145edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
{{Nowrap| 145 {{=}} 5 × 29 }}, and 145edo shares the same perfect fifth with [[29edo]]. It is generally a sharp-tending system, with [[prime harmonic]]s 3 to 23 all tuned sharp except for [[7/1|7]], which is slightly flat. It is [[consistent]] to the [[11-odd-limit]], or the no-13 no-15 [[23-odd-limit]], with [[13/7]], [[15/8]] and their [[octave complement]]s being the only intervals going over the line. | |||
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 [[ | As an equal temperament, 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]], [[364/363]], [[676/675]], [[847/845]], and [[1001/1000]] 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 [[major minthmic chords]], because it tempers out 364/363 it allows [[minor minthmic 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. | |||
The 145c val provides a tuning for [[magic]] which is nearly identical to the [[POTE tuning]]. | The 145c val provides a tuning for [[magic]] which is nearly identical to the [[POTE tuning]]. | ||
=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|145| | {{Harmonics in equal|145|intervals=prime}} | ||
=== Octave stretch === | |||
145edo's approximated harmonics 3, 5, 11, 13, 17, 19, and 23 can be improved at the cost of a little worse 7, and moreover the approximated harmonic 13 can be brought to consistency, if slightly [[stretched and compressed tuning|compressing the octave]] is acceptable. [[375ed6]] is about at the sweet spot for this. | |||
=== Subsets and supersets === | === Subsets and supersets === | ||
145edo contains [[5edo]] and [[29edo]] as subset edos. | |||
== Regular temperament properties == | == Regular temperament properties == | ||
| Line 28: | Line 33: | ||
| 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 40: | ||
| 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 47: | ||
| 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 54: | ||
| 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 61: | ||
| 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 | ||
| 5.85 | | 5.85 | ||
|- | |||
| 2.3.5.7.11.13.17.19 | |||
| 196/195, 256/255, 343/342, 352/351, 361/360, 364/363, 476/475 | |||
| {{Mapping| 145 230 337 407 502 537 593 616 }} | |||
| -0.565 | |||
| 0.486 | |||
| 5.87 | |||
|- | |||
| 2.3.5.7.11.13.17.19.23 | |||
| 196/195, 256/255, 276/275, 352/351, 361/360, 364/363, 460/459, 476/475 | |||
| {{Mapping| 145 230 337 407 502 537 593 616 656 }} | |||
| -0.519 | |||
| 0.476 | |||
| 5.75 | |||
|} | |} | ||