140edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
The '''140 equal divisions of the octave''' ('''140edo'''), or the '''140(-tone) equal temperament''' ('''140tet''', '''140et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 140 parts of about 8.57 [[cent]]s each.
{{EDO intro|140}}


== Theory ==
== Theory ==
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=== Prime harmonics ===
=== Prime harmonics ===
{{Primes in edo|140}}
{{Harmonics in equal|140}}


== 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]] (¢)
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{| 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<br>(Reduced)
! Cents<br>(reduced)
! Cents<br>(Reduced)
! Associated<br>ratio
! Associated<br>Ratio
! Temperaments
! Temperaments
|-
|-

Revision as of 07:09, 10 January 2023

← 139edo 140edo 141edo →
Prime factorization 22 × 5 × 7
Step size 8.57143 ¢ 
Fifth 82\140 (702.857 ¢) (→ 41\70)
Semitones (A1:m2) 14:10 (120 ¢ : 85.71 ¢)
Consistency limit 9
Distinct consistency limit 9

Template:EDO intro

Theory

In the 5-limit, 140et 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

Approximation of prime harmonics in 140edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.90 -0.60 -0.25 -2.75 -0.53 -2.10 +2.49 -2.56 -1.01 +3.54
Relative (%) +0.0 +10.5 -7.0 -3.0 -32.0 -6.2 -24.5 +29.0 -29.9 -11.7 +41.3
Steps
(reduced) 		140
(0) 		222
(82) 		325
(45) 		393
(113) 		484
(64) 		518
(98) 		572
(12) 		595
(35) 		633
(73) 		680
(120) 		694
(134)

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 8ve 		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
5 43\140
(13\140) 		368.57
(111.43) 		10125/8192
(16/15) 		Qintosec
10 29\140
(1\140) 		248.57
(8.57) 		15/13
(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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