145edo: Difference between revisions

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== 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~~.

{{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 [[~~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.

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]].

=== 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 ===

~~145 = 5 × 29,~~ and ~~145edo shares the same excellent fifth with~~ [[29edo]].

145edo contains [[5edo]] and [[29edo]] as subset edos.

== Regular temperament properties ==

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| 2.3.5

| 1600000/1594323, {{monzo| 28 -3 -10 }}

| [{{~~val~~| 145 230 337 }}]

| {{Mapping| 145 230 337 }}

| -0.695

| 0.498

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| 2.3.5.7

| 4375/4374, 5120/5103, 50421/50000

| [{{~~val~~| 145 230 337 407 }}]

| {{Mapping| 145 230 337 407 }}

| -0.472

| 0.578

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| 2.3.5.7.11

| 441/440, 896/891, 3388/3375, 4375/4374

| [{{~~val~~| 145 230 337 407 502 }}]

| {{Mapping| 145 230 337 407 502 }}

| -0.561

| 0.547

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| 2.3.5.7.11.13

| 196/195, 352/351, 364/363, 676/675, 4375/4374

| [{{~~val~~| 145 230 337 407 502 537 }}]

| {{Mapping| 145 230 337 407 502 537 }}

| -0.630

| 0.522

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| 2.3.5.7.11.13.17

| 196/195, 256/255, 352/351, 364/363, 676/675, 1156/1155

| [{{~~val~~| 145 230 337 407 502 537 593 }}]

| {{Mapping| 145 230 337 407 502 537 593 }}

| -0.632

| 0.484

| 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

|}

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|+Table of rank-2 temperaments by generator

! Periods per 8ve

! Generator~~ (Reduced)~~

! Generator*

! Cents~~ (Reduced)~~

! Cents*

! Associated Ratio

! Associated Ratio*

! Temperaments

|-

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|145}}
+{{ED intro}}
 == Theory ==
-et 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.
+{{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 [[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.
+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]].
 === Prime harmonics ===
-{{Harmonics in equal|145|columns=11}}
+{{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 ===
-= 5 × 29, and 145edo shares the same excellent fifth with [[29edo]].
+edo contains [[5edo]] and [[29edo]] as subset edos.
 == Regular temperament properties ==
@@ Line 28: / Line 33: @@
 | 2.3.5
 | 1600000/1594323, {{monzo| 28 -3 -10 }}
-| [{{val| 145 230 337 }}]
+| {{Mapping| 145 230 337 }}
 | -0.695
 | 0.498
@@ Line 35: / Line 40: @@
 | 2.3.5.7
 | 4375/4374, 5120/5103, 50421/50000
-| [{{val| 145 230 337 407 }}]
+| {{Mapping| 145 230 337 407 }}
 | -0.472
 | 0.578
@@ Line 42: / Line 47: @@
 | 2.3.5.7.11
 | 441/440, 896/891, 3388/3375, 4375/4374
-| [{{val| 145 230 337 407 502 }}]
+| {{Mapping| 145 230 337 407 502 }}
 | -0.561
 | 0.547
@@ Line 49: / Line 54: @@
 | 2.3.5.7.11.13
 | 196/195, 352/351, 364/363, 676/675, 4375/4374
-| [{{val| 145 230 337 407 502 537 }}]
+| {{Mapping| 145 230 337 407 502 537 }}
 | -0.630
 | 0.522
@@ Line 56: / Line 61: @@
 | 2.3.5.7.11.13.17
 | 196/195, 256/255, 352/351, 364/363, 676/675, 1156/1155
-| [{{val| 145 230 337 407 502 537 593 }}]
+| {{Mapping| 145 230 337 407 502 537 593 }}
 | -0.632
 | 0.484
 | 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
 |}
@@ Line 66: / Line 85: @@
 |+Table of rank-2 temperaments by generator
 ! Periods<br>per 8ve
-! Generator<br>(Reduced)
+! Generator*
-! Cents<br>(Reduced)
+! Cents*
-! Associated<br>Ratio
+! Associated<br>Ratio*
 ! Temperaments
 |-