540edo

← 539edo

540edo

541edo →

Prime factorization	22 × 33 × 5
Step size	2.22222 ¢
Fifth	316\540 (702.222 ¢) (→ 79\135)
Semitones (A1:m2)	52:40 (115.6 ¢ : 88.89 ¢)
Consistency limit	13
Distinct consistency limit	13

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

Approximation of prime harmonics in 540edo

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.

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.

23-odd-limit intervals in 540edo (direct approximation, even if inconsistent)

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

23-odd-limit intervals in 540edo (patent val mapping)

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