289edo: Difference between revisions
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289edo is the [[optimal patent val]] for [[13-limit]] [[History (temperament)|history]] temperament, which tempers out [[364/363]], [[441/440]] and [[676/675]], and provides a good tuning for the 11-limit version also, and is also the optimal patent val for [[sextilififths]] in both the 11- and 13-limit. It is uniquely [[consistent]] in the 9-odd-limit, and tempers out the [[schisma]], 32805/32768 in the 5-limit; [[4375/4374]] and 65625/65536 in the 7-limit; 441/440 and [[4000/3993]] in the 11-limit; and 364/363, 676/675, [[1001/1000]], [[1575/1573]] and [[2080/2079]] in the 13-limit. | 289edo is the [[optimal patent val]] for [[13-limit]] [[History (temperament)|history]] temperament, which tempers out [[364/363]], [[441/440]] and [[676/675]], and provides a good tuning for the 11-limit version also, and is also the optimal patent val for [[sextilififths]] in both the 11- and 13-limit. It is uniquely [[consistent]] in the 9-odd-limit, and tempers out the [[schisma]], 32805/32768 in the 5-limit; [[4375/4374]] and 65625/65536 in the 7-limit; 441/440 and [[4000/3993]] in the 11-limit; and 364/363, 676/675, [[1001/1000]], [[1575/1573]] and [[2080/2079]] in the 13-limit. | ||
Since 289 is square of 17, 289 = 17 × 17, 289edo [[support]]s the [[chlorine]] temperament, which tempers out the [[septendecima]] {{monzo|-52 -17 34}} and the ragisma 4375/4374. | Since 289 is square of 17<!-- How specifically the square property of 289 ensures it supports chlorine? -->, 289 = 17 × 17, 289edo [[support]]s the [[chlorine]] temperament, which tempers out the [[septendecima]] {{monzo|-52 -17 34}} and the ragisma 4375/4374. | ||
=== Prime harmonics === | === Prime harmonics === | ||
Revision as of 19:08, 16 July 2022
| ← 288edo | 289edo | 290edo → |
The 289 equal divisions of the octave (289edo), or the 289(-tone) equal temperament (289tet, 289et) when viewed from a regular temperament perspective, divides the octave into 289 equal parts of about 4.15 cents each.
Theory
289edo is the optimal patent val for 13-limit history temperament, which tempers out 364/363, 441/440 and 676/675, and provides a good tuning for the 11-limit version also, and is also the optimal patent val for sextilififths in both the 11- and 13-limit. It is uniquely consistent in the 9-odd-limit, and tempers out the schisma, 32805/32768 in the 5-limit; 4375/4374 and 65625/65536 in the 7-limit; 441/440 and 4000/3993 in the 11-limit; and 364/363, 676/675, 1001/1000, 1575/1573 and 2080/2079 in the 13-limit.
Since 289 is square of 17, 289 = 17 × 17, 289edo supports the chlorine temperament, which tempers out the septendecima [-52 -17 34⟩ and the ragisma 4375/4374.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.22 | -0.15 | -1.35 | +0.93 | -1.77 | -1.15 | +1.45 | -1.28 | +0.18 | +0.99 |
| Relative (%) | +0.0 | -5.4 | -3.7 | -32.6 | +22.4 | -42.7 | -27.7 | +34.9 | -30.9 | +4.3 | +23.7 | |
| Steps (reduced) |
289 (0) |
458 (169) |
671 (93) |
811 (233) |
1000 (133) |
1069 (202) |
1181 (25) |
1228 (72) |
1307 (151) |
1404 (248) |
1432 (276) | |