540edo
← 539edo | 540edo | 541edo → |
540 equal divisions of the octave (abbreviated 540edo), or 540-tone equal temperament (540tet), 540 equal temperament (540et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 540 equal parts of about 2.22 ¢ each. Each step represents a frequency ratio of 21/540, or the 540 root of 2.
Theory
Since 540 = 2 × 270 and 540 = 45 × 12, 540edo contains 270edo and 12edo as subsets, both being important zeta edos. It is enfactored in the 13-limit, with the same tuning as 270edo, but it makes for a reasonable 17-, 19- and 23-limit system, and perhaps beyond. It is, however, no longer consistent in the 15-odd-limit, all because of 15/13 being 1.14 cents sharp of just.
The equal temperament tempers out 1156/1155 and 2601/2600 in the 17-limit; 1216/1215, 1331/1330, 1445/1444 and 1729/1728 in the 19-limit; 1105/1104 and 1496/1495 in the 23-limit.
A step of 540edo is known as a dexl, proposed by Joseph Monzo in April 2023 as an interval size measure[1].
Prime harmonics
Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Error | absolute (¢) | +0.000 | +0.267 | +0.353 | +0.063 | -0.207 | -0.528 | -0.511 | +0.265 | +0.615 | -0.688 | -0.591 |
relative (%) | +0 | +12 | +16 | +3 | -9 | -24 | -23 | +12 | +28 | -31 | -27 | |
Steps (reduced) |
540 (0) |
856 (316) |
1254 (174) |
1516 (436) |
1868 (248) |
1998 (378) |
2207 (47) |
2294 (134) |
2443 (283) |
2623 (463) |
2675 (515) |
Subsets and supersets
540 is a very composite number. The prime factorization of 540 is 22 × 33 × 5. Its nontrivial divisors are 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 27, 30, 36, 45, 54, 60, 90, 108, 135, 180, and 270.
Regular temperament properties
Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
---|---|---|---|---|---|
Absolute (¢) | Relative (%) | ||||
2.3.5.7.11.13.17 | 676/675, 1001/1000, 1156/1155, 1716/1715, 3025/3024, 4096/4095 | [⟨540 856 1254 1516 1868 1998 2207]] | -0.0022 | 0.1144 | 5.15 |
2.3.5.7.11.13.17.19 | 676/675, 1001/1000, 1156/1155, 1216/1215, 1331/1330, 1445/1444, 1729/1728 | [⟨540 856 1254 1516 1868 1998 2207 2294]] | -0.0098 | 0.1088 | 4.90 |
2.3.5.7.11.13.17.19.23 | 676/675, 1001/1000, 1105/1104, 1156/1155, 1216/1215, 1331/1330, 1445/1444, 1496/1495 | [⟨540 856 1254 1516 1868 1998 2207 2294 2443]] | -0.024 | 0.1100 | 4.95 |