270edo
| ← 269edo | 270edo | 271edo → |
The 270 equal divisions of the octave (270edo), or the 270(-tone) equal temperament (270tet, 270et) when viewed from a regular temperament perspective, divides the octave into 270 equal parts of 4.4 cents each, a size close to 385/384, the keenanisma.
270edo's step size is called a tredek when used as an interval size unit.
Theory
270edo is an extremely strong 13-limit system, distinctly consistent through the 15-odd-limit with all intervals in the 15-odd-limit being approximated with less than 25% relative error with only the exception of 15/13 which barely misses (corresponding to the fact of tempering out 676/675). This results in it being a record edo for Pepper ambiguity in the 11-, 13- and 15-odd-limit. It is the 11th zeta gap edo, the 13th zeta integral edo, the 23rd zeta peak edo, and the 18th zeta peak integer edo, making it a strict zeta edo, and is the first non-trivial edo to be consistent in the 16-odd-prime-sum-limit.
In the 5-limit it tempers out the ennealimma, [1 -27 18⟩, the vulture comma, [24 -21 4⟩, and the vishnuzma (a.k.a. semisuper comma), [23 6 -14⟩.
In the 7-limit it tempers out the breedsma (2401/2400), the ragisma (4375/4374), the wizma (420175/419904), and the landscape comma (250047/250000), so that it supports ennealimmal temperament. It also tempers out the quasiorwellisma (29360128/29296875) and the garischisma (33554432/33480783).
In the 11-limit, it tempers out 3025/3024, 5632/5625, and 9801/9800, meaning it tempers out the four smallest superparticular commas in the 11-limit (2401/2400, 3025/3024, 4375/4374, and 9801/9800). In addition to these, it also tempers out both the nexus comma (1771561/1769472) and the quartisma (117440512/117406179), which, in turn means that the symbiotic comma (19712/19683) is tempered out as well.
Finally, in the 13-limit it is not quite as accurate but still very accurate, as it tempers out 676/675, 1001/1000, 1716/1715, and 2080/2079, making it an archipelago tuning, and the optimal patent val for some of the archipelago temperaments such as hemiennealimmal, vulture, eagle, and avicenna.
The excellent tuning accuracy does not bar it from the utility of essentially tempered chords, including sinbadmic chords in the 13-odd-limit, and island chords in the 15-odd-limit.
Beyond the 13-limit, the approximated harmonic 17 is more than 1/3-edostep sharp of just, and while the harmonic 19 is accurately tuned, the harmonic 23 is more than 1/3-edostep flat of just. 17/13, 23/15, and 23/17 are all the inconsistently approximated 23-odd-limit intervals, making 270edo a somewhat viable but tricky full 23-limit system. It tempers out 715/714, 936/935, 1089/1088, 1225/1224, 1701/1700, 2025/2023, 2058/2057, and 2431/2430 in the 17-limit; 1216/1215, 1331/1330, 1521/1520, 1540/1539, and 1729/1728 in the 19-limit; and 460/459, 529/528, 736/735, 897/896, 1288/1287, 1311/1309, and 1771/1768 in the 23-limit. The harmonics 29 and 31 are also more than 1/3-edostep sharp, but not as sharp as 17 to incur inconsistency with the lower primes. In fact, 270edo is consistent in the no-17 no-23 35-odd-limit. We may note that it tempers out 784/783, 900/899, and 1024/1023.
On top of this, its step size is so small as to arguably give a good enough approximation for any relatively simple JI consonance, as the maximum error is only 2.2¢. If, however, you want an edo for very high-limit use, the obvious alternative choice is 311edo, which is in many ways dual to 270edo as it emphasizes consistency and accuracy in very high-prime-limit and high-odd-limit situations at the expense of lower ones, and is a prime edo as opposed to a very composite one. While 270edo approximates the first 16 harmonics very accurately, 311edo approximates the first 42 but not as accurately – strongly favouring the approximation of as many harmonics as possible.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.267 | +0.353 | +0.063 | -0.207 | -0.528 | +1.711 | +0.265 | -1.608 | +1.534 | +1.631 | +1.989 | +2.049 | -0.407 | +1.160 | +2.051 |
| Relative (%) | +0.0 | +6.0 | +7.9 | +1.4 | -4.7 | -11.9 | +38.5 | +6.0 | -36.2 | +34.5 | +36.7 | +44.8 | +46.1 | -9.1 | +26.1 | +46.1 | |
| Steps (reduced) |
270 (0) |
428 (158) |
627 (87) |
758 (218) |
934 (124) |
999 (189) |
1104 (24) |
1147 (67) |
1221 (141) |
1312 (232) |
1338 (258) |
1407 (57) |
1447 (97) |
1465 (115) |
1500 (150) |
1547 (197) | |
Subsets and supersets
270 is a very composite number. The prime factorization is 270 = 2 × 33 × 5, with divisors 2, 3, 5, 6, 9, 10, 15, 18, 27, 30, 45, 54, 90, and 135. This means that 270edo can be conceptualised as the superset of, for example, 10edo and 27edo, which are both interesting and somewhat peculiar in their own right.
