Table of 270edo intervals

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This table of 270edo intervals assumes the 31-limit patent val of 270edo ⟨270 428 647 758 934 999 1104 1147 1221 1312 1338]. Prime harmonics are in bold; inconsistently mapped intervals are in italic. As is analysed in the main article, besides being a strong 13-limit system, 270edo can be used in the full 23-limit with a handful of inconsistencies into the 27-odd-limit, or as a no-17 no-23 31-limit system, fully consistent to the 31-odd-limit. For this reason, intervals of 17 and 23 are given in parentheses.

# Cents Marks 5-limit 7-limit 11-limit 13-limit 23-limit 31-limit
0 0.0 P1 1/1 " " " " "
1 4.4 32805/32768 3136/3125 385/384 " " "
2 8.8 15625/15552 225/224 176/175 196/195 " "
3 13.3 78732/78125 126/125 " " " "
4 17.7 2048/2025 " 99/98 " " "
5 22.2 81/80 " " 78/77 " "
6 26.6 20000/19683 64/63 " 65/64 " "
7 31.1 3125/3072 " 56/55 " " "
8 35.5 1594323/1562500 49/48 " " " "
9 40.0 128/125 " 45/44 " " "
10 44.4 6561/6400 525/512 77/75 40/39 39/38 "
11 48.8 250/243 36/35 " " " "
12 53.3 16875/16384 4096/3969 33/32 " " 32/31
13 57.7 sA1 262144/253125 405/392 125/121 91/88 " 30/29
14 62.2 648/625 28/27 " " " "
15 66.6 20480/19683 8192/7875 80/77 26/25 " "
16 71.1 25/24 " " " " "
17 75.5 273375/262144 256/245 " 117/112 (23/22) "
18 80.0 16384/15625 1875/1792 22/21 " " "
19 84.4 6561/6250 21/20 " " " "
20 88.8 m2 256/243 " 81/77 " 20/19 "
21 93.3 135/128 " 132/125 96/91 19/18 "
22 97.7 62500/59049 200/189 128/121 55/52 (18/17) "
23 102.2 78125/73728 3584/3375 35/33 " " "
24 106.6 524288/492075 625/588 " 117/110 (17/16) 33/31
25 111.1 16/15 " " " " "
26 115.5 A1 2187/2048 " 77/72 " " 31/29
27 120.0 3125/2916 15/14 " " " "
28 124.4 140625/131072 672/625 189/176 130/121 (102/95) 29/27
29 128.8 32768/30375 2205/2048 264/245 14/13 " "
30 133.3 27/25 " " " " "
31 137.7 64000/59049 1024/945 250/231 13/12 " "
32 142.2 625/576 243/224 88/81 " 38/35 "
33 146.6 n2 2125764/1953125 49/45 " " (25/23) "
34 151.1 2048/1875 " 12/11 " " "
35 155.5 2187/2000 35/32 " " (23/21) "
36 160.0 800/729 192/175 " 169/154 (56/51) (34/31)
37 164.4 1125/1024 " 11/10 " " "
38 168.8 390625/354294 54/49 " " " 32/29
39 173.3 3456/3125 448/405 243/220 " 21/19 "
40 177.7 65536/59049 567/512 256/231 72/65 " 31/28
41 182.2 10/9 " " " " "
42 186.6 18225/16384 4096/3675 49/44 39/35 " "
43 191.1 78125/69984 125/112 " " (19/17) 29/26
44 195.5 17496/15625 28/25 " " " "
45 200.0 4096/3645 " 55/49 " " "
46 204.4 M2 9/8 " " " " "
47 208.8 200000/177147 640/567 " 44/39 " 35/31
48 213.3 15625/13824 2025/1792 112/99 " (26/23) "
49 217.7 177147/156250 245/216 " 143/126 (17/15) "
50 222.2 256/225 " 25/22 " " "
51 226.6 729/640 " 154/135 " 57/50 "
52 231.1 2500/2187 8/7 " " " "
53 235.5 sA2 9375/8192 3584/3125 55/48 " (39/34) "
54 240.0 524288/455625 147/128 " " (23/20) 31/27
55 244.4 144/125 " 121/105 " 38/33 "
56 248.8 59049/51200 4725/4096 231/200 15/13 " "
57 253.3 125/108 81/70 " " 22/19 "
58 257.7 151875/131072 512/441 297/256 65/56 " 29/25
59 262.2 32768/28125 3125/2688 64/55 " 57/49 "
60 266.6 729/625 7/6 " " " "
61 271.1 2560/2187 1024/875 90/77 " 76/65 "
62 275.5 75/64 " " " " (34/29)
63 280.0 625000/531441 147/125 " " (20/17) "
64 284.4 18432/15625 7168/6075 33/28 " " "
65 288.8 59049/50000 189/160 " 13/11 " "
66 293.3 m3 32/27 " " " " "
67 297.7 1215/1024 " 196/165 108/91 19/16 "
68 302.2 15625/13122 25/21 " " " "
69 306.6 78125/65536 448/375 105/88 " (68/57) 31/26
70 311.1 65536/54675 1225/1024 176/147 140/117 91/76 "
71 315.5 6/5 " " " " "
72 320.0 19683/16384 2048/1701 77/64 65/54 " "
73 324.4 3125/2592 135/112 " " 76/63 35/29
74 328.8 472392/390625 98/81 " " (23/19) 29/24
75 333.3 4096/3375 " 40/33 " " "
76 337.7 243/200 175/144 147/121 " (17/14) "
77 342.2 8000/6561 128/105 " 39/32 (28/23) "
78 346.6 625/512 " 11/9 " " "
79 351.1 n3 1953125/1594323 49/40 " " " 38/31
80 355.5 768/625 " 27/22 " " "
81 360.0 19683/16000 315/256 154/125 16/13 " "
82 364.4 100/81 " " " (21/17) "
83 368.8 10125/8192 8192/6615 99/80 26/21 " "
84 373.3 390625/314928 243/196 150/121 " (57/46) 31/25
85 377.7 3888/3125 56/45 " " " "
86 382.2 8192/6561 5103/4096 96/77 81/65 " "
87 386.6 5/4 " " " " "
88 391.1 164025/131072 784/625 441/352 351/280 (64/51) "
89 395.5 78125/62208 1125/896 44/35 " " "
90 400.0 19683/15625 63/50 " " (34/27) 39/31
91 404.4 512/405 " 125/99 91/72 24/19 "
92 408.8 M3 81/64 " " " 19/15 "
93 413.3 25000/19683 80/63 " 33/26 " "
94 417.7 15625/12288 7168/5625 14/11 " " "
95 422.2 1048576/820125 125/98 " " (23/18) "
96 426.6 32/25 " " " " "
97 431.1 6561/5120 2625/2048 77/60 50/39 " "
98 435.5 625/486 9/7 " " " "
99 440.0 sd4 84375/65536 1568/1215 165/128 156/121 49/38 40/31
100 444.4 65536/50625 1323/1024 128/99 84/65 (22/17) 31/24
101 448.8 162/125 35/27 " " " "
102 453.3 25600/19683 2048/1575 100/77 13/10 " "
103 457.7 125/96 " " " 99/76 "
104 462.2 1366875/1048576 64/49 " " (30/23) "
105 466.6 4096/3125 " 55/42 " (17/13) 38/29
106 471.1 6561/5000 21/16 " " " "
107 475.5 320/243 " " 154/117 25/19 "
108 480.0 675/512 " 33/25 " " 29/22
109 484.4 78125/59049 250/189 160/121 143/108 (45/34) "
110 488.8 20736/15625 896/675 175/132 65/49 " "
111 493.3 131072/98415 1701/1280 512/385 117/88 (85/64) "
112 497.7 P4 4/3 " " " " "
113 502.2 10935/8192 8192/6125 147/110 " (91/68) "
114 506.6 15625/11664 75/56 " " " "
115 511.1 104976/78125 168/125 121/90 " (51/38) 39/29
116 515.5 8192/6075 " 66/49 35/26 " "
117 520.0 27/20 " " " (23/17) "
118 524.4 80000/59049 256/189 " 65/48 " 42/31
119 528.8 3125/2304 1215/896 110/81 " 19/14 "
120 533.3 531441/390625 49/36 " " (34/25) "
121 537.7 512/375 " 15/11 " " "
122 542.2 2187/1600 175/128 " 160/117 26/19 "
123 546.6 1000/729 48/35 " " " "
124 551.1 5625/4096 " 11/8 " " "
125 555.5 sA4 1048576/759375 135/98 " 91/66 (69/50) 40/29
126 560.0 864/625 112/81 " " 105/76 29/21
127 564.4 81920/59049 2835/2048 320/231 18/13 " "
128 568.8 25/18 " " " " "
129 573.3 91125/65536 1024/735 245/176 39/28 (32/23) "
130 577.7 65536/46875 625/448 88/63 " " "
