52edt
| ← 51edt | 52edt | 53edt → |
52 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 52edt or 52ed3), is a nonoctave tuning system that divides the interval of 3/1 into 52 equal parts of about 36.6 ¢ each. Each step represents a frequency ratio of 31/52, or the 52nd root of 3.
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 7-limit, so that it tempers out the same commas as 13edt. 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.
Harmonics
While 52edt does not improve much on 26edt's mapping of the odd primes, it still maps 11, 13, 19, and 29 slightly better than 26edt, which is notable for giving them (despite being nearly half a step off in the case of 11) a flat tendency in keeping with all the other odd primes, which allows 52edt to be the first edt that is consistent to the no-twos 29-throdd limit, which is a record unbeaten until 144edt.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +7.0 | +0.0 | -6.5 | -3.8 | -18.2 | -14.8 | -3.8 | -13.4 | -15.0 |
| Relative (%) | +19.2 | +0.0 | -17.9 | -10.5 | -49.8 | -40.5 | -10.3 | -36.7 | -41.1 | |
| Steps (reduced) |
33 (33) |
52 (0) |
76 (24) |
92 (40) |
113 (9) |
121 (17) |
134 (30) |
139 (35) |
148 (44) | |
| Harmonic | 25 | 27 | 29 | 31 | 33 | 35 | 37 | 39 | 41 | 43 | 45 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -13.1 | +0.0 | -14.0 | +16.9 | -18.2 | -10.4 | +3.2 | -14.8 | +8.3 | -1.0 | -6.5 |
| Relative (%) | -35.7 | +0.0 | -38.2 | +46.1 | -49.8 | -28.3 | +8.6 | -40.5 | +22.8 | -2.7 | -17.9 | |
| Steps (reduced) |
152 (48) |
156 (0) |
159 (3) |
163 (7) |
165 (9) |
168 (12) |
171 (15) |
173 (17) |
176 (20) |
178 (22) |
180 (24) | |
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