540edo, which divides the edostep in two, and 810edo, which divides the edostep in three, provide good correction for harmonics 17, 23, and beyond.
Intervals
Here may be found a table of 270edo intervals.
Approximation to JI
23-odd-limit interval mappings
The following tables show how 23-odd-limit intervals are represented in 270edo. Prime harmonics are in bold; inconsistent intervals are in italics.
| Interval and complement | Error (abs, ¢) | Error (rel, %) |
|---|---|---|
| 1/1, 2/1 | 0.000 | 0.0 |
| 19/12, 24/19 | 0.002 | 0.1 |
| 21/20, 40/21 | 0.023 | 0.5 |
| 7/4, 8/7 | 0.063 | 1.4 |
| 21/19, 38/21 | 0.065 | 1.5 |
| 5/3, 6/5 | 0.086 | 1.9 |
| 19/10, 20/19 | 0.088 | 2.0 |
| 9/5, 10/9 | 0.181 | 4.1 |
| 19/14, 28/19 | 0.202 | 4.5 |
| 7/6, 12/7 | 0.204 | 4.6 |
| 11/8, 16/11 | 0.207 | 4.7 |
| 19/16, 32/19 | 0.265 | 6.0 |
| 3/2, 4/3 | 0.267 | 6.0 |
| 19/18, 36/19 | 0.270 | 6.1 |
| 11/7, 14/11 | 0.270 | 6.1 |
| 7/5, 10/7 | 0.290 | 6.5 |
| 13/11, 22/13 | 0.321 | 7.2 |
| 21/16, 32/21 | 0.330 | 7.4 |
| 5/4, 8/5 | 0.353 | 7.9 |
| 19/15, 30/19 | 0.355 | 8.0 |
| 9/7, 14/9 | 0.471 | 10.6 |
| 19/11, 22/19 | 0.472 | 10.6 |
| 11/6, 12/11 | 0.474 | 10.7 |
| 13/8, 16/13 | 0.528 | 11.9 |
| 9/8, 16/9 | 0.534 | 12.0 |
| 21/11, 22/21 | 0.537 | 12.1 |
| 15/14, 28/15 | 0.557 | 12.5 |
| 11/10, 20/11 | 0.560 | 12.6 |
| 13/7, 14/13 | 0.591 | 13.3 |
| 15/8, 16/15 | 0.620 | 14.0 |
| 11/9, 18/11 | 0.741 | 16.7 |
| 19/13, 26/19 | 0.792 | 17.8 |
| 13/12, 24/13 | 0.795 | 17.9 |
| 15/11, 22/15 | 0.827 | 18.6 |
| 21/13, 26/21 | 0.858 | 19.3 |
| 13/10, 20/13 | 0.881 | 19.8 |
| 13/9, 18/13 | 1.062 | 23.9 |
| 23/13, 26/23 | 1.080 | 24.3 |
| 17/15, 30/17 | 1.091 | 24.5 |
| 23/17, 34/23 | 1.126 | 25.3 |
| 15/13, 26/15 | 1.148 | 25.8 |
| 17/9, 18/17 | 1.177 | 26.5 |
| 17/10, 20/17 | 1.358 | 30.6 |
| 21/17, 34/21 | 1.381 | 31.1 |
| 23/22, 44/23 | 1.401 | 31.5 |
| 17/12, 24/17 | 1.444 | 32.5 |
| 19/17, 34/19 | 1.446 | 32.5 |
| 23/16, 32/23 | 1.608 | 36.2 |
| 17/14, 28/17 | 1.648 | 37.1 |
| 23/14, 28/23 | 1.671 | 37.6 |
| 17/16, 32/17 | 1.711 | 38.5 |
| 23/19, 38/23 | 1.872 | 42.1 |
| 23/12, 24/23 | 1.875 | 42.2 |
| 17/11, 22/17 | 1.918 | 43.2 |
| 23/21, 42/23 | 1.938 | 43.6 |
| 23/20, 40/23 | 1.961 | 44.1 |
| 23/18, 36/23 | 2.142 | 48.2 |
| 17/13, 26/17 | 2.206 | 49.6 |
| 23/15, 30/23 | 2.217 | 49.9 |
| Interval and complement | Error (abs, ¢) | Error (rel, %) |
|---|---|---|
| 1/1, 2/1 | 0.000 | 0.0 |
| 19/12, 24/19 | 0.002 | 0.1 |
| 21/20, 40/21 | 0.023 | 0.5 |
| 7/4, 8/7 | 0.063 | 1.4 |
| 21/19, 38/21 | 0.065 | 1.5 |
| 5/3, 6/5 | 0.086 | 1.9 |
| 19/10, 20/19 | 0.088 | 2.0 |
