75
199 4259.15
200 4280.55
201 4301.96 12/1
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