145edo: Difference between revisions

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{{Infobox ET

~~| Prime factorization = 5 × 29~~

~~| Step size = 8.27586¢~~

~~| Fifth = 85\145 (703.45¢) (→ [[29edo|17\29]])~~

~~| Major 2nd = 25\145 (206.90¢)~~

~~| Semitones = 15:10 (124.14¢ : 82.76¢)~~

~~| Consistency = 11~~

}}

The '''145 equal divisions of the octave''' ('''145edo''') or '''145(-tone) equal temperament''' ('''145tet''', '''145et''') when viewed from a [[regular temperament]] perspective, is the tuning system derived by dividing the [[octave]] into 145 [[equal]] parts of 8.28 [[cent]]s each.

== 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 ~~triad~~]], 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 ===

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

== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

! rowspan="2" | Subgroup

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Comma list|Comma List]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve ~~stretch~~ (¢)

! rowspan="2" | Optimal 8ve Stretch (¢)

! colspan="2" | Tuning ~~error~~

! colspan="2" | Tuning Error

|-

! [[TE error|Absolute]] (¢)

Line 32:

Line 33:

| 2.3.5

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

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

| {{Mapping| 145 230 337 }}

| -0.695

| 0.498

Line 39:

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 46:

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 53:

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 60:

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

|}

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{| class="wikitable center-all left-5"

|+Table of rank-2 temperaments by generator

! Periods per ~~octave~~

! Periods per 8ve

! Generator~~ (reduced)~~

! Generator*

! Cents~~ (reduced)~~

! Cents*

! Associated ~~ratio~~

! Associated Ratio*

! Temperaments

|-

Line 84:

Line 99:

| 12\145

| 99.31

| ~~35/33,~~ 18/17

| 18/17

| [[Quinticosiennic]]

|-

Line 91:

Line 106:

| 115.86

| 77/72

| [[~~Mercy]] / [[countermiracle~~]]

| [[Countermiracle]]

|-

| 1

Line 97:

Line 112:

| 322.76

| 3087/2560

| [[~~Senior~~]] / ~~[[seniority]]~~

| [[Seniority]] / senator

|-

| 1

Line 103:

Line 118:

| 339.31

| 128/105

| [[Amity]]

| [[Amity]] / catamite

|-

| 5

Line 117:

Line 132:

| [[Mystery]]

|}

== Scales ==

* [[Magic7]]

* [[Magic10]]

* [[Magic13]]

* [[Magic16]]

* [[Magic19]]

* [[Magic22]]

== Music ==

* [http://micro.soonlabel.com/magic/daily20120113-piano-magic16-.mp3 Chromatic piece in magic 16] [~~http://www.chrisvaisvil.com/ Chris Vaisvil~~]

; [[Chris Vaisvil]] ([http://www.chrisvaisvil.com/ site])

* [http://micro.soonlabel.com/magic/daily20120113-piano-magic16-.mp3 ''Chromatic piece in magic 16''] – magic[16] in 145edo tuning

~~[[Category:145edo]]~~

~~[[Category:Equal divisions of the octave]]~~

[[Category:Mystery]]

[[Category:Pele]]

[[Category:Magic]]

[[Category:Listen]]

@@ Line 1: / Line 1: @@
-{{Infobox ET
+{{Infobox ET}}
-| Prime factorization = 5 × 29
+{{ED intro}}
-| Step size = 8.27586¢
-| Fifth = 85\145 (703.45¢) (→ [[29edo|17\29]])
-| Major 2nd = 25\145 (206.90¢)
-| Semitones = 15:10 (124.14¢ : 82.76¢)
-| Consistency = 11
-}}
-The '''145 equal divisions of the octave''' ('''145edo''') or '''145(-tone) equal temperament''' ('''145tet''', '''145et''') when viewed from a [[regular temperament]] perspective, is the tuning system derived by dividing the [[octave]] into 145 [[equal]] parts of 8.28 [[cent]]s each.
 == 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 triad]], 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 ===
-{{Primes in edo|145|columns=9}}
+{{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 ===
+edo contains [[5edo]] and [[29edo]] as subset edos.
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
-! rowspan="2" | Subgroup
+! rowspan="2" | [[Subgroup]]
-! rowspan="2" | [[Comma list]]
+! rowspan="2" | [[Comma list|Comma List]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br>8ve stretch (¢)
+! rowspan="2" | Optimal<br>8ve Stretch (¢)
-! colspan="2" | Tuning error
+! colspan="2" | Tuning Error
 |-
 ! [[TE error|Absolute]] (¢)
@@ Line 32: / Line 33: @@
 | 2.3.5
 | 1600000/1594323, {{monzo| 28 -3 -10 }}
-| [{{val| 145 230 337 }}]
+| {{Mapping| 145 230 337 }}
 | -0.695
 | 0.498
@@ Line 39: / 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 46: / 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 53: / 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 60: / 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 69: / Line 84: @@
 {| class="wikitable center-all left-5"
 |+Table of rank-2 temperaments by generator
-! Periods<br>per octave
+! Periods<br>per 8ve
-! Generator<br>(reduced)
+! Generator*
-! Cents<br>(reduced)
+! Cents*
-! Associated<br>ratio
+! Associated<br>Ratio*
 ! Temperaments
 |-
@@ Line 84: / Line 99: @@
 | 12\145
 | 99.31
-| 35/33, 18/17
+| 18/17
 | [[Quinticosiennic]]
 |-
@@ Line 91: / Line 106: @@
 | 115.86
 | 77/72
-| [[Mercy]] / [[countermiracle]]
+| [[Countermiracle]]
 |-
 | 1
@@ Line 97: / Line 112: @@
 | 322.76
 | 3087/2560
-| [[Senior]] / [[seniority]]
+| [[Seniority]] / senator
 |-
 | 1
@@ Line 103: / Line 118: @@
 | 339.31
 | 128/105
-| [[Amity]]
+| [[Amity]] / catamite
 |-
 | 5
@@ Line 117: / Line 132: @@
 | [[Mystery]]
 |}
+== Scales ==
+* [[Magic7]]
+* [[Magic10]]
+* [[Magic13]]
+* [[Magic16]]
+* [[Magic19]]
+* [[Magic22]]
 == Music ==
-* [http://micro.soonlabel.com/magic/daily20120113-piano-magic16-.mp3 Chromatic piece in magic 16] [http://www.chrisvaisvil.com/ Chris Vaisvil]
+; [[Chris Vaisvil]] ([http://www.chrisvaisvil.com/ site])
+* [http://micro.soonlabel.com/magic/daily20120113-piano-magic16-.mp3 ''Chromatic piece in magic 16''] – magic[16] in 145edo tuning
-[[Category:145edo]]
-[[Category:Equal divisions of the octave]]
 [[Category:Mystery]]
 [[Category:Pele]]
 [[Category:Magic]]
+[[Category:Listen]]