| 9/5, 10/9 | 0.181 | 4.1 |
| 19/14, 28/19 | 0.202 | 4.5 |
| 7/6, 12/7 | 0.204 | 4.6 |
| 11/8, 16/11 | 0.207 | 4.7 |
| 19/16, 32/19 | 0.265 | 6.0 |
| 3/2, 4/3 | 0.267 | 6.0 |
| 19/18, 36/19 | 0.270 | 6.1 |
| 11/7, 14/11 | 0.270 | 6.1 |
| 7/5, 10/7 | 0.290 | 6.5 |
| 13/11, 22/13 | 0.321 | 7.2 |
| 21/16, 32/21 | 0.330 | 7.4 |
| 5/4, 8/5 | 0.353 | 7.9 |
| 19/15, 30/19 | 0.355 | 8.0 |
| 9/7, 14/9 | 0.471 | 10.6 |
| 19/11, 22/19 | 0.472 | 10.6 |
| 11/6, 12/11 | 0.474 | 10.7 |
| 13/8, 16/13 | 0.528 | 11.9 |
| 9/8, 16/9 | 0.534 | 12.0 |
| 21/11, 22/21 | 0.537 | 12.1 |
| 15/14, 28/15 | 0.557 | 12.5 |
| 11/10, 20/11 | 0.560 | 12.6 |
| 13/7, 14/13 | 0.591 | 13.3 |
| 15/8, 16/15 | 0.620 | 14.0 |
| 11/9, 18/11 | 0.741 | 16.7 |
| 19/13, 26/19 | 0.792 | 17.8 |
| 13/12, 24/13 | 0.795 | 17.9 |
| 15/11, 22/15 | 0.827 | 18.6 |
| 21/13, 26/21 | 0.858 | 19.3 |
| 13/10, 20/13 | 0.881 | 19.8 |
| 13/9, 18/13 | 1.062 | 23.9 |
| 23/13, 26/23 | 1.080 | 24.3 |
| 17/15, 30/17 | 1.091 | 24.5 |
| 15/13, 26/15 | 1.148 | 25.8 |
| 17/9, 18/17 | 1.177 | 26.5 |
| 17/10, 20/17 | 1.358 | 30.6 |
| 21/17, 34/21 | 1.381 | 31.1 |
| 23/22, 44/23 | 1.401 | 31.5 |
| 17/12, 24/17 | 1.444 | 32.5 |
| 19/17, 34/19 | 1.446 | 32.5 |
| 23/16, 32/23 | 1.608 | 36.2 |
| 17/14, 28/17 | 1.648 | 37.1 |
| 23/14, 28/23 | 1.671 | 37.6 |
| 17/16, 32/17 | 1.711 | 38.5 |
| 23/19, 38/23 | 1.872 | 42.1 |
| 23/12, 24/23 | 1.875 | 42.2 |
| 17/11, 22/17 | 1.918 | 43.2 |
| 23/21, 42/23 | 1.938 | 43.6 |
| 23/20, 40/23 | 1.961 | 44.1 |
| 23/18, 36/23 | 2.142 | 48.2 |
| 23/15, 30/23 | 2.228 | 50.1 |
| 17/13, 26/17 | 2.239 | 50.4 |
| 23/17, 34/23 | 3.319 | 74.7 |
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5 | [23 6 -14⟩, [24 -21 4⟩ | [⟨270 428 627]] | −0.1069 | 0.0759 | 1.71 |
| 2.3.5.7 | 2401/2400, 4375/4374, 29360128/29296875 | [⟨270 428 627 758]] | −0.0858 | 0.0752 | 1.69 |
| 2.3.5.7.11 | 2401/2400, 3025/3024, 4375/4374, 5632/5625 | [⟨270 428 627 758 934]] | −0.0567 | 0.0889 | 2.00 |
| 2.3.5.7.11.13 | 676/675, 1001/1000, 1716/1715, 3025/3024, 4096/4095 | [⟨270 428 627 758 934 999]] | −0.0235 | 0.1100 | 2.48 |
| 2.3.5.7.11.13.19 | 676/675, 1001/1000, 1216/1215, 1331/1330, 1540/1539, 1729/1728 | [⟨270 428 627 758 934 999 1147]] | −0.0290 | 0.1028 | 2.31 |
| 2.3.5.7.11.13.17 | 676/675, 715/714, 936/935, 1001/1000, 1225/1224, 4096/4095 | [⟨270 428 627 758 934 999 1104]] | −0.0799 | 0.1718 | 3.86 |
| 2.3.5.7.11.13.17.19 | 676/675, 715/714, 936/935, 1001/1000, 1216/1215, 1225/1224, 1331/1330 | [⟨270 428 627 758 934 999 1104 1147]] | −0.0777 | 0.1608 | 3.62 |
