239edo

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← 238edo 239edo 240edo →
Prime factorization 239 (prime)
Step size 5.02092 ¢ 
Fifth 140\239 (702.929 ¢)
Semitones (A1:m2) 24:17 (120.5 ¢ : 85.36 ¢)
Consistency limit 11
Distinct consistency limit 11

239 equal divisions of the octave (abbreviated 239edo or 239ed2), also called 239-tone equal temperament (239tet) or 239 equal temperament (239et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 239 equal parts of about 5.02 ¢ each. Each step represents a frequency ratio of 21/239, or the 239th root of 2.

Theory

239edo has a sharp tendency, with prime harmonics 3 through 11 all tuned sharp. The equal temperament tempers out 2401/2400, 5120/5103, and 29360128/29296875 in the 7-limit, supporting the hemififths temperament and providing an excellent tuning. It also supports and provides a good tuning for quasiorwell and alphaquarter. In the 11-limit, it tempers out 3025/3024, 4000/3993, 5632/5625, and 12005/11979.

It also encompasses a large variety of higher primes, specifically commendable in the 2.3.5.7.11.17.29.31.37.43.53.59 subgroup.

Prime harmonics

Approximation of prime harmonics in 239edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.97 +0.30 +0.21 +0.98 -2.03 +0.48 -1.28 -0.66 -0.29 -0.27
Relative (%) +0.0 +19.4 +5.9 +4.2 +19.6 -40.5 +9.6 -25.5 -13.1 -5.7 -5.3
Steps
(reduced) 		239
(0) 		379
(140) 		555
(77) 		671
(193) 		827
(110) 		884
(167) 		977
(21) 		1015
(59) 		1081
(125) 		1161
(205) 		1184
(228)
Approximation of prime harmonics in 239edo (continued)
Harmonic 37 41 43 47 53 59 61 67 71 73 79
Error Absolute (¢) -0.30 -2.28 +0.62 +2.28 +0.14 +0.24 -2.24 +1.03 +1.06 -1.85 +1.99
Relative (%) -5.9 -45.5 +12.3 +45.3 +2.7 +4.8 -44.6 +20.5 +21.0 -36.8 +39.6
Steps
(reduced) 		1245
(50) 		1280
(85) 		1297
(102) 		1328
(133) 		1369
(174) 		1406
(211) 		1417
(222) 		1450
(16) 		1470
(36) 		1479
(45) 		1507
(73)

Subsets and supersets

239edo is the 52nd prime edo.

