540edo
| ← 539edo | 540edo | 541edo → |
540 equal divisions of the octave (abbreviated 540edo or 540ed2), also called 540-tone equal temperament (540tet) or 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 540th 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 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.
The approximated 29 and 31 are relatively weak, but 37, 41 and 43 are quite spot on, with the 43 coming from 270edo. For this reason, we may consider it as a full 43-limit system. For all the primes starting with 29, it removes the distinction of otonal and utonal superparticular pairs (e.g. 29/28 vs 30/29 for prime 29) by tempering out the corresponding square superparticulars, which is responsible for its slightly flat-tending tuning profile. Prime 47 does not have that privilege and falls practically halfway between, though the sharp mapping might be preferred to keep 47/46 wider than 48/47. As a compensation, you do get a spot-on prime 53 for free.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | +0.267 | +0.353 | +0.063 | -0.207 | -0.528 | -0.511 | +0.265 | +0.615 | -0.688 | -0.591 | -0.233 |
| Relative (%) | +0.0 | +12.0 | +15.9 | +2.8 | -9.3 | -23.7 | -23.0 | +11.9 | +27.7 | -31.0 | -26.6 | -10.5 | |
| Steps (reduced) |
540 (0) |
856 (316) |
1254 (174) |
1516 (436) |
1868 (248) |
1998 (378) |
2207 (47) |
2294 (134) |
2443 (283) |
2623 (463) |
2675 (515) |
2813 (113) | |
| Harmonic | 41 | 43 | 47 | 53 | 59 | 61 | 67 | 71 | 73 | 79 | 83 | 89 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -0.174 | -0.407 | -1.062 | -0.171 | +0.828 | +0.893 | +0.693 | +0.303 | +1.099 | -0.092 | +1.064 | +0.231 |
| Relative (%) | -7.8 | -18.3 | -47.8 | -7.7 | +37.3 | +40.2 | +31.2 | +13.7 | +49.5 | -4.2 | +47.9 | +10.4 | |
| Steps (reduced) |
2893 (193) |
2930 (230) |
2999 (299) |
3093 (393) |
3177 (477) |
3203 (503) |
3276 (36) |
3321 (81) |
3343 (103) |
3404 (164) |
3443 (203) |
3497 (257) | |
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].
Approximation to JI
The following tables show how 23-odd-limit intervals are represented in 540edo. 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 |
| 23/15, 30/23 | 0.006 | 0.3 |
| 17/13, 26/17 | 0.017 | 0.8 |
| 21/20, 40/21 | 0.023 | 1.0 |
| 7/4, 8/7 | 0.063 | 2.8 |
| 21/19, 38/21 | 0.065 | 2.9 |
| 23/18, 36/23 | 0.080 | 3.6 |
| 5/3, 6/5 | 0.086 | 3.9 |
| 19/10, 20/19 | 0.088 | 4.0 |
| 9/5, 10/9 | 0.181 | 8.2 |
| 19/14, 28/19 | 0.202 | 9.1 |
| 7/6, 12/7 | 0.204 | 9.2 |
| 11/8, 16/11 | 0.207 | 9.3 |
| 23/20, 40/23 | 0.262 | 11.8 |
| 19/16, 32/19 | 0.265 | 11.9 |
| 3/2, 4/3 | 0.267 | 12.0 |
| 19/18, 36/19 | 0.270 | 12.1 |
| 11/7, 14/11 | 0.270 | 12.1 |
| 23/21, 42/23 | 0.284 | 12.8 |
| 7/5, 10/7 | 0.290 | 13.0 |
| 17/11, 22/17 | 0.304 | 13.7 |
| 13/11, 22/13 | 0.321 | 14.4 |
| 21/16, 32/21 | 0.330 | 14.9 |
| 23/12, 24/23 | 0.347 | 15.6 |
| 23/19, 38/23 | 0.350 | 15.7 |
| 5/4, 8/5 | 0.353 | 15.9 |
| 19/15, 30/19 | 0.355 | 16.0 |
