140edo: Difference between revisions

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The '''140 equal divisions of the octave''' divides the [[octave]] into 140 parts of 8.57 [[cent]]s each.  
The '''140 equal divisions of the octave''' divides the [[octave]] into 140 parts of 8.57 [[cent]]s each.  


== Theory ==
In the 5-limit, it tempers out [[15625/15552]], making it a kleismic system, and the kwazy comma, {{monzo| -53 10 16 }}. It is most notable, however, in the 7-limit, where it tempers out [[2401/2400]], [[5120/5103]], [[10976/10935]] and [[65625/65536]]. It supports the 7-limit rank-2 temperaments [[tertiaseptal]], [[hemififths]], [[countercata]] and [[bisupermajor]], and is a good tuning recommendation for countercata, the 53&140 temperament tempering out 15625/15552 and 5120/5103, and provides the [[optimal patent val]] for 13-limit countercata. In the 11-limit it tempers out [[385/384]], [[1331/1323]], [[1375/1372]], [[5632/5625]], [[6250/6237]] and [[9801/9800]], and in the 13-limit [[325/324]], [[352/351]], [[625/624]], [[676/675]], [[847/845]], [[1001/1000]], [[1716/1715]] and [[2080/2079]].
In the 5-limit, it tempers out [[15625/15552]], making it a kleismic system, and the kwazy comma, {{monzo| -53 10 16 }}. It is most notable, however, in the 7-limit, where it tempers out [[2401/2400]], [[5120/5103]], [[10976/10935]] and [[65625/65536]]. It supports the 7-limit rank-2 temperaments [[tertiaseptal]], [[hemififths]], [[countercata]] and [[bisupermajor]], and is a good tuning recommendation for countercata, the 53&140 temperament tempering out 15625/15552 and 5120/5103, and provides the [[optimal patent val]] for 13-limit countercata. In the 11-limit it tempers out [[385/384]], [[1331/1323]], [[1375/1372]], [[5632/5625]], [[6250/6237]] and [[9801/9800]], and in the 13-limit [[325/324]], [[352/351]], [[625/624]], [[676/675]], [[847/845]], [[1001/1000]], [[1716/1715]] and [[2080/2079]].


If we use the [[val]] {{val| 140 223 325 394 }} (140bbd) we obtain a tuning for [[porcupine]] temperament; the generator 19\140 is 0.023 cents flat of the [[POTE generator]].
If we use the [[val]] {{val| 140 223 325 394 }} (140bbd) we obtain a tuning for [[porcupine]] temperament; the generator 19\140 is 0.023 cents flat of the [[POTE generator]].
=== Prime harmonics ===
{{Primes in edo|140}}
== 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
| 15625/15552, {{monzo| 35 -25 2 }}
| [{{val| 140 222 325 }}]
| -0.104
| 0.346
| 4.03
|-
| 2.3.5.7
| 2401/2400, 5120/5103, 15625/15552
| [{{val| 140 222 325 393 }}]
| -0.055
| 0.311
| 3.63
|-
| 2.3.5.7.11
| 385/384, 1331/1323, 1375/1372, 2200/2187
| [{{val| 140 222 325 393 484 }}]
| +0.115
| 0.439
| 5.12
|-
| 2.3.5.7.11.13
| 325/324, 352/351, 385/384, 625/624, 1331/1323
| [{{val| 140 222 325 393 484 518 }}]
| +0.119
| 0.401
| 4.68
|}
=== 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
| 9\140
| 77.14
| 22/21
| [[Tertiaseptal]] / [[tertia]]
|-
| 1
| 13\140
| 111.43
| 16/15
| [[Stockhausenic]]
|-
| 1
| 37\140
| 317.14
| 6/5
| [[Hanson]] / [[countercata]]
|-
| 1
| 41\140
| 351.43
| 49/40
| [[Hemififths]]
|-
| 1
| 53\140
| 454.29
| 13/10
| [[Fibo]]
|-
| 1
| 59\140
| 505.71
| 75/56
| [[Marfifths]]
|-
| 2
| 3\140
| 25.71
| 64/63
| [[Ketchup]]
|-
| 2
| 19\140
| 162.86
| 11/10
| [[Kwazy]] / [[bisupermajor]]
|-
| 2
| 41\140<br>(29\140)
| 351.43<br>(248.57)
| 49/40<br>(15/13)
| [[Semihemi]]
|-
| 4
| 37\140<br>(2\140)
| 317.14<br>(17.14)
| 6/5<br>(126/125)
| [[Quadritikleismic]]
|-
| 4
| 58\140<br>(12\140)
| 497.14<br>(102.86)
| 4/3<br>(35/33)
| [[Undim]]
|-
| 10
| 13\140<br>(1\140)
| 111.43<br>(8.57)
| 16/15<br>(176/175)
| [[Decoid]]
|-
| 20
| 54\140<br>(2\140)
| 497.14<br>(17.14)
| 4/3<br>(126/125)
| [[Degrees]]
|-
| 28
| 54\140<br>(2\140)
| 497.14<br>(17.14)
| 4/3<br>(126/125)
| [[Oquatonic]]
|}


