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'''88EDT''' is the [[Edt|equal division of the third harmonic]] into 88 parts of 21.6131 [[cent|cents]] each, corresponding to 55.5218 [[edo]] (similar to every second step of [[111edo]]). It is consistent to the no-twos 11-limit, tempering out 1331/1323, 16875/16807, and 216513/214375. In the 3.4.5.7.11 subgroup, it tempers out 176/175, 540/539, 1331/1323, and 5120/5103. | '''88EDT''' is the [[Edt|equal division of the third harmonic]] into 88 parts of 21.6131 [[cent|cents]] each, corresponding to 55.5218 [[edo]] (similar to every second step of [[111edo]]). It is consistent to the no-twos 11-limit, tempering out 1331/1323, 16875/16807, and 216513/214375. In the 3.4.5.7.11 subgroup, it tempers out 176/175, 540/539, 1331/1323, and 5120/5103. | ||
88EDT is the | 88EDT is the largest EDT to not correspond to a [[val]] of some [[EDO]] that has a [[5L 2s|diatonic]] fifth, instead corresponding to both the [[55edo|55b]] val, with [[5edo]]'s fifth, and the [[56edo|56b]] val, with [[7edo]]'s fifth. It is also a [[the Riemann zeta function and tuning#Removing primes|no-twos zeta peak EDT]]. | ||
== Harmonics == | == Harmonics == | ||
| Line 22: | Line 22: | ||
== Intervals == | == Intervals == | ||
{{Interval table}} | {{Interval table}} | ||
Latest revision as of 19:24, 1 August 2025
| ← 87edt | 88edt | 89edt → |
88EDT is the equal division of the third harmonic into 88 parts of 21.6131 cents each, corresponding to 55.5218 edo (similar to every second step of 111edo). It is consistent to the no-twos 11-limit, tempering out 1331/1323, 16875/16807, and 216513/214375. In the 3.4.5.7.11 subgroup, it tempers out 176/175, 540/539, 1331/1323, and 5120/5103.
88EDT is the largest EDT to not correspond to a val of some EDO that has a diatonic fifth, instead corresponding to both the 55b val, with 5edo's fifth, and the 56b val, with 7edo's fifth. It is also a no-twos zeta peak EDT.
Harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +10.34 | +0.00 | +1.78 | +2.82 | -1.60 | -9.84 | +1.22 | +3.18 | -3.38 | +5.97 | -1.43 |
| Relative (%) | +47.8 | +0.0 | +8.2 | +13.1 | -7.4 | -45.5 | +5.7 | +14.7 | -15.6 | +27.6 | -6.6 | |
| Steps (reduced) |
56 (56) |
88 (0) |
129 (41) |
156 (68) |
192 (16) |
205 (29) |
227 (51) |
236 (60) |
251 (75) |
270 (6) |
275 (11) | |
| Harmonic | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -5.15 | -9.96 | -5.97 | -8.66 | -0.53 | +8.32 | -6.17 | +4.32 | -9.62 | +7.13 | +0.06 |
| Relative (%) | -23.8 | -46.1 | -27.6 | -40.1 | -2.5 | +38.5 | -28.5 | +20.0 | -44.5 | +33.0 | +0.3 | |
| Steps (reduced) |
289 (25) |
297 (33) |
301 (37) |
