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.
Theory
270edo is an extremely strong 13-limit system, distinct and 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 (and which can be interpreted as the result of tempering 676/675). This results in it being a record edo for Pepper ambiguity in the 11-, 13- and 15-odd-limits. It is the 11th zeta gap edo, the 13th zeta integral edo, the 23rd zeta peak edo and the 18th zeta peak integer edo.
In the 5-limit it tempers out the ennealimma, [1 -27 18⟩, the vulture comma, [24 -21 4⟩, and the vishnuzma (aka semisuper comma), [23 6 -14⟩.
In the 7-limit it tempers out 2401/2400 and 4375/4374, so that it supports ennealimmal temperament; the wizma (420175/419904) and the landscape comma (250047/250000).
In the 11-limit, it tempers out 3025/3024, 5632/5625, and 9801/9800, meaning it tempers the 4 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 symbiosma (19712/19683) is tempered out as well.
Finally, in the 13-limit it isn't 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.
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 highly 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
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Divisors
270 is a very composite number. The prime factorization is: 270 = 2 × 33 × 5, with divisors 1, 2, 3, 5, 6, 9, 10, 15, 18, 27, 30, 45, 54, 90 and 135, and some of these form the periods of some rank two temperaments 270 supports; these include ennealimmal, hemiennealimmal and decitonic. This means that 270edo can be conceptualised as the superset/intersection of, for example, 10edo and 27edo, which are both interesting and somewhat peculiar in their own right.
Intervals
Here may be found a table of 270edo intervals.
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, 1540/1539, 1716/1715, 1729/1728 | [⟨270 428 627 758 934 999 1147]] | -0.0290 | 0.1028 | 2.31 |
Rank-2 temperaments
| Periods per octave |
Generator (reduced) |
Cents (reduced) |
Associated ratio |
Temperaments |
|---|---|---|---|---|
| 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 | 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) |
10125/8192 (16/15) |
Qintosec (5-limit) |
| 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 |