289edo: Difference between revisions
Clarify (there's no correlation between it being square and the temperaments it supports); adopt new template |
rank 2 temps |
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{{Harmonics in equal|289}} | {{Harmonics in equal|289}} | ||
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | [[Category:Equal divisions of the octave|###]] | ||
== Regular temperament properties == | |||
289edo has decent 11 and 13-limit interpretations despite not being consistent. | |||
=== Rank two temperaments by generator === | |||
{| class="wikitable center-all left-5" | |||
!Periods | |||
per octave | |||
!Generator | |||
(reduced) | |||
!Cents | |||
(reduced) | |||
!Associated | |||
ratio | |||
!Temperaments | |||
|- | |||
|1 | |||
|20\289 | |||
|83.045 | |||
|21/20 | |||
|[[Sextilififths]] | |||
|- | |||
|17 | |||
|93\289 | |||
(8\289) | |||
|386.159 | |||
(33.218) | |||
|[-23 5 9 -2⟩ | |||
(100352/98415) | |||
|[[Chlorine]] | |||
|}<!-- 3-digit number --> | |||
[[Category:History (temperament)]] | [[Category:History (temperament)]] | ||
[[Category:Sextilififths]] | [[Category:Sextilififths]] | ||
Revision as of 18:21, 15 September 2022
| ← 288edo | 289edo | 290edo → |
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.
289 is 17 squared. In light of containing 17edo as a subset, 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) | |
Regular temperament properties
289edo has decent 11 and 13-limit interpretations despite not being consistent.
Rank two temperaments by generator
| Periods
per octave |
Generator
(reduced) |
Cents
(reduced) |
Associated
ratio |
Temperaments |
|---|---|---|---|---|
| 1 | 20\289 | 83.045 | 21/20 | Sextilififths |
| 17 | 93\289
(8\289) |
386.159
(33.218) |
[-23 5 9 -2⟩
(100352/98415) |
Chlorine |