| 9/7, 14/9 | 0.471 | 21.2 |
| 19/11, 22/19 | 0.472 | 21.2 |
| 11/6, 12/11 | 0.474 | 21.3 |
| 17/16, 32/17 | 0.511 | 23.0 |
| 13/8, 16/13 | 0.528 | 23.7 |
| 9/8, 16/9 | 0.534 | 24.0 |
| 21/11, 22/21 | 0.537 | 24.2 |
| 23/14, 28/23 | 0.552 | 24.8 |
| 15/14, 28/15 | 0.557 | 25.1 |
| 11/10, 20/11 | 0.560 | 25.2 |
| 17/14, 28/17 | 0.574 | 25.8 |
| 13/7, 14/13 | 0.591 | 26.6 |
| 23/16, 32/23 | 0.615 | 27.7 |
| 15/8, 16/15 | 0.620 | 27.9 |
| 11/9, 18/11 | 0.741 | 33.4 |
| 19/17, 34/19 | 0.776 | 34.9 |
| 17/12, 24/17 | 0.778 | 35.0 |
| 19/13, 26/19 | 0.792 | 35.7 |
| 13/12, 24/13 | 0.795 | 35.8 |
| 23/22, 44/23 | 0.821 | 37.0 |
| 15/11, 22/15 | 0.827 | 37.2 |
| 21/17, 34/21 | 0.841 | 37.9 |
| 21/13, 26/21 | 0.858 | 38.6 |
| 17/10, 20/17 | 0.864 | 38.9 |
| 13/10, 20/13 | 0.881 | 39.6 |
| 17/9, 18/17 | 1.045 | 47.0 |
| 13/9, 18/13 | 1.062 | 47.8 |
| 15/13, 26/15 | 1.074 | 48.3 |
| 23/13, 26/23 | 1.080 | 48.6 |
| 17/15, 30/17 | 1.091 | 49.1 |
| 23/17, 34/23 | 1.097 | 49.4 |
| Interval and complement | Error (abs, ¢) | Error (rel, %) |
|---|---|---|
| 1/1, 2/1 | 0.000 | 0.0 |
| 19/12, 24/19 | 0.002 | 0.1 |
| 23/15, 30/23 | 0.006 | 0.3 |
| 17/13, 26/17 | 0.017 | 0.8 |
| 21/20, 40/21 | 0.023 | 1.0 |
| 7/4, 8/7 | 0.063 | 2.8 |
| 21/19, 38/21 | 0.065 | 2.9 |
| 23/18, 36/23 | 0.080 | 3.6 |
| 5/3, 6/5 | 0.086 | 3.9 |
| 19/10, 20/19 | 0.088 | 4.0 |
| 9/5, 10/9 | 0.181 | 8.2 |
| 19/14, 28/19 | 0.202 | 9.1 |
| 7/6, 12/7 | 0.204 | 9.2 |
| 11/8, 16/11 | 0.207 | 9.3 |
| 23/20, 40/23 | 0.262 | 11.8 |
| 19/16, 32/19 | 0.265 | 11.9 |
| 3/2, 4/3 | 0.267 | 12.0 |
| 19/18, 36/19 | 0.270 | 12.1 |
| 11/7, 14/11 | 0.270 | 12.1 |
| 23/21, 42/23 | 0.284 | 12.8 |
| 7/5, 10/7 | 0.290 | 13.0 |
| 17/11, 22/17 | 0.304 | 13.7 |
| 13/11, 22/13 | 0.321 | 14.4 |
| 21/16, 32/21 | 0.330 | 14.9 |
| 23/12, 24/23 | 0.347 | 15.6 |
| 23/19, 38/23 | 0.350 | 15.7 |
| 5/4, 8/5 | 0.353 | 15.9 |
| 19/15, 30/19 | 0.355 | 16.0 |
| 9/7, 14/9 | 0.471 | 21.2 |
| 19/11, 22/19 | 0.472 | 21.2 |
| 11/6, 12/11 | 0.474 | 21.3 |
| 17/16, 32/17 | 0.511 | 23.0 |
| 13/8, 16/13 | 0.528 | 23.7 |
| 9/8, 16/9 | 0.534 | 24.0 |
| 21/11, 22/21 | 0.537 | 24.2 |
| 23/14, 28/23 | 0.552 | 24.8 |
| 15/14, 28/15 | 0.557 | 25.1 |
| 11/10, 20/11 | 0.560 | 25.2 |
| 17/14, 28/17 | 0.574 | 25.8 |
| 13/7, 14/13 | 0.591 | 26.6 |
| 23/16, 32/23 | 0.615 | 27.7 |
| 15/8, 16/15 | 0.620 | 27.9 |
| 11/9, 18/11 | 0.741 | 33.4 |
| 19/17, 34/19 | 0.776 | 34.9 |
| 17/12, 24/17 | 0.778 | 35.0 |
| 19/13, 26/19 | 0.792 | 35.7 |
| 13/12, 24/13 | 0.795 | 35.8 |
| 23/22, 44/23 | 0.821 | 37.0 |
| 15/11, 22/15 | 0.827 | 37.2 |
| 21/17, 34/21 | 0.841 | 37.9 |
| 21/13, 26/21 | 0.858 | 38.6 |
| 17/10, 20/17 | 0.864 | 38.9 |
| 13/10, 20/13 | 0.881 | 39.6 |
| 17/9, 18/17 | 1.045 | 47.0 |
| 13/9, 18/13 | 1.062 | 47.8 |
| 23/17, 34/23 | 1.126 | 50.6 |
| 17/15, 30/17 | 1.131 | 50.9 |
| 23/13, 26/23 | 1.142 | 51.4 |
| 15/13, 26/15 | 1.148 | 51.7 |
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 |