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145edo: Difference between revisions

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== 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.  
{{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]], [[15/14]], [[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 ([[amity comma]]) and {{monzo| 28 -3 -10 }} ([[mabila comma]]) in the [[5-limit]]; 4375/4374 ([[ragisma]]), 5120/5103 ([[aberschisma]]), and 50421/50000 ([[trimyna comma]]) 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.  
It provides an excellent [[optimal patent val]] for the 5-limit [[mabila]] temperament, 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. 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]].  
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=== Octave stretch ===
=== 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.  
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 ===
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{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! 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]] (¢)
! [[TE error|Absolute]] (¢)
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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 }}
| {{Mapping| 145 230 337 }}
| -0.695
| -0.695
| 0.498
| 0.498
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| 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 }}
| {{Mapping| 145 230 337 407 }}
| -0.472
| -0.472
| 0.578
| 0.578
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| 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 }}
| {{Mapping| 145 230 337 407 502 }}
| -0.561
| -0.561
| 0.547
| 0.547
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| 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 }}
| {{Mapping| 145 230 337 407 502 537 }}
| -0.630
| -0.630
| 0.522
| 0.522
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| 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 }}
| {{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
|}
|}


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! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br>Ratio*
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
|-
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| 77/72
| 77/72
| [[Countermiracle]]
| [[Countermiracle]]
|-
| 1
| 32\145
| 264.83
| 7/6
| [[Hemimabila]]
|-
|-
| 1
| 1
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| 339.31
| 339.31
| 128/105
| 128/105
| [[Amity]] / catamite
| [[Amity]] / stalagmite
|-
| 1
| 64\145
| 529.66
| 512/375
| [[Mabila]]
|-
|-
| 5
| 5
| 67\145<br>(9\145)
| 9\145
| 554.48<br>(74.48)
| 74.48
| 11/8<br>(25/24)
| 25/24
| [[Trisedodge]] / [[countdown]]
| [[Trisedodge]] / [[countdown]]
|-
|-
| 29
| 29
| 60\145<br>(2\145)
| 2\145
| 496.55<br>(16.55)
| 16.55
| 4/3<br>(100/99)
| 100/99
| [[Mystery]]
| [[Mystery]]
|}
|}
<nowiki/>* In [[Normal forms #Minimal-generator form|minimal-generator form]]


== Scales ==
== Scales ==
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== Music ==
== Music ==
; [[Chris Vaisvil]] ([http://www.chrisvaisvil.com/ site])
; [[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
* [http://micro.soonlabel.com/magic/daily20120113-piano-magic16-.mp3 ''Chromatic piece in magic 16''] – Magic[16] in 145edo tuning


[[Category:Mabila]]
[[Category:Mystery]]
[[Category:Mystery]]
[[Category:Pele]]
[[Category:Pele]]
[[Category:Magic]]
[[Category:Magic]]
[[Category:Listen]]
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/145edo"