140edo
| ← 139edo | 140edo | 141edo → |
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.
It is very strong as a high-limit/no-limit system, performing generally very well for its size in extremely high odd-limits like 125 but also being a good choice for all odd limits 41 through 51. The main flaw is inconsistency; the cost of getting so much right is that there is a lot of things it maps inconsistently (so with more than 1\140 = ~8.57 ¢ of error), even though there is far more that it gets right. It is especially notable as a tuning of degrees (with 1\20 period), decoid (with 1\10 period) and thunderclysmic (with 1\5 period), all extending to high limits (largely) through the wealth of interpretations of intervals of 5edo.
Other info
35edo and 28edo are both interesting subsets worth exploring as a precursory assessment of the tuning qualities of 140edo; despite the fact that their 11-limit is for the most part highly damaged, their sound (depending on who you ask) may be heard to cohere in unexpected nuanced ways, which could be explained as part of the high-limit behavior of 140edo. The same is true for 70edo, which interestingly provides a dual-5's and dual-7's system of at least the 17-limit. These are also interesting because their sound is generally rather unlike that of 7edo which is the subset edo common to all of them.
140edo can be seen as a hemipyth analogue of 70edo, which has no exact semifourth or semisixth despite admitting interseptimal intervals. The slightly sharpened approximation of Pythagorean tuning given by 70edo is itself interesting for the peculiar property of being the first/smallest edo to not yield a better approximation of the fifth after 53edo when approximating log2(3/2)/log2(4/3) = ~1.409 as sqrt(2) = ~1.414..., though the theoretical significance is unclear.
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
| 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
| 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) |
Quadritikleismic |
| 4 | 58\140 (12\140) |
497.14 (102.86) |
4/3 (35/33) |
Undim |
| 5 | 37\140 (9\140) |
317.14 (77.14) |
6/5 (24/23) |
Thunderclysmic |
| 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