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The ''145 equal division'' divides the octave into 145 equal parts of 8.276 cents each. It is the [[Optimal_patent_val|optimal patent val]] for 11-limit [[Hemifamity_temperaments|mystery temperament]] and 11-limit rank three temperament [[Hemifamity_family|pele temperament]]. It 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. The 145c val provides a tuning for [[Magic|magic]] which is nearly identical to the POTE tuning. It also supports and provides a good tuning for 13-limit mystery, and because it tempers out 441/440 it allows [[werckismic_chords|werckismic chords]], because it tempers out 196/195 it allows [[mynucumic_chords|mynucumic chords]], because it tempers out 352/351 it allows [[minthmic_chords|minthmic chords]], because it tempers out 364/363 it allows [[gentle_chords|gentle chords]], and because it tempers out 847/845 it allows the [[cuthbert_triad|cuthbert triad]], making it a very flexible harmonic system. The same is true of [[232edo|232edo]], the optimal patent val for 13-limit mystery.
{{Infobox ET}}
{{ED intro}}


=Music=
== Theory ==
[http://micro.soonlabel.com/magic/daily20120113-piano-magic16-.mp3 Chromatic piece in magic 16] [http://www.chrisvaisvil.com/ Chris Vaisvil]
{{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]].
 
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|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 ===
145edo contains [[5edo]] and [[29edo]] as subset edos.
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| 1600000/1594323, {{monzo| 28 -3 -10 }}
| {{Mapping| 145 230 337 }}
| -0.695
| 0.498
| 6.02
|-
| 2.3.5.7
| 4375/4374, 5120/5103, 50421/50000
| {{Mapping| 145 230 337 407 }}
| -0.472
| 0.578
| 6.99
|-
| 2.3.5.7.11
| 441/440, 896/891, 3388/3375, 4375/4374
| {{Mapping| 145 230 337 407 502 }}
| -0.561
| 0.547
| 6.61
|-
| 2.3.5.7.11.13
| 196/195, 352/351, 364/363, 676/675, 4375/4374
| {{Mapping| 145 230 337 407 502 537 }}
| -0.630
| 0.522
| 6.32
|-
| 2.3.5.7.11.13.17
| 196/195, 256/255, 352/351, 364/363, 676/675, 1156/1155
| {{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
|}
 
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>Ratio*
! Temperaments
|-
| 1
| 2\145
| 16.55
| 100/99
| [[Quincy]]
|-
| 1
| 12\145
| 99.31
| 18/17
| [[Quinticosiennic]]
|-
| 1
| 14\145
| 115.86
| 77/72
| [[Countermiracle]]
|-
| 1
| 39\145
| 322.76
| 3087/2560
| [[Seniority]] / senator
|-
| 1
| 41\145
| 339.31
| 128/105
| [[Amity]] / catamite
|-
| 5
| 67\145<br>(9\145)
| 554.48<br>(74.48)
| 11/8<br>(25/24)
| [[Trisedodge]] / [[countdown]]
|-
| 29
| 60\145<br>(2\145)
| 496.55<br>(16.55)
| 4/3<br>(100/99)
| [[Mystery]]
|}
 
== Scales ==
* [[Magic7]]
* [[Magic10]]
* [[Magic13]]
* [[Magic16]]
* [[Magic19]]
* [[Magic22]]
 
== Music ==
; [[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:Mystery]]
[[Category:Pele]]
[[Category:Magic]]
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/145edo"