131 582.2 4374/3125 7/5 " " " "
132 586.6 d5 1024/729 " 108/77 " 80/57 "
133 591.1 45/32 " " " 38/27 "
134 595.5 250000/177147 343/243 " 55/39 (24/17) 31/22
135 600.0 78125/55296 10125/7168 99/70 " " "
136 604.4 177147/125000 486/343 343/242 78/55 (17/12) 44/31
137 608.8 64/45 " " " 27/19 "
138 613.3 A4 729/512 " 77/54 " 57/40 "
139 617.7 3125/2187 10/7 " " " "
140 622.2 46875/32768 896/625 63/44 " " "
141 626.6 131072/91125 735/512 352/245 56/39 (23/16) "
142 631.1 36/25 " " " " "
143 635.5 59049/40960 4096/2835 231/160 13/9 " "
144 640.0 625/432 81/56 " " " 42/29
145 644.4 sd5 759375/524288 196/135 " 132/91 (100/69) 29/20
146 648.8 8192/5625 " 16/11 " " "
147 653.3 729/500 35/24 " " " "
148 657.7 3200/2187 256/175 225/154 117/80 19/13 "
149 662.2 375/256 " 22/15 " " "
150 666.6 781250/531441 72/49 " " (25/17) "
151 671.1 4608/3125 1792/1215 81/55 " 28/19 "
152 675.5 59049/40000 189/128 " 65/44 " 31/21
153 680.0 40/27 " " " (34/23) "
154 684.4 6075/4096 " 49/33 " " "
155 688.8 78125/52488 125/84 " " (76/51) 58/39
156 693.3 23328/15625 112/75 " " (85/57) "
157 697.7 16384/10935 6125/4096 220/147 175/117 (136/91) "
158 702.2 P5 3/2 " " " " "
159 706.6 98415/65536 2560/1701 385/256 176/117 (128/85) "
160 711.1 15625/10368 675/448 264/175 98/65 95/63 "
161 715.5 118098/78125 189/125 121/80 " (68/45) "
162 720.0 1024/675 " 50/33 " " 44/29
163 724.4 243/160 " " 117/77 38/25 "
164 728.8 10000/6561 32/21 " " " "
165 733.3 3125/2048 " 55/36 " (26/17) 29/19
166 737.7 2097152/1366875 49/32 " " (23/15) "
167 742.2 192/125 " 135/88 " " "
168 746.6 19683/12800 1575/1024 77/50 20/13 " "
169 751.1 125/81 54/35 " " " "
170 755.5 50625/32768 2048/1323 99/64 65/42 (17/11) 48/31
171 760.0 sA5 131072/84375 1215/784 256/165 121/78 76/49 31/20
172 764.4 972/625 14/9 " " " "
173 768.8 10240/6561 4096/2625 120/77 39/25 " "
174 773.3 25/16 " " " " "
175 777.7 820125/524288 196/125 " " (36/23) "
176 782.2 24576/15625 5625/3584 11/7 " " "
177 786.6 19683/12500 63/40 " 52/33 " "
178 791.1 m6 128/81 " " " 30/19 "
179 795.5 405/256 " 198/125 144/91 19/12 "
180 800.0 31250/19683 100/63 " " (27/17) 62/39
181 804.4 78125/49152 1792/1125 35/22 " " "
182 808.8 262144/164025 625/392 " 351/220 (51/32) "
183 813.3 8/5 " " " " "
184 817.7 6561/4096 " 77/48 " " "
185 822.2 3125/1944 45/28 " " " "
186 826.6 421875/262144 392/243 121/75 " (92/57) 29/18
187 831.1 16384/10125 6615/4096 160/99 21/13 " "
188 835.5 81/50 " " " (34/21) "
189 840.0 32000/19683 512/315 125/77 13/8 " "
190 844.4 625/384 " 44/27 " " "
191 848.8 n6 3188646/1953125 49/30 " " " 31/19
192 853.3 1024/625 " 18/11 " " "
193 857.7 6561/4000 105/64 " 64/39 (23/14) "
194 862.2 400/243 288/175 242/147 " (28/17) "
195 866.6 3375/2048 " 33/20 " " "
196 871.1 390625/236196 81/49 " " (38/23) 48/29
197 875.5 5184/3125 224/135 " " 63/38 58/35
198 880.0 32768/19683 1701/1024 128/77 108/65 " "
199 884.4 5/3 " " " " "
200 888.8 54675/32768 2048/1225 147/88 117/70 " "
201 893.3 78125/46656 375/224 176/105 " (57/34) 52/31