| 2.3.5.7.11.13.17.19.23 | 460/459, 529/528, 676/675, 715/714, 736/735, 936/935, 1001/1000, 1216/1215 | [⟨270 428 627 758 934 999 1104 1147 1221]] | −0.0296 | 0.2037 | 4.58 |
- 270et has lower relative errors than any previous equal temperaments in the 11-, 13-, 19-, and 23-limit. It is the first to beat 72 in the 11-limit, 224 in the 13-limit, and 217 in the 19- and 23-limit. The next equal temperament that does better in terms of either absolute or relative error in the 11-limit is 342, in the 13-limit 494, in the 23-limit 282, and in the 19-limit, 311 for absolute error and 581 for relative error.
- It is even more prominent in the 2.3.5.7.11.13.19 subgroup. Not until 552 do we reach a better equal temperament in terms of absolute error, and not until 2190 do we reach one in terms of relative error.
- It is also prominent in the 17-limit, with lower absolute errors than any previous equal temperaments, despite inconsistency in the corresponding odd limit.
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperament |
|---|---|---|---|---|
| 1 | 1\270 | 4.44 | 385/384 | Keenanose |
| 1 | 29\270 | 128.89 | 14/13 | Tertiathirds |
| 1 | 61\270 | 271.11 | 90/77 | Quasiorwell |
| 1 | 71\270 | 315.56 | 6/5 | Acrokleismic / counteracro |
| 1 | 79\270 | 351.11 | 49/40 | Newt |
| 1 | 97\270 | 431.11 | 77/60 | Lockerbie |
| 1 | 107\270 | 475.56 | 25/19 | Vulture |
| 2 | 14\270 | 62.22 | 28/27 | Eagle |
| 2 | 16\270 | 71.11 | 25/24 | Vishnu / ananta / acyuta |
| 2 | 28\270 | 124.44 | 275/256 | Semivulture |
| 2 | 47\270 | 208.89 | 44/39 | Abigail |
| 2 | 52\270 | 231.11 | 8/7 | Orga |
| 2 | 131\270 (4\270) |
582.22 (17.78) |
7/5 (99/98) |
Quarvish |
| 3 | 17\270 | 75.56 | 24/23 | Terture |
| 3 | 31\270 | 137.78 | 13/12 | Avicenna |
| 5 | 83\270 (25\270) |
368.89 (111.11) |
1024/891 (16/15) |
Quintosec |
| 6 | 112\270 (4\270) |
497.78 (97.78) |
4/3 (128/121) |
Sextile |
| 9 | 71\270 (11\270) |
315.56 (48.89) |
6/5 (36/35) |
Ennealimmal / ennealimmia |
| 10 | 16\270 (11\270) |
71.11 (48.89) |
25/24 (36/35) |
Decavish |
| 10 | 56\270 (2\270) |
248.89 (8.89) |
15/13 (176/175) |
Decoid |
| 10 | 71\270 (10\270) |
315.56 (44.44) |
6/5 (40/39) |
Deca |
| 18 | 71\270 (4\270) |
248.89 (17.78) |
15/13 (99/98) |
Hemiennealimmal |
| 18 | 71\270 (2\270) |
475.56 (8.89) |
1053/800 (1287/1280) |
Semihemiennealimmal |
| 27 | 61\270 (1\270) |
271.11 (4.44) |
1375/1176 (385/384) |
Trinealimmal |
| 30 | 82\270 (1\270) |
364.44 (4.44) |
216/175 (385/384) |
Zinc |
| 45 | 59\270 (1\270) |
262.22 (4.44) |
64/55 (385/384) |
Rhodium |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct