289edo: Difference between revisions
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== Theory == | == Theory == | ||
289edo has decent 11- and 13-limit interpretations despite not being [[consistent]]. | 289edo has decent 11- and 13-limit interpretations despite not being [[consistent]]. The equal temperament [[tempering out|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. | ||
It is the [[optimal patent val]] for the [[13-limit]] rank-5 temperament tempering out 364/363, and the 13-limit [[History (temperament)|history]] temperament, which tempers out 364/363, 441/440 and 676/675. It provides a good tuning for the 11-limit version also. It is also the optimal patent val for [[sextilififths]] in both the 11- and 13-limit, and for [[quintaschis]] in both the 11- and 13-limit. | It is the [[optimal patent val]] for the [[13-limit]] rank-5 temperament tempering out 364/363, and the 13-limit [[History (temperament)|history]] temperament, which tempers out 364/363, 441/440 and 676/675. It provides a good tuning for the 11-limit version also. It is also the optimal patent val for [[sextilififths]] in both the 11- and 13-limit, and for [[quintaschis]] in both the 11- and 13-limit. | ||
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{{Harmonics in equal|289}} | {{Harmonics in equal|289}} | ||
=== | === Subsets and supersets === | ||
289 is 17 squared. In light of containing [[17edo]] as a subset, 289edo [[support]]s the [[chlorine]] temperament, which tempers out the [[septendecima]] {{monzo|-52 -17 34}} and the ragisma 4375/4374. | 289 is 17 squared. In light of containing [[17edo]] as a subset, 289edo [[support]]s the [[chlorine]] temperament, which tempers out the [[septendecima]] {{monzo| -52 -17 34 }} and the ragisma 4375/4374. | ||
== Regular temperament properties == | == Regular temperament properties == | ||
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| 2.3 | | 2.3 | ||
| {{monzo| -458 289 }} | | {{monzo| -458 289 }} | ||
| | | {{mapping| 289 458 }} | ||
| +0.0709 | | +0.0709 | ||
| 0.0710 | | 0.0710 | ||
| Line 33: | Line 33: | ||
| 2.3.5 | | 2.3.5 | ||
| 32805/32768, {{monzo| 7 41 -31 }} | | 32805/32768, {{monzo| 7 41 -31 }} | ||
| | | {{mapping| 289 458 671 }} | ||
| +0.0695 | | +0.0695 | ||
| 0.0580 | | 0.0580 | ||
| Line 40: | Line 40: | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 4375/4374, 32805/32768, 235298/234375 | | 4375/4374, 32805/32768, 235298/234375 | ||
| | | {{mapping| 289 458 671 811 }} | ||
| +0.1725 | | +0.1725 | ||
| 0.1854 | | 0.1854 | ||
| Line 47: | Line 47: | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 441/440, 4000/3993, 4375/4374, 32805/32768 | | 441/440, 4000/3993, 4375/4374, 32805/32768 | ||
| | | {{mapping| 289 458 671 811 1000 }} | ||
| +0.0841 | | +0.0841 | ||
| 0.2423 | | 0.2423 | ||
| Line 54: | Line 54: | ||
| 2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
| 364/363, 441/440, 676/675, 4375/4374, 19773/19712 | | 364/363, 441/440, 676/675, 4375/4374, 19773/19712 | ||
| | | {{mapping| 289 458 671 811 1000 1069 }} | ||
| +0.1500 | | +0.1500 | ||
| 0.2657 | | 0.2657 | ||
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{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
! Periods<br>per 8ve | ! Periods<br>per 8ve | ||
! Generator | ! Generator* | ||
! Cents | ! Cents* | ||
! Associated<br>Ratio | ! Associated<br>Ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
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| [[Chlorine]] | | [[Chlorine]] | ||
|} | |} | ||
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | |||
[[Category:Minor minthmic]] | [[Category:Minor minthmic]] | ||