2460edo

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2460edo

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Prime factorization	22 × 3 × 5 × 41
Step size	0.487805 ¢
Fifth	1439\2460 (701.951 ¢)
Semitones (A1:m2)	233:185 (113.7 ¢ : 90.24 ¢)
Consistency limit	27
Distinct consistency limit	27

Theory

2460edo is distinctly consistent through to the 27-odd-limit, which is not very remarkable in itself (388edo is the first such system), but what is remarkable is the degree of accuracy to which it represents the 27-odd-limit intervals (see below). It is also a zeta peak edo, and it has been used in Sagittal notation to define the olympian level of JI notation.

As a micro- (or nano-) temperament, it is a landscape system in the 7-limit, tempering out 250047/250000, and in the 11-limit it tempers out 9801/9800. Beyond that, 10648/10647 in the 13-limit, 12376/12375 in the 17-limit, 5929/5928 and 6860/6859 in the 19-limit and 8281/8280 in the 23-limit.

Prime harmonics

Approximation of prime harmonics in 2460edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.000	-0.004	+0.028	-0.045	-0.098	-0.040	-0.077	+0.048	+0.018	+0.179	-0.158
	Relative (%)	+0.0	-0.8	+5.7	-9.3	-20.2	-8.2	-15.9	+9.8	+3.8	+36.7	-32.3
Steps (reduced)		2460 (0)	3899 (1439)	5712 (792)	6906 (1986)	8510 (1130)	9103 (1723)	10055 (215)	10450 (610)	11128 (1288)	11951 (2111)	12187 (2347)

JI approximation in the 27-odd-limit

The following table shows how 27-odd-limit intervals are represented in 2460edo. Prime harmonics are in bold.

As 2460edo is consistent in the 27-odd-limit, the mappings by direct approximation and through the patent val are identical.

27-odd-limit intervals in 2460edo

Interval and complement	Error (abs, ¢)	Error (rel, %)
1/1, 2/1	0.000	0.0
3/2, 4/3	0.004	0.8
13/7, 14/13	0.006	1.1
23/15, 30/23	0.006	1.2
25/19, 38/25	0.008	1.5
9/8, 16/9	0.008	1.6
21/13, 26/21	0.009	1.9
23/20, 40/23	0.009	1.9
27/16, 32/27	0.011	2.3
23/16, 32/23	0.018	3.8
19/10, 20/19	0.020	4.1
17/11, 22/17	0.021	4.3
23/12, 24/23	0.022	4.5
15/8, 16/15	0.024	4.9
19/15, 30/19	0.024	4.9
23/18, 36/23	0.026	5.3
5/4, 8/5	0.028	5.7
21/17, 34/21	0.028	5.8
27/26, 52/27	0.029	5.8
23/19, 38/23	0.030	6.1
27/23, 46/27	0.030	6.1
5/3, 6/5	0.032	6.5
17/14, 28/17	0.032	6.5
13/9, 18/13	0.032	6.6
27/14, 28/27	0.034	7.0
9/5, 10/9	0.035	7.2
13/12, 24/13	0.036	7.4
25/23, 46/25	0.037	7.6
17/13, 26/17	0.038	7.7
9/7, 14/9	0.038	7.8
27/20, 40/27	0.039	8.0
13/8, 16/13	0.040	8.2
7/6, 12/7	0.042	8.5
7/4, 8/7	0.045	9.3
19/16, 32/19	0.048	9.8
21/16, 32/21	0.049	10.1
21/11, 22/21	0.049	10.1
19/12, 24/19	0.052	10.6
11/7, 14/11	0.053	10.9
25/16, 32/25	0.055	11.4
19/18, 36/19	0.056	11.4
23/13, 26/23	0.058	11.9
13/11, 22/13	0.059	12.0
25/24, 48/25	0.059	12.2
27/19, 38/27	0.059	12.2
25/18, 36/25	0.063	12.9
23/14, 28/23	0.064	13.1
15/13, 26/15	0.064	13.1
27/17, 34/27	0.066	13.5
27/25, 50/27	0.067	13.7
23/21, 42/23	0.068	13.8
13/10, 20/13	0.068	13.9
15/14, 28/15	0.069	14.2
17/9, 18/17	0.070	14.3
7/5, 10/7	0.073	15.0
17/12, 24/17	0.074	15.1
21/20, 40/21	0.077	15.8
17/16, 32/17	0.077	15.9
27/22, 44/27	0.087	17.9
19/13, 26/19	0.088	18.0
11/9, 18/11	0.091	18.6
19/14, 28/19	0.093	19.1
11/6, 12/11	0.095	19.4
25/13, 26/25	0.095	19.5
23/17, 34/23	0.096	19.6
21/19, 38/21	0.097	19.9
11/8, 16/11	0.098	20.2
25/14, 28/25	0.101	20.7
17/15, 30/17	0.101	20.8
25/21, 42/25	0.105	21.5
17/10, 20/17	0.105	21.5
23/22, 44/23	0.117	23.9
15/11, 22/15	0.122	25.1
19/17, 34/19	0.125	25.7
11/10, 20/11	0.126	25.9
25/17, 34/25	0.133	27.2
19/11, 22/19	0.146	30.0
25/22, 44/25	0.154	31.6

Subsets and supersets

2460 is divisible by 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 41, 60, 82, 123, 164, 205, 246, 410, 492, 615, 820, and 1230, and its abundancy index is 1.868. Of the divisors, 12edo is too well-known to need any introduction, 41edo is an important system, and 205edo has proponents such as Aaron Andrew Hunt, who uses it as the default tuning for Hi-pi Instruments (and as a unit: mem). Aside from these, 15edo is notable for use by Easley Blackwood Jr., 60edo is a highly composite edo. In addition, 2460edo maps the schisma to an exact fraction of the octave, 4 steps. However, such mapping does not hold in 615edo.

In light of having a large amount of divisors and precise approximation of just intonation, 2460edo has been proposed as the basis for a unit, the mina, which could be used in place of the cent. Moreover, a cent is exactly 2.05 minas, and a mem, 1\205, is exactly 12 minas.

2460edo is also notable for being the smallest EDO that is a multiple of 12 to be purely consistent in the 15-odd-limit.

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3	[-3899 4320⟩	[⟨2460 3899]]	+0.001	0.001	0.24
2.3.5	[91 -12 -31⟩, [-70  72 -19⟩	[⟨2460 3899 5712]]	−0.003	0.006	1.29
2.3.5.7	250047/250000, [3 -24 3 10⟩, [-48 0 11 8⟩	[⟨2460 3899 5712 6096]]	+0.002	0.010	2.05
2.3.5.7.11	9801/9800, 151263/151250, [24 -10 -5  0 1⟩, [-3 -16 -1 6 4⟩	[⟨2460 3899 5712 6096 8510]]	+0.007	0.014	2.86
2.3.5.7.11.13	9801/9800, 10648/10647, 105644/105625, 196625/196608, 1063348/1063125	[⟨2460 3899 5712 6096 8510 9103]]	+0.008	0.013	2.63
2.3.5.7.11.13.17	9801/9800, 10648/10647, 12376/12375, 31213/31212, 37180/37179, 221221/221184	[⟨2460 3899 5712 6096 8510 9103 10055]]	+0.009	0.013	2.56

• 2460edo has lower 23-limit relative error than any edo until 8269. Also it has a lower 23-limit TE logflat badness than any smaller edo and less than any until 16808.
• In addition, it has the lowest relative error in the 19-limit, being only bettered by 3395edo.

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperaments
1	271\2460	132.195	[-38 5 13⟩	Astro
1	1219\2460	594.634	[-70 72 -19⟩	Gaster
10	583\2460 (91\2460)	284.390 (44.390)	[10 29 -24⟩ (?)	Neon
12	1021\2460 (4\2460)	498.049 (1.951)	4/3 (32805/32768)	Atomic
20	353\2460 (16\2460)	172.195 (7.805)	169/153 (?)	Calcium
30	747\2460 (9\2460)	364.390 (4.390)	216/175 (385/384)	Zinc
41	1021\2460 (1\2460)	498.049 (0.488)	4/3 ([215 -121 -10⟩)	Niobium
60	747\2460 (9\2460)	364.390 (4.390)	216/175 (385/384)	Neodymium / neodymium magnet
60	1021\2460 (4\2460)	498.049 (1.951)	4/3 (32805/32768)	Minutes

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct