540edo: Difference between revisions
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=== Prime harmonics === | === Prime harmonics === | ||
{{ | {{Harmonics in equal|540|columns=11}} | ||
=== Divisors === | === Divisors === | ||
540 is a very composite number. The prime factorization of 540 is 2<sup>2</sup> × 3<sup>3</sup> × 5. Its 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. | 540 is a very composite number. The prime factorization of 540 is 2<sup>2</sup> × 3<sup>3</sup> × 5. Its 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 == | |||
{| class="wikitable center-4 center-5 center-6" | |||
! rowspan="2" | Subgroup | |||
! rowspan="2" | [[Comma list]] | |||
! rowspan="2" | [[Mapping]] | |||
! rowspan="2" | Optimal<br>8ve stretch (¢) | |||
! colspan="2" | Tuning error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |||
| 2.3.5.7.11.13.17 | |||
| 676/675, 1001/1000, 1156/1155, 1716/1715, 3025/3024, 4096/4095 | |||
| [{{val| 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 | |||
| [{{val| 540 856 1254 1516 1868 1998 2207 2294 }}] | |||
| -0.0098 | |||
| 0.1088 | |||
| 4.90 | |||
|} | |||
[[Category:Equal divisions of the octave]] | [[Category:Equal divisions of the octave]] | ||
Revision as of 19:55, 17 March 2022
The 540 equal divisions of the octave (540edo), or the 540(-tone) equal temperament (540tet, 540et), divides the octave in 540 equal steps of about 2.22 cents each.
Theory
Since 540 = 2 × 270 and 540 = 45 × 12, it contains 270edo and 12edo as subsets, both belonging to the zeta peak edos, zeta integral edos and zeta gap edos sequences. It is enfactored in the 13-limit, with the same tuning as 270edo, but it makes for a reasonable 17- and 19-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.
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 (%) | +0.0 | +12.0 | +15.9 | +2.8 | -9.3 | -23.7 | -23.0 | +11.9 | +27.7 | -31.0 | -26.6 | |
| Steps (reduced) |
540 (0) |
856 (316) |
1254 (174) |
1516 (436) |
1868 (248) |
1998 (378) |
2207 (47) |
2294 (134) |
2443 (283) |
2623 (463) |
2675 (515) | |
Divisors
540 is a very composite number. The prime factorization of 540 is 22 × 33 × 5. Its 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 |