140edo
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 & 87 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
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | +0.90 | -0.60 | -0.25 | -2.75 | -0.53 | -2.10 | +2.49 | -2.56 | -1.01 |
| Relative (%) | +0.0 | +10.5 | -7.0 | -3.0 | -32.0 | -6.2 | -24.5 | +29.0 | -29.9 | -11.7 | |
| Steps (reduced) |
140 (0) |
222 (82) |
325 (45) |
393 (113) |
484 (64) |
518 (98) |
572 (12) |
595 (35) |
633 (73) |
680 (120) | |
| Harmonic | 31 | 37 | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +3.54 | -2.77 | -0.49 | +2.77 | +3.06 | +0.78 | +3.69 | -2.60 | -2.16 | +0.30 |
| Relative (%) | +41.3 | -32.3 | -5.7 | +32.3 | +35.8 | +9.1 | +43.0 | -30.3 | -25.2 | +3.5 | |
| Steps (reduced) |
694 (134) |
729 (29) |
750 (50) |
760 (60) |
778 (78) |
802 (102) |
824 (124) |
830 (130) |
849 (9) |
861 (21) | |
Subsets and supersets
Since 140 factors into 22 × 5 × 7, 140edo has subset edos 2, 4, 5, 7, 10, 14, 20, 28, 35, and 70.
Regular temperament properties
Template:Comma basis begin |- | 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 |- | 2.3.5.7.11.13.17 | 289/288, 325/324, 352/351, 385/384, 442/441, 625/624 | [⟨140 222 325 393 484 518 572]] | +0.176 | 0.396 | 4.62 Template:Comma basis end
- 140et has lower absolute errors than any previous equal temperaments in the 17-, 19-, and 23-limit, and perhaps beyond. In the 17-limit it is the first to beat 121 and is superseded by 171. In the 19- and 23-limit it is the first to beat 130 and is superseded by 152fg.
Rank-2 temperaments
Template:Rank-2 begin
|-
| 1
| 9\140
| 77.14
| 22/21
| Tertiaseptal / tertia
|-
| 1
| 13\140
| 111.43
| 16/15
| Stockhausenic
|-
| 1
| 37\140
| 317.14
| 6/5
| 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
| 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)
| 1024/891
(16/15)
| Quintosec
|-
| 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
Template:Rank-2 end
Template:Orf