52edt

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The 52 equal division of 3, the tritave, divides it into 52 equal parts of 36.576 cents each, corresponding to 32.808 edo. It is something of a curiosity as it really needs to be considered as a 29-limit no-twos system. While not super-accurate, it gets the entire no-twos 29-limit to within 18 cents. It is distinctly flat, in the sense that 5, 7, 11, 13, 17, 19, 23 and 29 are all flat, so using something other than pure-threes tuning might be advisable. It is contorted in the 11-limit, so that it tempers out the same commas as 26edt in the 11-limit and 13edt in the 7-limit. Other commas it tempers out includes 121/119, 209/207, 247/245, 275/273, 299/297, 325/323, 345/343, 363/361, 377/375, 437/435, 495/493, 627/625, 665/663, 667/665, 847/845, 1127/1125, 1311/1309 and 1617/1615. It is the eleventh no-twos zeta peak edt.

Intervals

Steps Cents Hekts BP nonatonic degree Diatonic degree Corresponding JI intervals Comments Generator for...
1 36.6 25 Qa1/3d2 Sa1 50/49~33/32~49/48
2 73.15 50 Sa1/sd2 A1/dd2 25/24~28/27~22/21~27/26~24/23~21/20~29/28
3 109.7 75 3A1/qd2 A+1/d-2 15/14~16/15~29/27~121/112
4 146.3 100 A1/m2 AA1/sm2 27/25~25/23~49/45~13/12~14/13~11/10~169/162
5 182.9 125 Sm2 sm+2 10/9
6 219.5 150 N2 m2 9/8~8/7~44/39
7 256.0 175 sM2 N2 147/128~7/6
8 292.6 200 M2/d3 M2 32/27~25/21~13/11~27/23
9 329.2 225 Qa2/3d3 SM-2/d-2 6/5 11/9-
10 365.8 250 Sa2/sd3 SM2/dd3 5/4~16/13 11/9+
11 402.3 275 3A3/qd3 SM+2 81/64~63/50~33/26~23/18
12 438.9 300 A2/P3/d4 AA2/sm3 32/25~9/7~14/11~104/81~13/10
13 475.5 325 Qa3/3d4 sm+3 21/16~98/75
14 512.1 350 Sa3/sd4 m3 4/3~27/20~162/121
15 548.6 375 3A3/qd4 N3 11/8~243/169 18/13-
16 585.2 400 A3/m4/d5 M3 7/5~25/18~112/81~88/63~32/23~29/21 18/13+
17 621.8 425 Sm4/3d5 SM-3 10/7~36/25~81/56~63/44~23/16~42/29 13/9-
18 658.4 450 N4/sd5 SM3/dd4 16/11~338/243 13/9+
19 694.95 475 sM4/qd5 SM+3/d-4 3/2~40/27~121/81
20 731.5 500 M4/m5 AA3/d4 32/21~75/49
21 768.1 525 Qa4/Sm5 d+4 25/16~14/9~11/7~81/52
22 804.7 550 Sa4/N5 P4 8/5~36/23
23 841.25 575 3A4/sM5 A-4 13/8
24 877.8 600 A4/M5/d6 A4 5/3
25 914.4 625 Qa5/3d6 A+4 27/16~42/25~22/13~46/27
26 951 650 Sa5/sd6 AA4/dd5 125/72
27 987.55 675 3A5/qd6 d-5 16/9~8/7~39/22~75/46
28 1024.1 700 A5/m6/d7 d5 9/5
29 1060.7 725 Sm6/3d7 d+5 50/27~46/25~90/49~24/13~13/7~20/11
30 1097.3 750 N6/sd7 P5 15/8
31 1133.9 775 sM6/qd7 A-5 48/25~27/14~21/11~52/27~23/12~40/21~56/29
32 1170.4 800 M6/m7 A5/dd6 49/25~64/33~96/49
33 1207.0 825 Qa6/Sm7 A+5 2/1
34 1243.6 850 Sa6/N7 AA5/sm6 33/16~100/49~49/24~729/338
35 1280.2 875 3A6/sM7 sm+6 25/12~56/27~44/21~27/13
36 1316.7 900 A6/M7/d8 m6 15/7~32/15~58/27
37 1353.3 925 Qa7/3d8 N6 54/25~50/23~98/45~13/6~169/81
38 1389.9 950 Sa7/sd8 M6 20/9~9/4-
39 1426.5 975 3A7/qd8 SM-6 9/4+~16/7
40 1463.0 1000 A7/P8/d9 SM6/dd7 147/64~7/3
41 1499.6 1025 Qa8/3d9 SM+6/sm-7 64/27~50/21~26/11~81/23
42 1536.2 1050 Sa8/sd9 AA6/sm7 12/5 22/9-
43 1572.7 1075 3A8/qd9 sm-7 5/2~32/13 22/9+
44 1609.3 1100 A8/m9 m7 81/32~63/25~33/13~23/9
45 1645.9 1125 Sm9 N7 64/45~18/7~28/11~208/81~13/5
46 1682.5 1150 N9 M7 21/8~196/75
47 1719.1 1175 sM9 SM-7 8/3~27/10
48 1755.65 1200 M9/d10 SM7/dd8 69/25~135/49 36/13-
49 1792.2 1225 Qa9/3d10 SM+7/d-8 14/5~25/9~224/81~176/63~64/23~58/21 36/13+
50 1828.8 1250 Sa9/sd10 A7/d8 20/7~72/25~81/28~63/22~23/8~84/29 26/9-
51 1865.4 1275 3A9/qd10 P-8 147/50~32/11~338/81~144/49 26/9+
52 1902.0 1300 A9/P10 P8 3/1 Tritave

It is a weird coincidence how 52edt intones any 52edo intervals within plus or minus 6.5 cents when it is supposed to have nothing to do with this other tuning:

52edt 52edo Discrepancy
365.761 369.231 -3.47
512.065 507.692 +4.373
877.825 876.923 +0.902
1243.586 1246.154 -2.168
1389.89 1384.615 +5.275
1755.651 1753.846 +1.805
2121.411 2123.077 -1.666
2633.476 2630.769 +2.647

…and so on