145edo: Difference between revisions

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'''145edo''' divides the octave into 145 equal parts of 8.276 cents each.  
'''145edo''' divides the octave into 145 equal parts of 8.276 cents each.  


145et is the [[optimal patent val]] for 11-limit [[Hemifamity temperaments #Mystery|mystery temperament]] and 11-limit rank-3 temperament [[Hemifamity family #Pele|pele temperament]].  
== Theory ==
145et is the [[optimal patent val]] for the 11-limit [[mystery]] temperament and the 11-limit rank-3 [[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.  
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.  
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It also supports 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.  
It also supports 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.  


{{Primes in edo|145|columns=9|prec=3}}
=== Prime harmonics ===
{{Primes in edo|145|columns=9}}
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" | Subgroup
! rowspan="2" | [[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 }}
| [{{val| 145 230 337 }}]
| -0.695
| 0.498
| 6.02
|-
| 2.3.5.7
| 4375/4374, 5120/5103, 50421/50000
| [{{val| 145 230 337 407 }}]
| -0.472
| 0.578
| 6.99
|-
| 2.3.5.7.11
| 441/440, 896/891, 3388/3375, 4375/4374
| [{{val| 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
| [{{val| 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
| [{{val| 145 230 337 407 502 537 593 }}]
| -0.632
| 0.484
| 5.85
|}
 
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods<br>per octave
! Generator<br>(reduced)
! Cents<br>(reduced)
! Associated<br>ratio
! Temperaments
|-
| 1
| 2\145
| 16.55
| 100/99
| [[Quincy]]
|-
| 1
| 14\145
| 115.86
| 77/72
| [[Countermiracle]]
|-
| 1
| 39\145
| 322.76
| 3087/2560
| [[Senior]] / [[seniority]]
|-
| 1
| 41\145
| 339.31
| 243/200
| [[Amity]]
|-
| 29
| 60\145<br>(2\145)
| 496.55<br>(16.55)
| 4/3<br>(100/99)
| [[Mystery]]
|}


== Music ==
== Music ==

Revision as of 05:41, 13 July 2021

145edo divides the octave into 145 equal parts of 8.276 cents each.

Theory

145et is the optimal patent val for the 11-limit mystery temperament and the 11-limit rank-3 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 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, 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.

Prime harmonics

Script error: No such module "primes_in_edo".

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3.5 1600000/1594323, [28 -3 -10⟩ [⟨145 230 337]] -0.695 0.498 6.02
2.3.5.7 4375/4374, 5120/5103, 50421/50000 [⟨145 230 337 407]] -0.472 0.578 6.99
2.3.5.7.11 441/440, 896/891, 3388/3375, 4375/4374 [⟨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 [⟨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 [⟨145 230 337 407 502 537 593]] -0.632 0.484 5.85

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per octave 		Generator
(reduced) 		Cents
(reduced) 		Associated
ratio 		Temperaments
1 2\145 16.55 100/99 Quincy
1 14\145 115.86 77/72 Countermiracle
1 39\145 322.76 3087/2560 Senior / seniority
1 41\145 339.31 243/200 Amity
29 60\145
(2\145) 		496.55
(16.55) 		4/3
(100/99) 		Mystery

Music

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