202 897.7 26244/15625 42/25 " " " "
203 902.2 2048/1215 " 165/98 91/54 32/19 "
204 906.6 M6 27/16 " " " " "
205 911.1 100000/59049 320/189 " 22/13 " "
206 915.5 15625/9216 6075/3584 56/33 " (39/23) "
207 920.0 531441/312500 245/144 " 143/84 (17/10) "
208 924.4 128/75 " 75/44 " " (29/17)
209 928.8 2187/1280 875/512 77/45 " 65/38 "
210 933.3 1250/729 12/7 " " " "
211 937.7 sd7 28125/16384 5376/3125 55/32 " " "
212 942.2 262144/151875 441/256 " 112/65 (69/40) 31/18
213 946.6 216/125 140/81 121/70 " 19/11 "
214 951.1 102400/59049 8192/4725 343/198 26/15 " "
215 955.5 125/72 " " " 33/19 "
216 960.0 455625/262144 256/147 " 195/112 (40/23) 54/31
217 964.4 16384/9375 3125/1792 96/55 " (68/39) "
218 968.8 2187/1250 7/4 " " " "
219 973.3 1280/729 " 135/77 " 100/57 "
220 977.7 225/128 " 44/25 " " "
221 982.2 312500/177147 432/245 " 143/81 (30/17) "
222 986.6 27648/15625 3584/2025 99/56 " (23/13) "
223 991.1 177147/100000 567/320 " 39/22 " "
224 995.5 m7 16/9 " " " " "
225 1000.0 3645/2048 " 98/55 " 57/32 "
226 1004.4 15625/8748 25/14 " " " "
227 1008.8 139968/78125 224/125 " " (34/19) 52/29
228 1013.3 32768/18225 3675/2048 88/49 70/39 " "
229 1017.7 9/5 " " " " "
230 1022.2 59049/32768 1024/567 231/128 65/36 " 56/31
231 1026.6 3125/1728 405/224 " " 38/21 "
232 1031.1 708588/390625 49/27 " " " 29/16
233 1035.5 2048/1125 " 20/11 " " "
234 1040.0 729/400 175/96 " " (51/28) (31/17)
235 1044.4 4000/2187 64/35 " " (42/23) "
236 1048.8 1875/1024 " 11/6 " " "
237 1053.3 n7 1953125/1062882 90/49 " " (46/25) "
238 1057.7 1152/625 448/243 81/44 " 35/19 "
239 1062.2 59049/32000 945/512 231/125 24/13 " "
240 1066.6 50/27 " " " " "
241 1071.1 30375/16384 4096/2205 245/132 13/7 " "
242 1075.5 262144/140625 625/336 225/121 121/65 (95/51) 54/29
243 1080.0 5832/3125 28/15 " " " "
244 1084.4 4096/2187 " 144/77 " " 58/31
245 1088.8 15/8 " " " " "
246 1093.3 492075/262144 1176/625 847/450 220/117 (32/17) 62/33
247 1097.7 78125/41472 3375/1792 66/35 49/26 " "
248 1102.2 59049/31250 189/100 121/64 104/55 (17/9) "
249 1106.6 256/135 " 125/66 91/48 36/19 "
250 1111.1 M7 243/128 " 154/81 " 19/10 "
251 1115.5 12500/6561 40/21 " " " "
252 1120.0 15625/8192 3584/1875 21/11 " " "
253 1124.4 524288/273375 245/128 " 224/117 (23/12) "
254 1128.8 48/25 " " " " "
255 1133.3 19683/10240 7875/4096 77/40 25/13 " "
256 1137.7 625/324 27/14 " " " "
257 1142.2 sd8 253125/131072 784/405 242/125 176/91 (85/44) 29/15
258 1146.6 32768/16875 3969/2048 64/33 " (33/17) 31/16
259 1151.1 243/125 35/18 " " " "
260 1155.5 12800/6561 1024/525 150/77 39/20 " "
261 1160.0 125/64 " 88/45 " " "
262 1164.4 3125000/1594323 49/25 " " (45/23) "
263 1168.8 6144/3125 " 55/28 " (51/26) "
264 1173.3 19683/10000 63/32 " " " "
265 1177.7 160/81 " " 77/39 75/38 "
266 1182.2 2025/1024 " 99/50 " (91/46) 87/44
267 1186.6 78125/39366 125/63 " " " "
268 1191.1 31104/15625 448/225 175/88 " " "
269 1195.5 65536/32805 3125/1568 539/270 351/176 (255/128) "
270 1200.0 P8 2/1 " " " " "
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