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{{Infobox ET}}
{{Infobox ET}}
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.
{{ED intro}}


== Theory ==
== 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]].  
The 145c val provides a tuning for [[magic]] which is nearly identical to the [[POTE tuning]].  


=== Prime harmonics ===
=== Prime harmonics ===
{{Primes in edo|145|columns=9}}
{{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 ==
== Regular temperament properties ==
{| 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]]
! rowspan="2" | [[Comma list|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]] (¢)
Line 25: Line 33:
| 2.3.5
| 2.3.5
| 1600000/1594323, {{monzo| 28 -3 -10 }}
| 1600000/1594323, {{monzo| 28 -3 -10 }}
| [{{val| 145 230 337 }}]
| {{Mapping| 145 230 337 }}
| -0.695
| -0.695
| 0.498
| 0.498
Line 32: Line 40:
| 2.3.5.7
| 2.3.5.7
| 4375/4374, 5120/5103, 50421/50000
| 4375/4374, 5120/5103, 50421/50000
| [{{val| 145 230 337 407 }}]
| {{Mapping| 145 230 337 407 }}
| -0.472
| -0.472
| 0.578
| 0.578
Line 39: Line 47:
| 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
| [{{val| 145 230 337 407 502 }}]
| {{Mapping| 145 230 337 407 502 }}
| -0.561
| -0.561
| 0.547
| 0.547
Line 46: Line 54:
| 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
| [{{val| 145 230 337 407 502 537 }}]
| {{Mapping| 145 230 337 407 502 537 }}
| -0.630
| -0.630
| 0.522
| 0.522
Line 53: Line 61:
| 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
| [{{val| 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
|}
|}


Line 62: Line 84:
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+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
! Temperaments
|-
|-
Line 77: Line 99:
| 12\145
| 12\145
| 99.31
| 99.31
| 35/33, 18/17
| 18/17
| [[Quinticosiennic]]
| [[Quinticosiennic]]
|-
|-
Line 84: Line 106:
| 115.86
| 115.86
| 77/72
| 77/72
| [[Mercy]] / [[countermiracle]]
| [[Countermiracle]]
|-
|-
| 1
| 1
Line 90: Line 112:
| 322.76
| 322.76
| 3087/2560
| 3087/2560
| [[Senior]] / [[seniority]]
| [[Seniority]] / senator
|-
|-
| 1
| 1
Line 96: Line 118:
| 339.31
| 339.31
| 128/105
| 128/105
| [[Amity]]
| [[Amity]] / catamite
|-
|-
| 5
| 5
Line 110: Line 132:
| [[Mystery]]
| [[Mystery]]
|}
|}
== Scales ==
* [[Magic7]]
* [[Magic10]]
* [[Magic13]]
* [[Magic16]]
* [[Magic19]]
* [[Magic22]]


== Music ==
== 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|###]] <!-- 3-digit number -->
[[Category:Mystery]]
[[Category:Mystery]]
[[Category:Pele]]
[[Category:Pele]]
[[Category:Magic]]
[[Category:Magic]]
[[Category:Listen]]

Latest revision as of 13:49, 24 March 2025

← 144edo 145edo 146edo →
Prime factorization 5 × 29
Step size 8.27586 ¢ 
Fifth 85\145 (703.448 ¢) (→ 17\29)
Semitones (A1:m2) 15:10 (124.1 ¢ : 82.76 ¢)
Consistency limit 11
Distinct consistency limit 11

145 equal divisions of the octave (abbreviated 145edo or 145ed2), also called 145-tone equal temperament (145tet) or 145 equal temperament (145et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 145 equal parts of about 8.28 ¢ each. Each step represents a frequency ratio of 21/145, or the 145th root of 2.

Theory

145 = 5 × 29, and 145edo shares the same perfect fifth with 29edo. It is generally a sharp-tending system, with prime harmonics 3 to 23 all tuned sharp except for 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 complements being the only intervals going over the line.

As an equal temperament, 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, 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 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 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

Approximation of prime harmonics in 145edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +1.49 +2.65 -0.55 +3.16 +3.61 +2.63 +0.42 +0.69 -3.37 -2.97
Relative (%) +0.0 +18.0 +32.0 -6.6 +38.2 +43.6 +31.8 +5.1 +8.4 -40.7 -35.8
Steps
(reduced) 		145
(0) 		230
(85) 		337
(47) 		407
(117) 		502
(67) 		537
(102) 		593
(13) 		616
(36) 		656
(76) 		704
(124) 		718
(138)

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 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

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
2.3.5.7.11.13.17.19 196/195, 256/255, 343/342, 352/351, 361/360, 364/363, 476/475 [⟨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 [⟨145 230 337 407 502 537 593 616 656]] -0.519 0.476 5.75

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
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
(9\145) 		554.48
(74.48) 		11/8
(25/24) 		Trisedodge / countdown
29 60\145
(2\145) 		496.55
(16.55) 		4/3
(100/99) 		Mystery

Scales

Music

Chris Vaisvil (site)
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