58edo: Difference between revisions

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→Intervals: Removed pions column
Line 16: Line 16:
| | degree of 58edo
| | degree of 58edo
| | cents value
| | cents value
|pions
|7mus
|7mus
| | ratios
| | ratios
|-
|-
| colspan="4"| 0
| colspan="3" | 0
| | 1/1
| | 1/1
|-
|-
| | 1
| | 1
| | 20.69
| | 20.69
|21.93
|26.48 (1A.7C<sub>16</sub>)
|26.48 (1A.7C<sub>16</sub>)
| | 56/55, 64/63, 81/80, 128/125
| | 56/55, 64/63, 81/80, 128/125
Line 31: Line 29:
| | 2
| | 2
| | 41.38
| | 41.38
|43.86
|52.97 (34.F7<sub>16</sub>)
|52.97 (34.F7<sub>16</sub>)
| | 36/35, 49/48, 50/49, 55/54
| | 36/35, 49/48, 50/49, 55/54
Line 37: Line 34:
| | 3
| | 3
| | 62.07
| | 62.07
|65.79
|79.45 (4F.72<sub>16</sub>)
|79.45 (4F.72<sub>16</sub>)
| | 25/24, 26/25, 27/26, 28/27, 33/32
| | 25/24, 26/25, 27/26, 28/27, 33/32
Line 43: Line 39:
| | 4
| | 4
| | 82.76
| | 82.76
|87.72
|105.93 (69.EE<sub>16</sub>)
|105.93 (69.EE<sub>16</sub>)
| | 21/20, 22/21
| | 21/20, 22/21
Line 49: Line 44:
| | 5
| | 5
| | 103.45
| | 103.45
|109.655
|132.41 (84.69<sub>16</sub>)
|132.41 (84.69<sub>16</sub>)
| | 16/15, 17/16, 18/17
| | 16/15, 17/16, 18/17
Line 55: Line 49:
| | 6
| | 6
| | 124.14
| | 124.14
|131.59
|158.9 (9E.E58<sub>16</sub>)
|158.9 (9E.E58<sub>16</sub>)
| | 14/13, 15/14, 27/25
| | 14/13, 15/14, 27/25
Line 61: Line 54:
| | 7
| | 7
| | 144.83
| | 144.83
|153.52
|185.38 (B9.61<sub>16</sub>)
|185.38 (B9.61<sub>16</sub>)
| | 12/11, 13/12
| | 12/11, 13/12
Line 67: Line 59:
| | 8
| | 8
| | 165.52
| | 165.52
|175.45
|211.86 (D3.DD<sub>16</sub>)
|211.86 (D3.DD<sub>16</sub>)
| | 11/10
| | 11/10
Line 73: Line 64:
| | 9
| | 9
| | 186.21
| | 186.21
|197.38
|238.345 (EE.58<sub>16</sub>)
|238.345 (EE.58<sub>16</sub>)
| | 10/9
| | 10/9
Line 79: Line 69:
| | 10
| | 10
| | 206.9
| | 206.9
|219.31
|264.83 (108.D3<sub>16</sub>)
|264.83 (108.D3<sub>16</sub>)
| | 9/8, 17/15
| | 9/8, 17/15
Line 85: Line 74:
| | 11
| | 11
| | 227.59
| | 227.59
|241.24
|291.31 (123.4F<sub>16</sub>)
|291.31 (123.4F<sub>16</sub>)
| | 8/7
| | 8/7
Line 91: Line 79:
| | 12
| | 12
| | 248.28
| | 248.28
|263.17
|317.79 (13D.CB<sub>16</sub>)
|317.79 (13D.CB<sub>16</sub>)
| | 15/13
| | 15/13
Line 97: Line 84:
| | 13
| | 13
| | 268.97
| | 268.97
|285.1
|344.28 (158.47<sub>16</sub>)
|344.28 (158.47<sub>16</sub>)
| | 7/6
| | 7/6
Line 103: Line 89:
| | 14
| | 14
| | 289.655
| | 289.655
|307.035
|370.76 (172.C2<sub>16</sub>)
|370.76 (172.C2<sub>16</sub>)
| | 13/11, 20/17
| | 13/11, 20/17
Line 109: Line 94:
| | 15
| | 15
| | 310.345
| | 310.345
|328.97
|397.24 (18D.3E<sub>16</sub>)
|397.24 (18D.3E<sub>16</sub>)
| | 6/5
| | 6/5
Line 115: Line 99:
| | 16
| | 16
| | 331.035
| | 331.035
|350.9
|423.72 (1A7.B9<sub>16</sub>)
|423.72 (1A7.B9<sub>16</sub>)
| | 17/14
| | 17/14
Line 121: Line 104:
| | 17
| | 17
| | 351.72
| | 351.72
|372.83
|450.21 (1C2.35<sub>16</sub>)
|450.21 (1C2.35<sub>16</sub>)
| | 11/9, 16/13
| | 11/9, 16/13
Line 127: Line 109:
| | 18
| | 18
| | 372.41
| | 372.41
|394.76
|476.69 (1DC.B1<sub>16</sub>)
|476.69 (1DC.B1<sub>16</sub>)
| | 21/17
| | 21/17
Line 133: Line 114:
| | 19
| | 19
| | 393.1
| | 393.1
|416.69
|503.17 (1F7.2C<sub>16</sub>)
|503.17 (1F7.2C<sub>16</sub>)
| | 5/4
| | 5/4
Line 139: Line 119:
| | 20
| | 20
| | 413.79
| | 413.79
|438.62
|529.655 (211.A7<sub>16</sub>)
|529.655 (211.A7<sub>16</sub>)
| | 14/11
| | 14/11
Line 145: Line 124:
| | 21
| | 21
| | 434.48
| | 434.48
|460.55
|556.14 (22C.23<sub>16</sub>)
|556.14 (22C.23<sub>16</sub>)
| | 9/7
| | 9/7
Line 151: Line 129:
| | 22
| | 22
| | 455.17
| | 455.17
|482.48
|582.62 (246.9F<sub>16</sub>)
|582.62 (246.9F<sub>16</sub>)
| | 13/10, 17/13, 22/17
| | 13/10, 17/13, 22/17
Line 157: Line 134:
| | 23
| | 23
| | 475.86
| | 475.86
|504.41
|609.1 (261.1A8<sub>16</sub>)
|609.1 (261.1A8<sub>16</sub>)
| | 21/16
| | 21/16
Line 163: Line 139:
| | 24
| | 24
| | 496.55
| | 496.55
|526.345
|635.59 (27B.96<sub>16</sub>)
|635.59 (27B.96<sub>16</sub>)
| | 4/3
| | 4/3
Line 169: Line 144:
| | 25
| | 25
| | 517.24
| | 517.24
|548.28
|662.07 (296.12<sub>16</sub>)
|662.07 (296.12<sub>16</sub>)
| | 27/20
| | 27/20
Line 175: Line 149:
| | 26
| | 26
| | 537.93
| | 537.93
|570.21
|688.55 (2B0.5D<sub>16</sub>)
|688.55 (2B0.5D<sub>16</sub>)
| | 15/11
| | 15/11
Line 181: Line 154:
| | 27
| | 27
| | 558.62
| | 558.62
|592.14
|715.035 (2CB.09<sub>16</sub>)
|715.035 (2CB.09<sub>16</sub>)
| | 11/8, 18/13
| | 11/8, 18/13
Line 187: Line 159:
| | 28
| | 28
| | 579.31
| | 579.31
|614.07
|741.52 (2F5.84<sub>16</sub>)
|741.52 (2F5.84<sub>16</sub>)
