145edo: Difference between revisions
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== Theory == | == Theory == | ||
145et tempers out [[1600000/1594323]] in the [[5-limit]]; [[4375/4374]] and [[5120/5103]] in the [[7-limit]]; [[441/440]] and [[896/891]] in the 11-limit; [[196/195]], [[352/351]] and [[364/363]] in the 13-limit; [[595/594]] in the 17-limit; [[343/342]] and [[476/475]] in the 19-limit. | 145et [[tempering out|tempers out]] [[1600000/1594323]] in the [[5-limit]]; [[4375/4374]] and [[5120/5103]] in the [[7-limit]]; [[441/440]] and [[896/891]] in the 11-limit; [[196/195]], [[352/351]] and [[364/363]] in the 13-limit; [[595/594]] in the 17-limit; [[343/342]] and [[476/475]] in the 19-limit. | ||
It is the [[optimal patent val]] for the 11-limit [[mystery]] temperament and the 11-limit rank-3 [[pele]] temperament. It also [[support]]s and provides a good tuning for 13-limit mystery, and because it tempers out 441/440 it allows [[werckismic chords]], because it tempers out 196/195 it allows [[mynucumic chords]], because it tempers out 352/351 it allows [[minthmic chords]], because it tempers out 364/363 it allows [[gentle chords]], and because it tempers out 847/845 it allows the [[cuthbert chords]], making it a very flexible harmonic system. The same is true of [[232edo]], the optimal patent val for 13-limit mystery. | It is the [[optimal patent val]] for the 11-limit [[mystery]] temperament and the 11-limit rank-3 [[pele]] temperament. It also [[support]]s and provides a good tuning for 13-limit mystery, and because it tempers out 441/440 it allows [[werckismic chords]], because it tempers out 196/195 it allows [[mynucumic chords]], because it tempers out 352/351 it allows [[minthmic chords]], because it tempers out 364/363 it allows [[gentle chords]], and because it tempers out 847/845 it allows the [[cuthbert chords]], making it a very flexible harmonic system. The same is true of [[232edo]], the optimal patent val for 13-limit mystery. | ||
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| 2.3.5 | | 2.3.5 | ||
| 1600000/1594323, {{monzo| 28 -3 -10 }} | | 1600000/1594323, {{monzo| 28 -3 -10 }} | ||
| | | {{mapping| 145 230 337 }} | ||
| -0.695 | | -0.695 | ||
| 0.498 | | 0.498 | ||
| Line 35: | Line 35: | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 4375/4374, 5120/5103, 50421/50000 | | 4375/4374, 5120/5103, 50421/50000 | ||
| | | {{mapping| 145 230 337 407 }} | ||
| -0.472 | | -0.472 | ||
| 0.578 | | 0.578 | ||
| Line 42: | Line 42: | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 441/440, 896/891, 3388/3375, 4375/4374 | | 441/440, 896/891, 3388/3375, 4375/4374 | ||
| | | {{mapping| 145 230 337 407 502 }} | ||
| -0.561 | | -0.561 | ||
| 0.547 | | 0.547 | ||
| Line 49: | Line 49: | ||
| 2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
| 196/195, 352/351, 364/363, 676/675, 4375/4374 | | 196/195, 352/351, 364/363, 676/675, 4375/4374 | ||
| | | {{mapping| 145 230 337 407 502 537 }} | ||
| -0.630 | | -0.630 | ||
| 0.522 | | 0.522 | ||
| Line 56: | Line 56: | ||
| 2.3.5.7.11.13.17 | | 2.3.5.7.11.13.17 | ||
| 196/195, 256/255, 352/351, 364/363, 676/675, 1156/1155 | | 196/195, 256/255, 352/351, 364/363, 676/675, 1156/1155 | ||
| | | {{mapping| 145 230 337 407 502 537 593 }} | ||
| -0.632 | | -0.632 | ||
| 0.484 | | 0.484 | ||
| Line 66: | Line 66: | ||
|+Table of rank-2 temperaments by generator | |+Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | ! Periods<br>per 8ve | ||
! Generator | ! Generator* | ||
! Cents | ! Cents* | ||
! Associated<br>Ratio | ! Associated<br>Ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||