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 == | ||
| Line 26: | Line 26: | ||
| 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 | ||
|- | |- | ||
| Line 128: | Line 128: | ||
| [[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]] | ||
Revision as of 09:22, 4 March 2024
| ← 288edo | 289edo | 290edo → |
Theory
289edo has decent 11- and 13-limit interpretations despite not being consistent. The equal temperament 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, 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.
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) | |
Subsets and supersets
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.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-458 289⟩ | [⟨289 458]] | +0.0709 | 0.0710 | 1.71 |
| 2.3.5 | 32805/32768, [7 41 -31⟩ | [⟨289 458 671]] | +0.0695 | 0.0580 | 1.40 |
| 2.3.5.7 | 4375/4374, 32805/32768, 235298/234375 | [⟨289 458 671 811]] | +0.1725 | 0.1854 | 4.46 |
| 2.3.5.7.11 | 441/440, 4000/3993, 4375/4374, 32805/32768 | [⟨289 458 671 811 1000]] | +0.0841 | 0.2423 | 5.83 |
| 2.3.5.7.11.13 | 364/363, 441/440, 676/675, 4375/4374, 19773/19712 | [⟨289 458 671 811 1000 1069]] | +0.1500 | 0.2657 | 6.40 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated Ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 4\289 | 16.61 | 100/99 | Quincy |
| 1 | 13\289 | 53.98 | 33/32 | Tridecafifths |
| 1 | 20\289 | 83.04 | 21/20 | Sextilififths |
| 1 | 24\289 | 99.65 | 18/17 | Quintaschis |
| 1 | 76\289 | 315.57 | 6/5 | Acrokleismic |
| 1 | 86\289 | 357.09 | 768/625 | Dodifo |
| 1 | 108\289 | 448.44 | 35/27 | Semidimfourth |
| 1 | 120\289 | 498.27 | 4/3 | Pontiac |
| 1 | 135\289 | 560.55 | 864/625 | Whoosh |
| 17 | 93\289 (8\289) |
386.16 (33.22) |
[-23 5 9 -2⟩ (100352/98415) |
Chlorine |
* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct