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
140edo is related to 70edo, from which it inherits the slightly sharp tuning of the 3rd harmonic and the slightly flat tunings of the 11th, 13th and 17th harmonics, but the 5th and 7th harmonics are much improved, enabling it to approximate JI of various limits. Its peak form is in the 17-, 19- and 23-limit, despite the obvious lack of consistency in the corresponding odd limits. In fact, the only inconsistently mapped intervals in the 17-odd-limit are 11/9 and its octave complement, though with the 23-odd-limit, 19/11, 19/17, 23/18, 23/19 and their octave complements are also added to that list.
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.
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 primes as 2 × 5 × 7, 140edo has subset edos 2, 4, 5, 7, 10, 14, 20, 28, 35, and 70.
28edo and 35edo 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, neutral third or semisixth despite admitting both interseptimal intervals and neutral intervals. The slightly sharpened approximation of Pythagorean tuning given by 70edo is itself interesting for the peculiar property of being the first edo to not yield a better approximation of the fifth after 53edo when approximating log2(3/2)/log2(4/3) = ~1.409 as √2 = ~1.414…, though the theoretical significance is unclear.
Miscellany
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.
Interval table
Due to its inconsistencies in higher limits (as discussed in #Higher-limit JI), this table focuses primarily on the 29-limit add-41 add-53 and especially the 17-limit, with some more accurate intervals of 37 and 43 included. It was generated partially algorithmically with Godtone's code. Many additions to this interpretation from higher limits are possible; specifically, by observing omitted odds early on in the table, it's possible to guess where and why the inconsistencies arise.
| Degree | Cents | Approximate ratios* |
|---|---|---|
| 1 | 8.57 | S15, S14, S14, S13, S11 |
| 2 | 17.14 | S12, 105/104, 100/99, 99/98, 91/90, 85/84 |
| 3 | 25.71 | 78/77, 66/65, 65/64, 64/63, 55/54 |
| 4 | 34.29 | 56/55, 54/53, 53/52, 52/51, 51/50, 50/49, 49/48, 46/45 |
| 5 | 42.86 | 45/44, 44/43, 43/42, 128/125, 42/41, 41/40, 40/39, 39/38 |
| 6 | 51.43 | 36/35, 35/34, 34/33, 33/32 |
| 7 | 60.0 | 30/29, 29/28, 28/27 |
| 8 | 68.57 | 27/26, 53/51, 26/25, 51/49, 25/24 |
| 9 | 77.14 | 24/23, 23/22, 68/65, 22/21 |
| 10 | 85.71 | 21/20, 104/99, 41/39, 20/19 |
| 11 | 94.29 | 96/91, 19/18, 56/53, 37/35, 55/52 |
| 12 | 102.86 | 18/17, 35/33, 52/49, 17/16 |
| 13 | 111.43 | 16/15 |
| 14 | 120.0 | 15/14 |
| 15 | 128.57 | 14/13 |
| 16 | 137.14 | 27/25, 40/37, 53/49, 13/12 |
| 17 | 145.71 | 38/35, 25/23, 99/91, 37/34, 49/45 |
| 18 | 154.29 | 12/11, 35/32, 58/53, 23/21 |
| 19 | 162.86 | 45/41, 56/51, 11/10 |
| 20 | 171.43 | 54/49, 75/68, 32/29, 85/77, 53/48, 21/19 |
| 21 | 180.0 | 72/65, 41/37, 51/46, 10/9 |
| 22 | 188.57 | 49/44, 39/35, 29/26 |
| 23 | 197.14 | 19/17, 28/25, 102/91, 46/41, 37/33, 55/49 |
| 24 | 205.71 | 9/8, 44/39 |
| 25 | 214.29 | 26/23, 60/53, 43/38, 17/15 |
| 26 | 222.86 | 25/22, 58/51, 33/29, 41/36 |
| 27 | 231.43 | 8/7, 55/48 |
| 28 | 240.0 | 63/55, 39/34, 147/128, 85/74, 23/20 |
| 29 | 248.57 | 38/33, 53/46, 15/13, 52/45, 22/19 |
| 30 | 257.14 | 51/44, 29/25, 65/56 |
| 31 | 265.71 | 64/55, 99/85, 7/6 |
| 32 | 274.29 | 90/77, 48/41, 41/35, 75/64, 34/29, 88/75 |
| 33 | 282.86 | 27/23, 20/17, 53/45, 33/28, 46/39 |
| 34 | 291.43 | 13/11, 58/49, 45/38, 32/27 |
| 35 | 300.0 | 19/16, 44/37, 25/21 |
| 36 | 308.57 | 43/36, 49/41, 55/46 |
| 37 | 317.14 | 6/5, 77/64 |
| 38 | 325.71 | 53/44, 41/34, 35/29, 64/53, 29/24, 23/19 |
| 39 | 334.29 | 40/33, 91/75, 17/14 |
| 40 | 342.86 | 28/23, 39/32, 50/41, 128/105, 11/9 |
| 41 | 351.43 | 60/49, 49/40 |
| 42 | 360.0 | 27/22, 16/13 |
| 43 | 368.57 | 21/17, 68/55, 99/80, 26/21 |
| 44 | 377.14 | 36/29, 41/33, 46/37, 51/41, 56/45, 66/53 |
| 45 | 385.71 | 96/77, 5/4 |
| 46 | 394.29 | 64/51, 54/43, 49/39, 44/35, 34/27 |
| 47 | 402.86 | 63/50, 29/23, 53/42, 24/19 |
| 48 | 411.43 | 19/15, 52/41, 33/26, 80/63 |
| 49 | 420.0 | 108/85, 14/11, 65/51, 51/40, 23/18 |
| 50 | 428.57 | 32/25, 41/32, 50/39, 68/53, 77/60 |