308 (44) |
318 (54) |
327 (63) |
329 (65) |
337 (73) |
341 (77) |
344 (80) |
350 (86) | |
Intervals
| Steps | Cents | Hekts | Approximate ratios |
|---|---|---|---|
| 0 | 0 | 0 | 1/1 |
| 1 | 21.6 | 14.8 | |
| 2 | 43.2 | 29.5 | |
| 3 | 64.8 | 44.3 | 27/26 |
| 4 | 86.5 | 59.1 | 41/39 |
| 5 | 108.1 | 73.9 | 33/31 |
| 6 | 129.7 | 88.6 | |
| 7 | 151.3 | 103.4 | |
| 8 | 172.9 | 118.2 | 21/19 |
| 9 | 194.5 | 133 | 19/17 |
| 10 | 216.1 | 147.7 | 17/15 |
| 11 | 237.7 | 162.5 | 31/27 |
| 12 | 259.4 | 177.3 | 29/25 |
| 13 | 281 | 192 | |
| 14 | 302.6 | 206.8 | 25/21, 31/26 |
| 15 | 324.2 | 221.6 | 35/29 |
| 16 | 345.8 | 236.4 | 11/9 |
| 17 | 367.4 | 251.1 | 21/17, 26/21 |
| 18 | 389 | 265.9 | |
| 19 | 410.6 | 280.7 | 19/15, 33/26 |
| 20 | 432.3 | 295.5 | 9/7 |
| 21 | 453.9 | 310.2 | |
| 22 | 475.5 | 325 | 25/19, 29/22 |
| 23 | 497.1 | 339.8 | |
| 24 | 518.7 | 354.5 | 31/23 |
| 25 | 540.3 | 369.3 | 26/19 |
| 26 | 561.9 | 384.1 | |
| 27 | 583.6 | 398.9 | 7/5 |
| 28 | 605.2 | 413.6 | |
| 29 | 626.8 | 428.4 | 33/23 |
| 30 | 648.4 | 443.2 | |
| 31 | 670 | 458 | 25/17 |
| 32 | 691.6 | 472.7 | |
| 33 | 713.2 | 487.5 | |
| 34 | 734.8 | 502.3 | 26/17, 29/19 |
| 35 | 756.5 | 517 | 17/11 |
| 36 | 778.1 | 531.8 | |
| 37 | 799.7 | 546.6 | 27/17 |
| 38 | 821.3 | 561.4 | 37/23 |
| 39 | 842.9 | 576.1 | |
| 40 | 864.5 | 590.9 | |
| 41 | 886.1 | 605.7 | 5/3 |
| 42 | 907.8 | 620.5 | |
| 43 | 929.4 | 635.2 | |
| 44 | 951 | 650 | 26/15 |
| 45 | 972.6 | 664.8 | |
| 46 | 994.2 | 679.5 | |
| 47 | 1015.8 | 694.3 | 9/5 |
| 48 | 1037.4 | 709.1 | 31/17 |
| 49 | 1059 | 723.9 | 35/19 |
| 50 | 1080.7 | 738.6 | |
| 51 | 1102.3 | 753.4 | 17/9 |
| 52 | 1123.9 | 768.2 | |
| 53 | 1145.5 | 783 | 33/17 |
| 54 | 1167.1 | 797.7 | |
| 55 | 1188.7 | 812.5 | |
| 56 | 1210.3 | 827.3 | |
| 57 | 1231.9 | 842 | |
| 58 | 1253.6 | 856.8 | |
| 59 | 1275.2 | 871.6 | 23/11 |
| 60 | 1296.8 | 886.4 | |
| 61 | 1318.4 | 901.1 | 15/7 |
| 62 | 1340 | 915.9 | |
| 63 | 1361.6 | 930.7 | |
| 64 | 1383.2 | 945.5 | |
| 65 | 1404.9 | 960.2 | |
| 66 | 1426.5 | 975 | |
| 67 | 1448.1 | 989.8 | |
| 68 | 1469.7 | 1004.5 | 7/3 |
| 69 | 1491.3 | 1019.3 | 26/11 |
| 70 | 1512.9 | 1034.1 | |
| 71 | 1534.5 | 1048.9 | 17/7 |
| 72 | 1556.1 | 1063.6 | 27/11 |
| 73 | 1577.8 | 1078.4 | |
| 74 | 1599.4 | 1093.2 | |
| 75 | 1621 | 1108 | |
| 76 | 1642.6 | 1122.7 | |
| 77 | 1664.2 | 1137.5 | |
| 78 | 1685.8 | 1152.3 | |
| 79 | 1707.4 | 1167 | |
| 80 | 1729.1 | 1181.8 | 19/7 |
| 81 | 1750.7 | 1196.6 | |
| 82 | 1772.3 | 1211.4 | |
| 83 | 1793.9 | 1226.1 | 31/11 |
| 84 | 1815.5 | 1240.9 | |
| 85 | 1837.1 | 1255.7 | 26/9 |
| 86 | 1858.7 | 1270.5 | |
| 87 | 1880.3 | 1285.2 | |
| 88 | 1902 | 1300 | 3/1 |