98304edo
| ← 98303edo | 98304edo | 98305edo → |
98304 equal divisions of the octave (abbreviated 98304edo or 98304ed2), also called 98304-tone equal temperament (98304tet) or 98304 equal temperament (98304et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 98304 equal parts of about 0.0122 ¢ each. Each step represents a frequency ratio of 21/98304, or the 98304th root of 2.
Theory
98304edo is consistent to the 19-odd-limit. The equal temperament tempers out [45 2 -28 6⟩, [-54 15 -16 24⟩, and [-29 135 -18 -51⟩ in the 7-limit; [-4 15 -14 7 -2⟩, [-3 28 2 -9 -6⟩, [-50 0 -2 17 2⟩, and [30 10 8 -2 -17⟩ in the 11-limit; 123201/123200, 32427005625/32426652544, 278924131584/278916015625, 37744795080531/37744172597248, and 156905298045000/156904157228819 in the 13-limit; 2000033/2000000, 154002541/154001250, 303464448/303460625, 338676338/338671875, 791249550/791243563, and 176846618624/176846076825 in the 17-limit; 89376/89375, 104976/104975, 709632/709631, 5836831/5836800, 494190983/494190000, 1206902781/1206878450, and 21867094832/21867015625 in the 19-limit.
As a tuning standard
A step of 98304edo is known as a tridecamu (thirteenth MIDI-resolution unit, 13mu, 213 = 8192 equal divisions of the 12edo semitone). The internal data structure of the 13mu requires two bytes, with the first bits of each byte reserved as flags to indicate the byte's status as data, and one bit in the first byte to indicate the sign (+ or −) showing the direction of the pitch-bend up or down; all bits are used. The first data byte transmitted is the Least Significant Byte (LSB), equivalent to a fine-tuning. The second data byte transmitted is the Most Significant Byte (MSB), equivalent to a coarse-tuning.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | absolute (¢) | +0.00000 | -0.00188 | +0.00220 | -0.00266 | +0.00042 | -0.00032 | +0.00065 | -0.00325 | -0.00286 | +0.00044 | -0.00383 |
| relative (%) | +0 | -15 | +18 | -22 | +3 | -3 | +5 | -27 | -23 | +4 | -31 | |
| Steps (reduced) |
98304 (0) |
155808 (57504) |
228255 (31647) |
275974 (79366) |
340076 (45164) |
363768 (68856) |
401814 (8598) |
417588 (24372) |
444684 (51468) |
477559 (84343) |
487017 (93801) | |
Selected intervals
| # | Cents | JI Interval | Error (Cents) | |
|---|---|---|---|---|
| Ratio | Cents | |||
| 0 | 0.0000 | 1/1 | 0.0000 | ±0.00000 |
| 3995 | 48.7671 | 36/35 | 48.7704 | −0.00329 |
| 4111 | 50.1831 | 35/34 | 50.1842 | −0.00111 |
| 4234 | 51.6846 | 34/33 | 51.6825 | +0.00210 |
| 4364 | 53.2715 | 33/32 | 53.2729 | −0.00146 |
| 5158 | 62.9639 | 28/27 | 62.9609 | +0.00296 |
| 5352 | 65.3320 | 27/26 | 65.3373 | −0.00531 |
| 5562 | 67.8955 | 26/25 | 67.9002 | −0.00473 |
| 6598 | 80.5420 | 22/21 | 80.5370 | +0.00496 |
| 7275 | 88.8062 | 20/19 | 88.8007 | +0.00545 |
| 7668 | 93.6035 | 19/18 | 93.6030 | +0.00050 |
| 8106 | 98.9502 | 18/17 | 98.9546 | −0.00440 |
| 8598 | 104.9561 | 17/16 | 104.9554 | +0.00065 |
| 9153 | 111.7310 | 16/15 | 111.7313 | −0.00033 |
| 9785 | 119.4458 | 15/14 | 119.4428 | +0.00299 |
| 10510 | 128.2959 | 14/13 | 128.2982 | −0.00235 |
| 11352 | 138.5742 | 13/12 | 138.5727 | +0.00156 |
| 12340 | 150.6348 | 12/11 | 150.6371 | −0.00229 |
| 13517 | 165.0024 | 11/10 | 165.0042 | −0.00179 |
| 14943 | 182.4097 | 10/9 | 182.4037 | +0.00596 |
| 15774 | 192.5537 | 19/17 | 192.5576 | −0.00390 |
| 16704 | 203.9063 | 9/8 | 203.9100 | −0.00375 |
| 17751 | 216.6870 | 17/15 | 216.6867 | +0.00032 |
| 18938 | 231.1768 | 8/7 | 231.1741 | +0.00266 |
| 20295 | 247.7417 | 15/13 | 247.7411 | +0.00065 |
| 20792 | 253.8086 | 22/19 | 253.8049 | +0.00367 |
| 21862 | 266.8701 | 7/6 | 266.8709 | −0.00079 |
| 23049 | 281.3599 | 20/17 | 281.3583 | +0.00156 |
| 23692 | 289.2090 | 13/11 | 289.2097 | −0.00074 |
| 24372 | 297.5098 | 19/16 | 297.5130 | −0.00325 |
| 25857 | 315.6372 | 6/5 | 315.6413 | −0.00408 |
| 27536 | 336.1328 | 17/14 | 336.1295 | +0.00331 |
| 28460 | 347.4121 | 11/9 | 347.4079 | +0.00417 |
| 29448 | 359.4727 | 16/13 | 359.4723 | +0.00032 |
| 31647 | 386.3159 | 5/4 | 386.3137 | +0.00220 |
| 33132 | 404.4434 | 24/19 | 404.4420 | +0.00137 |
| 33525 | 409.2407 | 19/15 | 409.2443 | −0.00358 |
| 34202 | 417.5049 | 14/11 | 417.5080 | −0.00308 |
| 35642 | 435.0830 | 9/7 | 435.0841 | −0.00109 |
| 36566 | 446.3623 | 22/17 | 446.3625 | −0.00023 |
| 37209 | 454.2114 | 13/10 | 454.2139 | −0.00252 |
| 38046 | 464.4287 | 17/13 | 464.4277 | +0.00096 |
| 38566 | 470.7764 | 21/16 | 470.7809 | −0.00454 |
| 40800 | 498.0469 | 4/3 | 498.0450 | +0.00188 |
| 43310 | 528.6865 | 19/14 | 528.6871 | −0.00059 |
| 43987 | 536.9507 | 15/11 | 536.9508 | −0.00009 |
| 44484 | 543.0176 | 26/19 | 543.0146 | +0.00293 |
| 45164 | 551.3184 | 11/8 | 551.3179 | +0.00042 |
| 46152 | 563.3789 | 18/13 | 563.3823 | −0.00343 |
| 47719 | 582.5073 | 7/5 | 582.5122 | −0.00487 |
| 48906 | 596.9971 | 24/17 | 596.9996 | −0.00252 |
| 49398 | 603.0029 | 17/12 | 603.0004 | +0.00252 |
| 50585 | 617.4927 | 10/7 | 617.4878 | +0.00487 |
| 57504 | 701.9531 | 3/2 | 701.9550 | −0.00188 |
| 62662 | 764.9170 | 14/9 | 764.9159 | +0.00109 |
| 66657 | 813.6841 | 8/5 | 813.6863 | −0.00220 |
| 72447 | 884.3628 | 5/3 | 884.3587 | +0.00408 |
| 76442 | 933.1299 | 12/7 | 933.1291 | +0.00079 |
| 79366 | 968.8232 | 7/4 | 968.8259 | −0.00266 |
| 98304 | 1200.0000 | 2/1 | 1200.0000 | ±0.00000 |
See also
- Equal multiplications of MIDI-resolution units