289edo: Difference between revisions
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=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|289}} | {{Harmonics in equal|289}} | ||
== Regular temperament properties == | == Regular temperament properties == | ||
289edo has decent 11 and 13-limit interpretations despite not being consistent. | 289edo has decent 11 and 13-limit interpretations despite not being consistent. | ||
=== Rank | === Rank-2 temperaments === | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
!Periods | ! Periods<br>per Octave | ||
per | ! Generator<br>(Reduced) | ||
!Generator | ! Cents<br>(Reduced) | ||
( | ! Associated<br>Ratio | ||
!Cents | ! Temperaments | ||
( | |||
!Associated | |||
!Temperaments | |||
|- | |- | ||
|1 | | 1 | ||
|20\289 | | 20\289 | ||
|83.045 | | 83.045 | ||
|21/20 | | 21/20 | ||
|[[Sextilififths]] | | [[Sextilififths]] | ||
|- | |- | ||
|17 | | 17 | ||
|93\289<br>(8\289) | | 93\289<br>(8\289) | ||
|386.159<br>(33.218) | | 386.159<br>(33.218) | ||
|{{monzo|-23 5 9 -2}}<br>(100352/98415) | | {{monzo|-23 5 9 -2}}<br>(100352/98415) | ||
|[[Chlorine]] | | [[Chlorine]] | ||
|} | |} | ||
<!-- 3-digit number --> | [[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | ||
[[Category:History (temperament)]] | [[Category:History (temperament)]] | ||
[[Category:Sextilififths]] | [[Category:Sextilififths]] | ||
Revision as of 03:45, 2 October 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-2 temperaments
| 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 |