2460edo
| ← 2459edo | 2460edo | 2461edo → |
2460 equal divisions of the octave (abbreviated 2460edo or 2460ed2), also called 2460-tone equal temperament (2460tet) or 2460 equal temperament (2460et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 2460 equal parts of about 0.488 ¢ each. Each step represents a frequency ratio of 21/2460, or the 2460th root of 2.
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 and zeta peak integer 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
| 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.
| 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.
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
| 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 it is distinct