| | 7/5
| | 7/5
Line 193: Line 164:
| | 29
| | 29
| | 600
| | 600
|636
|768 (300<sub>16</sub>)
|768 (300<sub>16</sub>)
| | 17/12, 24/17
| | 17/12, 24/17
Line 199: Line 169:
| | 30
| | 30
| | 620.69
| | 620.69
|657.93
|794.48 (31A.7C<sub>16</sub>)
|794.48 (31A.7C<sub>16</sub>)
| | 10/7
| | 10/7
Line 205: Line 174:
| | 31
| | 31
| | 641.38
| | 641.38
|679.86
|820.97 (334.F7<sub>16</sub>)
|820.97 (334.F7<sub>16</sub>)
| | 13/9, 16/11
| | 13/9, 16/11
Line 211: Line 179:
| | 32
| | 32
| | 662.07
| | 662.07
|701.79
|847.45 (34F.72<sub>16</sub>)
|847.45 (34F.72<sub>16</sub>)
| | 22/15
| | 22/15
Line 217: Line 184:
| | 33
| | 33
| | 682.76
| | 682.76
|723.72
|873.93 (369.EE<sub>16</sub>)
|873.93 (369.EE<sub>16</sub>)
| | 40/27
| | 40/27
Line 223: Line 189:
| | 34
| | 34
| | 703.45
| | 703.45
|745.655
|900.41 (384.69<sub>16</sub>)
|900.41 (384.69<sub>16</sub>)
| | 3/2
| | 3/2
Line 229: Line 194:
| | 35
| | 35
| | 724.14
| | 724.14
|767.59
|926.9 (39E.E58<sub>16</sub>)
|926.9 (39E.E58<sub>16</sub>)
| | 32/21
| | 32/21
Line 235: Line 199:
| | 36
| | 36
| | 744.83
| | 744.83
|789.52
|953.38 (3B9.61<sub>16</sub>)
|953.38 (3B9.61<sub>16</sub>)
| | 20/13, 26/17, 17/11
| | 20/13, 26/17, 17/11
Line 241: Line 204:
| | 37
| | 37
| | 765.52
| | 765.52
|811.45
|979.86 (3D3.DD<sub>16</sub>)
|979.86 (3D3.DD<sub>16</sub>)
| | 14/9
| | 14/9
Line 247: Line 209:
| | 38
| | 38
| | 786.21
| | 786.21
|833.38
|1006.345 (3EE.58<sub>16</sub>)
|1006.345 (3EE.58<sub>16</sub>)
| | 11/7
| | 11/7
Line 253: Line 214:
| | 39
| | 39
| | 806.9
| | 806.9
|855.31
|1032.83 (408.D3<sub>16</sub>)
|1032.83 (408.D3<sub>16</sub>)
| | 8/5
| | 8/5
Line 259: Line 219:
| | 40
| | 40
| | 827.59
| | 827.59
|877.24
|1059.31 (423.4F<sub>16</sub>)
|1059.31 (423.4F<sub>16</sub>)
| | 34/21
| | 34/21
Line 265: Line 224:
| | 41
| | 41
| | 848.28
| | 848.28
|899.17
|1085.79 (43D.CB<sub>16</sub>)
|1085.79 (43D.CB<sub>16</sub>)
| | 13/8, 18/11
| | 13/8, 18/11
Line 271: Line 229:
| | 42
| | 42
| | 868.97
| | 868.97
|921.1
|1112.28 (458.47<sub>16</sub>)
|1112.28 (458.47<sub>16</sub>)
| | 28/17
| | 28/17
Line 277: Line 234:
| | 43
| | 43
| | 889.655
| | 889.655
|943.035
|1138.76 (472.C2<sub>16</sub>)
|1138.76 (472.C2<sub>16</sub>)
| | 5/3
| | 5/3
Line 283: Line 239:
| | 44
| | 44
| | 910.345
| | 910.345
|964.97
|1165.24 (48D.3E<sub>16</sub>)
|1165.24 (48D.3E<sub>16</sub>)
| | 22/13, 17/10
| | 22/13, 17/10
Line 289: Line 244:
| | 45
| | 45
| | 931.035
| | 931.035
|986.9
|1191.72 (4A7.B9<sub>16</sub>)
|1191.72 (4A7.B9<sub>16</sub>)
| | 12/7
| | 12/7
Line 295: Line 249:
| | 46
| | 46
| | 951.72
| | 951.72
|1008.83
|1218.21 (4C2.35<sub>16</sub>)
|1218.21 (4C2.35<sub>16</sub>)
| | 26/15
| | 26/15
Line 301: Line 254:
| | 47
| | 47
| | 972.41
| | 972.41
|1030.76
|1244.69 (4DC.B1<sub>16</sub>)
|1244.69 (4DC.B1<sub>16</sub>)
| | 7/4
| | 7/4
Line 307: Line 259:
| | 48
| | 48
| | 993.1
| | 993.1
|1052.69
|1271.17 (4F7.2C<sub>16</sub>)
|1271.17 (4F7.2C<sub>16</sub>)
| | 16/9
| | 16/9
Line 313: Line 264:
| | 49
| | 49
| | 1013.79
| | 1013.79
|1074.62
|1297.655 (511.A7<sub>16</sub>)
|1297.655 (511.A7<sub>16</sub>)
| | 9/5
| | 9/5
Line 319: Line 269:
| | 50
| | 50
| | 1034.48
| | 1034.48
|1096.55
|1324.14 (52C.23<sub>16</sub>)
|1324.14 (52C.23<sub>16</sub>)
| | 20/11
| | 20/11
Line 325: Line 274:
| | 51
| | 51
| | 1055.17
| | 1055.17
|1118.48
|1350.62 (546.9F<sub>16</sub>)
|1350.62 (546.9F<sub>16</sub>)
| | 11/6, 24/13
| | 11/6, 24/13
Line 331: Line 279:
| | 52
| | 52
| | 1075.86
| | 1075.86
|1140.41
|1377.1 (561.1A8<sub>16</sub>)
|1377.1 (561.1A8<sub>16</sub>)
| | 13/7, 28/15
| | 13/7, 28/15
Line 337: Line 284:
| | 53
| | 53
| | 1096.55
| | 1096.55
|1162.345
|1403.59 (57B.96<sub>16</sub>)
|1403.59 (57B.96<sub>16</sub>)
| | 15/8, 32/17, 17/9
| | 15/8, 32/17, 17/9
Line 343: Line 289:
| | 54
| | 54
| | 1117.24
| | 1117.24
|1184.28
|1430.07 (596.12<sub>16</sub>)
|1430.07 (596.12<sub>16</sub>)
| | 40/21, 21/11
| | 40/21, 21/11
Line 349: Line 294:
| | 55
| | 55
| | 1137.93
| | 1137.93
|1206.21
|1456.55 (5B0.5D<sub>16</sub>)
|1456.55 (5B0.5D<sub>16</sub>)
| |48/25, 25/13, 52/27, 27/14, 64/33
| |48/25, 25/13, 52/27, 27/14, 64/33
Line 355: Line 299:
| |56
| |56
| | 1158.62
| | 1158.62
|1228.14
|1483.035 (5CB.09<sub>16</sub>)
|1483.035 (5CB.09<sub>16</sub>)
| |35/18, 96/49, 49/25, 108/55  
| |35/18, 96/49, 49/25, 108/55  
Line 361: Line 304:
| |57
| |57
| | 1179.31
| | 1179.31
|1250.07
|1509.52 (5F5.84<sub>16</sub>)
|1509.52 (5F5.84<sub>16</sub>)
| |55/28, 63/32, 160/81, 125/64
| |55/28, 63/32, 160/81, 125/64

Revision as of 13:20, 12 December 2019

The 58 equal temperament, often abbreviated 58-tET, 58-EDO, or 58-ET, is the scale derived by dividing the octave into 58 equally-sized steps. Each step represents a frequency ratio of 20.69 cents. It tempers out 2048/2025, 126/125, 1728/1715, 144/143, 176/175, 896/891, 243/242, 5120/5103, 351/350, 364/363, 441/440, and 540/539, and is a strong system in the 11, 13 and 17-limits. It is the smallest equal temperament which is consistent through the 17-limit, and is also the first et to map the entire 11-limit tonality diamond to distinct scale steps, and hence the first et which can define a version of the famous 43-note Genesis scale of Harry Partch. It supports hemififths, myna, diaschismic, harry, mystery, buzzard and thuja temperaments, and supplies the optimal patent val for 7-, 11- and 13-limit diaschismic, 11- and 13-limit hemififths, 11- and 13-limit thuja, and 13-limit myna. It also supplies the optimal patent val for the 13-limit rank three temperaments thrush, bluebird, aplonis and jofur.

While the 17th harmonic is a cent and a half flat, the harmonics below it are all a little sharp, giving it the sound of a sharp system. 58 = 2*29, and 58 shares the same excellent fifth with 29edo.

Scales

hemif7

hemif10

hemif17

Intervals

degree of 58edo cents value 7mus ratios
0 1/1
1 20.69 26.48 (1A.7C16) 56/55, 64/63, 81/80, 128/125
2 41.38 52.97 (34.F716) 36/35, 49/48, 50/49, 55/54
3 62.07 79.45 (4F.7216) 25/24, 26/25, 27/26, 28/27, 33/32
4 82.76 105.93 (69.EE16) 21/20, 22/21
5 103.45 132.41 (84.6916) 16/15, 17/16, 18/17
6 124.14 158.9 (9E.E5816) 14/13, 15/14, 27/25
7 144.83 185.38 (B9.6116) 12/11, 13/12
8 165.52 211.86 (D3.DD16) 11/10
9 186.21 238.345 (EE.5816) 10/9
10 206.9 264.83 (108.D316) 9/8, 17/15
11 227.59 291.31 (123.4F16) 8/7
12 248.28 317.79 (13D.CB16) 15/13
13 268.97 344.28 (158.4716) 7/6
14 289.655 370.76 (172.C216) 13/11, 20/17
15 310.345 397.24 (18D.3E16) 6/5
16 331.035 423.72 (1A7.B916) 17/14
17 351.72 450.21 (1C2.3516) 11/9, 16/13
18 372.41 476.69 (1DC.B116) 21/17
19 393.1 503.17 (1F7.2C16) 5/4
20 413.79 529.655 (211.A716) 14/11
21 434.48 556.14 (22C.2316) 9/7
22 455.17 582.62 (246.9F16) 13/10, 17/13, 22/17
23 475.86 609.1 (261.1A816) 21/16
24 496.55 635.59 (27B.9616) 4/3
25 517.24 662.07 (296.1216) 27/20
26 537.93 688.55 (2B0.5D16) 15/11
27 558.62 715.035 (2CB.0916) 11/8, 18/13
28 579.31 741.52 (2F5.8416) 7/5
29 600 768 (30016) 17/12, 24/17
30 620.69 794.48 (31A.7C16) 10/7
31 641.38 820.97 (334.F716) 13/9, 16/11
32 662.07 847.45 (34F.7216) 22/15
33 682.76 873.93 (369.EE16) 40/27
34 703.45 900.41 (384.6916) 3/2
35 724.14 926.9 (39E.E5816) 32/21
36 744.83 953.38 (3B9.6116) 20/13, 26/17, 17/11
37 765.52 979.86 (3D3.DD16) 14/9
38 786.21 1006.345 (3EE.5816) 11/7
39 806.9 1032.83 (408.D316) 8/5
40 827.59 1059.31 (423.4F16) 34/21
41 848.28 1085.79 (43D.CB16) 13/8, 18/11
42 868.97 1112.28 (458.4716) 28/17
43 889.655 1138.76 (472.C216) 5/3
44 910.345 1165.24 (48D.3E16) 22/13, 17/10
45 931.035 1191.72 (4A7.B916) 12/7
46 951.72 1218.21 (4C2.3516) 26/15
47 972.41 1244.69 (4DC.B116) 7/4
48 993.1 1271.17 (4F7.2C16) 16/9
49 1013.79 1297.655 (511.A716) 9/5
50 1034.48 1324.14 (52C.2316) 20/11
51 1055.17 1350.62 (546.9F16) 11/6, 24/13
52 1075.86 1377.1 (561.1A816) 13/7, 28/15
53 1096.55 1403.59 (57B.9616) 15/8, 32/17, 17/9
54 1117.24 1430.07 (596.1216) 40/21, 21/11
55 1137.93 1456.55 (5B0.5D16) 48/25, 25/13, 52/27, 27/14, 64/33
56 1158.62 1483.035 (5CB.0916) 35/18, 96/49, 49/25, 108/55
57 1179.31 1509.52 (5F5.8416) 55/28, 63/32, 160/81, 125/64

Rank two temperaments

Period Generator Name
1\1 1\58
3\58
5\58
7\58
9\58
11\58 Gorgik
13\58
15\58 Myna
17\58 Hemififths
19\58
21\58
23\58 Buzzard
25\58
27\58 Thuja
1\2 1\58
2\58
3\58
4\58 Harry
5\58 Srutal/Diaschismic
6\58
7\58
8\58 Echidna, Supers
9\58 Secant
10\58
11\58
12\58 Sruti
13\58
14\58
1\29 1\58 Mystery
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