# 140edo

 ← 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

140 equal divisions of the octave (abbreviated 140edo or 140ed2), also called 140-tone equal temperament (140tet) or 140 equal temperament (140et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 140 equal parts of about 8.57 ¢ each. Each step represents a frequency ratio of 21/140, or the 140th root of 2.

## 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

Approximation of prime harmonics in 140edo
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)
Approximation of prime harmonics in 140edo (continued)
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

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
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
• 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

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* 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 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)
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

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct