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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 15th [[ | 88EDT is the 15th [[the Riemann zeta function and tuning#Removing primes|no-twos zeta peak EDT]]. | ||
==Harmonics== | == Harmonics == | ||
{{Harmonics in equal | {{Harmonics in equal | ||
| steps = 88 | | steps = 88 | ||
| Line 20: | Line 20: | ||
}} | }} | ||
==Intervals== | == Intervals == | ||
{{Interval table}} | {{Interval table}} | ||
[[Category:Edt]] | [[Category:Edt]] | ||
[[Category:Edonoi]] | [[Category:Edonoi]] | ||
Revision as of 13:15, 10 April 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 15th 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 |