| 51 | 437.14 | 9/7, 85/66, 58/45, 49/38 |
| 52 | 445.71 | 84/65, 53/41, 128/99, 75/58, 22/17, 136/105, 35/27 |
| 53 | 454.29 | 100/77, 13/10 |
| 54 | 462.86 | 30/23, 64/49, 98/75, 17/13, 55/42 |
| 55 | 471.43 | 72/55, 38/29, 21/16, 130/99, 46/35, 25/19 |
| 56 | 480.0 | 29/22, 33/25, 70/53 |
| 57 | 488.57 | 45/34, 53/40, 65/49, 85/64 |
| 58 | 497.14 | 4/3 |
| 59 | 505.71 | 91/68, 75/56, 55/41, 51/38 |
| 60 | 514.29 | 35/26, 66/49, 85/63 |
| 61 | 522.86 | 27/20, 104/77, 88/65, 65/48 |
| 62 | 531.43 | 19/14, 72/53, 53/39, 34/25, 49/36 |
| 63 | 540.0 | 15/11, 56/41, 41/30, 26/19 |
| 64 | 548.57 | 48/35, 70/51, 136/99, 11/8 |
| 65 | 557.14 | 51/37, 40/29, 29/21 |
| 66 | 565.71 | 18/13, 104/75, 68/49, 25/18 |
| 67 | 574.29 | 32/23, 39/28, 46/33, 53/38, 60/53 |
| 68 | 582.86 | 7/5 |
| 69 | 591.43 | 45/32, 128/91, 55/39 |
| 70 | 600.0 | 24/17, 41/29, 140/99, 99/70, 58/41, 17/12 |
* As a no-31's no-47's 53-limit temperament.
Approximation to JI
Higher-limit JI
140edo is very strong as a high-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 53. 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 0.5\140 = ~4.28 ¢ of error, but almost always less 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.
In fact, in the full 125-odd-limit, according to the 113-limit patent val, there is only 10 interval pairs that are mapped with more than 1\140 of error, and they are all intervals of 112 = 121, due to 11 being relatively flat. They are: 121/118, 121/114, 121/109, 121/93, 121/89, 121/83, 121/81, 121/79, 121/73, 121/62 (and their octave-complements). This is remarkable because there is an astounding 1600 interval pairs in the 125-odd-limit. As for inconsistencies, there are 374 inconsistent interval pairs in the 125-odd-limit out of 1600, or around 23%. If we omit intervals of 121, it drops to 329/1543, or around 21%.
Tonality diamond
Godtone has algorithmically determined a tonality diamond 140edo does well in using his function orderedapproximator to find odd harmonics in the order in which they are most form-fitting relative to each-other. By using odds 1 through 15 as a starting point (accepting 140edo's 13-limit patent val), the worst odds are 31, 47, 57, 59, 73, 79, 81, 83, 89, 93, 109, 121. Omitting these from the 125-odd-limit gives us that less than 7% of interval pairs are inconsistent (68/987), with only two, 55/54 and 54/53, having more than 6 ¢ of error (meaning being more than 70% of a step of 140edo out of tune). If we explicitly assume an approximately just tendency on average, by having only odd 1 in the starting set, we get this same set of 12 odds as being the most out of tune.
The reason for picking 12, rather than less or more, is that odd 27, which we want to include, is included fairly late; according to orderedapproximator via both starting sets, it is the 14th worst odd w.r.t. the odds in the 125-odd-limit aggregated so far, with the 13th worst being 43, which can be reasoned as favourable to include, as it is only slightly more sharp than 27 so that 43/27 is very accurate (less than 0.1 ¢ off).
The reason for not picking less than 12 is that immediately after 43 we add an odd we intuitively expect to be bad: 31 × 3 = 93, as 31 is relatively very sharp so that we do not want to compound it with the slightly sharp 3, and it is very complex as well so that this damage is more likely to matter. After that is 47, a large prime that is quite sharp, and after that is 121, which we deduced as being very bad, and after that is 83 and 79 (two very large primes which are relatively very off), and after that is 3 × 19 = 57, another case of a sharp prime we do not want to composite with 3. So it seems to, in both directions, be a fairly natural cutoff points.
Zeta peak index
| Tuning | Strength | Octave (cents) | Integer limit | |||||||
|---|---|---|---|---|---|---|---|---|---|---|
| ZPI | Steps per 8ve |
Step size (cents) |
Height | Integral | Gap | Size | Stretch | Consistent | Distinct | |
| Tempered | Pure | |||||||||
| 872zpi | 139.990541 | 8.572008 | 10.076688 | 9.983474 | 1.548424 | 19.514765 | 1200.081082 | 0.081082 | 10 | 10 |
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 distinct