145edo: Difference between revisions

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== Theory ==

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, 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.

{{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.

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.

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.

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=== Prime harmonics ===

=== 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 ===

@@ Line 3: / Line 3: @@
 == Theory ==
-= 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, 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.
+{{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.
-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.
+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.
@@ Line 13: / Line 13: @@
 === Prime harmonics ===
 {{Harmonics in equal|145|intervals=prime}}
+=== Octave stretch ===
+edo'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 ===