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

Revision as of 17:43, 12 July 2021

The 140 equal divisions of the octave divides the octave into 140 parts of 8.57 cents each.

Theory

In the 5-limit, it tempers out 15625/15552, making it a kleismic system, and the kwazy comma, [-53 10 16⟩. It is most notable, however, in the 7-limit, where it tempers out 2401/2400, 5120/5103, 10976/10935 and 65625/65536. It supports the 7-limit rank-2 temperaments tertiaseptal, hemififths, countercata and bisupermajor, and is a good tuning recommendation for countercata, the 53&140 temperament tempering out 15625/15552 and 5120/5103, and provides the optimal patent val for 13-limit countercata. In the 11-limit it tempers out 385/384, 1331/1323, 1375/1372, 5632/5625, 6250/6237 and 9801/9800, and in the 13-limit 325/324, 352/351, 625/624, 676/675, 847/845, 1001/1000, 1716/1715 and 2080/2079.

If we use the val ⟨140 223 325 394] (140bbd) we obtain a tuning for porcupine temperament; the generator 19\140 is 0.023 cents flat of the POTE generator.

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 15625/15552, [35 -25 2⟩ [⟨140 222 325]] -0.104 0.346 4.03
2.3.5.7 2401/2400, 5120/5103, 15625/15552 [⟨140 222 325 393]] -0.055 0.311 3.63
2.3.5.7.11 385/384, 1331/1323, 1375/1372, 2200/2187 [⟨140 222 325 393 484]] +0.115 0.439 5.12
2.3.5.7.11.13 325/324, 352/351, 385/384, 625/624, 1331/1323 [⟨140 222 325 393 484 518]] +0.119 0.401 4.68

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per octave 		Generator
(reduced) 		Cents
(reduced) 		Associated
ratio 		Temperaments
1 9\140 77.14 22/21 Tertiaseptal / tertia
1 13\140 111.43 16/15 Stockhausenic
1 37\140 317.14 6/5 Hanson / countercata
1 41\140 351.43 49/40 Hemififths
1 53\140 454.29 13/10 Fibo
1 59\140 505.71 75/56 Marfifths
2 3\140 25.71 64/63 Ketchup
2 19\140 162.86 11/10 Kwazy / bisupermajor
2 41\140
(29\140) 		351.43
(248.57) 		49/40
(15/13) 		Semihemi
4 37\140
(2\140) 		317.14
(17.14) 		6/5
(126/125) 		Quadritikleismic
4 58\140
(12\140) 		497.14
(102.86) 		4/3
(35/33) 		Undim
10 13\140
(1\140) 		111.43
(8.57) 		16/15
(176/175) 		Decoid
20 54\140
(2\140) 		497.14
(17.14) 		4/3
(126/125) 		Degrees
28 54\140
(2\140) 		497.14
(17.14) 		4/3
(126/125) 		Oquatonic
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