540edo: Difference between revisions
m Adopt template: Factorization; misc. cleanup |
Compare with 270edo |
||
| Line 5: | Line 5: | ||
Since 540 = 2 × 270 and 540 = 45 × 12, 540edo contains [[270edo]] and [[12edo]] as subsets, both being important [[zeta edo]]s. It is [[enfactoring|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. | Since 540 = 2 × 270 and 540 = 45 × 12, 540edo contains [[270edo]] and [[12edo]] as subsets, both being important [[zeta edo]]s. It is [[enfactoring|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 [[tempering out|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. | The equal temperament [[tempering out|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. Although it does quite well in these limits, it is not as ''efficient'' as 270edo's original mappings, as it has greater relative errors (→ [[#Regular temperament properties]]). It is therefore a question of whether one thinks these tuning improvements and differently supplied [[essentially tempered chord]]s are worth the load of all the extra notes. | ||
=== Prime harmonics === | === Prime harmonics === | ||
| Line 14: | Line 12: | ||
=== Subsets and supersets === | === Subsets and supersets === | ||
540 is a very composite number. The [[prime factorization]] of 540 is {{factorization|540}}. Its nontrivial divisors are {{EDOs| 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 {{factorization|540}}. Its nontrivial divisors are {{EDOs| 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 27, 30, 36, 45, 54, 60, 90, 108, 135, 180, and 270 }}. | ||
A step of 540edo is known as a '''dexl''', proposed by [[Joseph Monzo]] in April 2023 as an [[interval size measure]]<ref>[http://tonalsoft.com/enc/d/dexl.aspx Tonalsoft Encyclopedia | Dexl / 540edo]</ref>. | |||
== Regular temperament properties == | == Regular temperament properties == | ||
Revision as of 10:58, 15 July 2024
| ← 539edo | 540edo | 541edo → |
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. Although it does quite well in these limits, it is not as efficient as 270edo's original mappings, as it has greater relative errors (→ #Regular temperament properties). It is therefore a question of whether one thinks these tuning improvements and differently supplied essentially tempered chords are worth the load of all the extra notes.
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) | |
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.
A step of 540edo is known as a dexl, proposed by Joseph Monzo in April 2023 as an interval size measure[1].
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 |