Intervals

Steps Cents Approximate ratios Ups and downs notation
0 0 1/1 D
1 5.02 ^D, ^8E♭♭
2 10.04 ^^D, ^9E♭♭
3 15.06 ^3D, ^10E♭♭
4 20.08 ^4D, ^11E♭♭
5 25.1 69/68, 70/69 ^5D, v12E♭
6 30.13 57/56, 58/57 ^6D, v11E♭
7 35.15 49/48, 50/49, 51/50 ^7D, v10E♭
8 40.17 43/42, 44/43 ^8D, v9E♭
9 45.19 39/38, 77/75 ^9D, v8E♭
10 50.21 35/34 ^10D, v7E♭
11 55.23 32/31 ^11D, v6E♭
12 60.25 29/28 ^12D, v5E♭
13 65.27 v11D♯, v4E♭
14 70.29 25/24 v10D♯, v3E♭
15 75.31 47/45 v9D♯, vvE♭
16 80.33 22/21 v8D♯, vE♭
17 85.36 21/20 v7D♯, E♭
18 90.38 39/37 v6D♯, ^E♭
19 95.4 37/35 v5D♯, ^^E♭
20 100.42 v4D♯, ^3E♭
21 105.44 17/16 v3D♯, ^4E♭
22 110.46 vvD♯, ^5E♭
23 115.48 31/29, 77/72 vD♯, ^6E♭
24 120.5 74/69 D♯, ^7E♭
25 125.52 43/40 ^D♯, ^8E♭
26 130.54 55/51, 69/64 ^^D♯, ^9E♭
27 135.56 40/37 ^3D♯, ^10E♭
28 140.59 51/47 ^4D♯, ^11E♭
29 145.61 37/34, 62/57 ^5D♯, v12E
30 150.63 12/11 ^6D♯, v11E
31 155.65 35/32 ^7D♯, v10E
32 160.67 34/31 ^8D♯, v9E
33 165.69 11/10 ^9D♯, v8E
34 170.71 32/29 ^10D♯, v7E
35 175.73 31/28 ^11D♯, v6E
36 180.75 ^12D♯, v5E
37 185.77 49/44, 69/62 v11D𝄪, v4E
38 190.79 48/43, 77/69 v10D𝄪, v3E
39 195.82 28/25 v9D𝄪, vvE
40 200.84 55/49, 64/57 v8D𝄪, vE
41 205.86 E
42 210.88 35/31 ^E, ^8F♭
43 215.9 17/15, 77/68 ^^E, ^9F♭
44 220.92 25/22 ^3E, ^10F♭
45 225.94 49/43 ^4E, ^11F♭
46 230.96 8/7 ^5E, v12F
47 235.98 55/48, 63/55 ^6E, v11F
48 241 54/47 ^7E, v10F
49 246.03 ^8E, v9F
50 251.05 37/32 ^9E, v8F
51 256.07 29/25, 51/44 ^10E, v7F
52 261.09 50/43, 57/49 ^11E, v6F
53 266.11 7/6 ^12E, v5F
54 271.13 76/65 v11E♯, v4F
55 276.15 34/29 v10E♯, v3F
56 281.17 20/17 v9E♯, vvF
57 286.19 46/39 v8E♯, vF
58 291.21 58/49 F
59 296.23 51/43 ^F, ^8G♭♭
60 301.26 25/21, 69/58 ^^F, ^9G♭♭
61 306.28 37/31, 68/57 ^3F, ^10G♭♭
62 311.3 ^4F, ^11G♭♭
63 316.32 6/5 ^5F, v12G♭
64 321.34 ^6F, v11G♭
65 326.36 35/29 ^7F, v10G♭
66 331.38 23/19 ^8F, v9G♭
67 336.4 17/14 ^9F, v8G♭
68 341.42 28/23 ^10F, v7G♭
69 346.44 ^11F, v6G♭
70 351.46 49/40, 60/49 ^12F, v5G♭
71 356.49 43/35, 70/57 v11F♯, v4G♭
72 361.51 69/56 v10F♯, v3G♭
73 366.53 21/17, 68/55 v9F♯, vvG♭
74 371.55 31/25, 57/46 v8F♯, vG♭
75 376.57 46/37 v7F♯, G♭
76 381.59 v6F♯, ^G♭
77 386.61 5/4 v5F♯, ^^G♭
78 391.63 v4F♯, ^3G♭
79 396.65 39/31, 44/35 v3F♯, ^4G♭
80 401.67 29/23 vvF♯, ^5G♭
81 406.69 43/34, 62/49 vF♯, ^6G♭
82 411.72 52/41 F♯, ^7G♭
83 416.74 14/11 ^F♯, ^8G♭
84 421.76 37/29 ^^F♯, ^9G♭
85 426.78 32/25, 55/43 ^3F♯, ^10G♭
86 431.8 77/60 ^4F♯, ^11G♭
87 436.82 ^5F♯, v12G
88 441.84 40/31 ^6F♯, v11G
89 446.86 22/17 ^7F♯, v10G
90 451.88 74/57 ^8F♯, v9G
91 456.9 56/43 ^9F♯, v8G
92 461.92 47/36, 64/49 ^10F♯, v7G
93 466.95 55/42, 72/55 ^11F♯, v6G
94 471.97 ^12F♯, v5G
95 476.99 v11F𝄪, v4G
96 482.01 37/28 v10F𝄪, v3G
97 487.03 49/37 v9F𝄪, vvG
98 492.05 v8F𝄪, vG
99 497.07 G
100 502.09 ^G, ^8A♭♭
101 507.11 63/47 ^^G, ^9A♭♭
102 512.13 39/29, 43/32 ^3G, ^10A♭♭
103 517.15 31/23 ^4G, ^11A♭♭
104 522.18 50/37 ^5G, v12A♭
105 527.2 ^6G, v11A♭
106 532.22 34/25 ^7G, v10A♭
107 537.24 15/11 ^8G, v9A♭
108 542.26 26/19 ^9G, v8A♭
109 547.28 48/35 ^10G, v7A♭
110 552.3 ^11G, v6A♭
111 557.32 40/29, 69/50 ^12G, v5A♭
112 562.34 v11G♯, v4A♭
113 567.36 43/31, 68/49 v10G♯, v3A♭
114 572.38 32/23 v9G♯, vvA♭
115 577.41 60/43 v8G♯, vA♭
116 582.43 7/5 v7G♯, A♭
117 587.45 66/47 v6G♯, ^A♭
118 592.47 69/49 v5G♯, ^^A♭
119 597.49 24/17 v4G♯, ^3A♭
120 602.51 17/12 v3G♯, ^4A♭
121 607.53 vvG♯, ^5A♭
122 612.55 47/33, 57/40 vG♯, ^6A♭
123 617.57 10/7 G♯, ^7A♭
124 622.59 43/30 ^G♯, ^8A♭
125 627.62 23/16 ^^G♯, ^9A♭
126 632.64 49/34, 62/43 ^3G♯, ^10A♭
127 637.66 ^4G♯, ^11A♭
128 642.68 29/20 ^5G♯, v12A
129 647.7 ^6G♯, v11A
130 652.72 35/24 ^7G♯, v10A
131 657.74 19/13 ^8G♯, v9A
132 662.76 22/15 ^9G♯, v8A
133 667.78 25/17 ^10G♯, v7A
134 672.8 ^11G♯, v6A
135 677.82 37/25 ^12G♯, v5A
136 682.85 46/31 v11G𝄪, v4A
137 687.87 58/39, 64/43 v10G𝄪, v3A
138 692.89 v9G𝄪, vvA
139 697.91 v8G𝄪, vA
140 702.93 A
141 707.95 ^A, ^8B♭♭
142 712.97 74/49, 77/51 ^^A, ^9B♭♭
143 717.99 56/37 ^3A, ^10B♭♭
144 723.01 ^4A, ^11B♭♭
145 728.03 ^5A, v12B♭
146 733.05 55/36 ^6A, v11B♭
147 738.08 49/32, 72/47 ^7A, v10B♭
148 743.1 43/28 ^8A, v9B♭
149 748.12 57/37, 77/50 ^9A, v8B♭
150 753.14 17/11 ^10A, v7B♭
151 758.16 31/20 ^11A, v6B♭
152 763.18 ^12A, v5B♭
153 768.2 v11A♯, v4B♭
154 773.22 25/16 v10A♯, v3B♭
155 778.24 58/37, 69/44 v9A♯, vvB♭
156 783.26 11/7 v8A♯, vB♭
157 788.28 41/26 v7A♯, B♭
158 793.31 49/31, 68/43 v6A♯, ^B♭
159 798.33 46/29, 65/41 v5A♯, ^^B♭
160 803.35 35/22, 62/39 v4A♯, ^3B♭
161 808.37 75/47 v3A♯, ^4B♭
162 813.39 8/5 vvA♯, ^5B♭
163 818.41 69/43, 77/48 vA♯, ^6B♭
164 823.43 37/23 A♯, ^7B♭
165 828.45 50/31 ^A♯, ^8B♭
166 833.47 34/21, 55/34 ^^A♯, ^9B♭
167 838.49 ^3A♯, ^10B♭
168 843.51 57/35, 70/43 ^4A♯, ^11B♭
169 848.54 49/30 ^5A♯, v12B
170 853.56 ^6A♯, v11B
171 858.58 23/14 ^7A♯, v10B
172 863.6 28/17 ^8A♯, v9B
173 868.62 38/23 ^9A♯, v8B
174 873.64 58/35 ^10A♯, v7B
175 878.66 ^11A♯, v6B
176 883.68 5/3 ^12A♯, v5B
177 888.7 v11A𝄪, v4B
178 893.72 57/34, 62/37 v10A𝄪, v3B
179 898.74 42/25 v9A𝄪, vvB
180 903.77 v8A𝄪, vB
181 908.79 49/29 B
182 913.81 39/23 ^B, ^8C♭
183 918.83 17/10 ^^B, ^9C♭
184 923.85 29/17, 75/44 ^3B, ^10C♭
185 928.87 65/38 ^4B, ^11C♭
186 933.89 12/7 ^5B, v12C
187 938.91 43/25 ^6B, v11C
188 943.93 50/29, 69/40 ^7B, v10C
189 948.95 64/37 ^8B, v9C
190 953.97 ^9B, v8C
191 959 47/27 ^10B, v7C
192 964.02 ^11B, v6C
193 969.04 7/4 ^12B, v5C
194 974.06 v11B♯, v4C
195 979.08 44/25 v10B♯, v3C
196 984.1 30/17 v9B♯, vvC
197 989.12 62/35 v8B♯, vC
198 994.14 C
199 999.16 57/32 ^C, ^8D♭♭
200 1004.18 25/14 ^^C, ^9D♭♭
201 1009.21 43/24, 77/43 ^3C, ^10D♭♭
202 1014.23 ^4C, ^11D♭♭
203 1019.25 ^5C, v12D♭
204 1024.27 56/31 ^6C, v11D♭
205 1029.29 29/16 ^7C, v10D♭
206 1034.31 20/11 ^8C, v9D♭
207 1039.33 31/17 ^9C, v8D♭
208 1044.35 64/35 ^10C, v7D♭
209 1049.37 11/6 ^11C, v6D♭
210 1054.39 57/31, 68/37 ^12C, v5D♭
211 1059.41 v11C♯, v4D♭
212 1064.44 37/20 v10C♯, v3D♭
213 1069.46 v9C♯, vvD♭
214 1074.48 v8C♯, vD♭
215 1079.5 69/37 v7C♯, D♭
216 1084.52 58/31 v6C♯, ^D♭
217 1089.54 v5C♯, ^^D♭
218 1094.56 32/17 v4C♯, ^3D♭
219 1099.58 v3C♯, ^4D♭
220 1104.6 70/37 vvC♯, ^5D♭
221 1109.62 74/39 vC♯, ^6D♭
222 1114.64 40/21 C♯, ^7D♭
223 1119.67 21/11 ^C♯, ^8D♭
224 1124.69 ^^C♯, ^9D♭
225 1129.71 48/25 ^3C♯, ^10D♭
226 1134.73 77/40 ^4C♯, ^11D♭
227 1139.75 56/29 ^5C♯, v12D
228 1144.77 31/16 ^6C♯, v11D
229 1149.79 68/35 ^7C♯, v10D
230 1154.81 76/39 ^8C♯, v9D
231 1159.83 43/22 ^9C♯, v8D
232 1164.85 49/25 ^10C♯, v7D
233 1169.87 57/29 ^11C♯, v6D
234 1174.9 69/35 ^12C♯, v5D
235 1179.92 v11C𝄪, v4D
236 1184.94 v10C𝄪, v3D
237 1189.96 v9C𝄪, vvD
238 1194.98 v8C𝄪, vD
239 1200 2/1 D

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [379 -239⟩ [⟨239 379]] −0.307 0.307 6.12
2.3.5 [3 -18 11⟩, [32 -7 -9⟩ [⟨239 379 555]] −0.247 0.265 5.27
2.3.5.7 2401/2400, 5120/5103, 29360128/29296875 [⟨239 379 555 671]] −0.204 0.241 4.80
2.3.5.7.11 2401/2400, 3025/3024, 4000/3993, 5120/5103 [⟨239 379 555 671 827]] −0.220 0.218 4.34

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 3\239 15.06 121/120 Yarman I (239)
1 7\239 35.15 1990656/1953125 Gammic (5-limit)
1 9\239 45.19 250/243 Quartonic (5-limit)
1 11\239 55.23 33/32 Escapade / alphaquarter
1 35\239 175.73 72/65 Quadrafifths (239f)
1 54\239 271.13 90/77 Quasiorwell (239)
1 70\239 351.46 49/40 Hemififths (7-limit)
1 79\239 396.65 44/35 Squarschmidt
1 83\239 416.74 14/11 Unthirds (239f)
1 116\239 582.43 7/5 Neptune (7-limit)

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

Music

Francium
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