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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|2460}}
{{ED intro}}


== Theory ==
== Theory ==
2460edo is uniquely [[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 [[The Riemann Zeta Function and Tuning #Zeta EDO lists|zeta peak and zeta peak integer edo]], and it has been used in [[Sagittal notation]] to define the ''olympian level'' of JI notation.
2460edo is [[consistency|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 [[#Approximation to JI]]). 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.
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, it tempers out [[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]].


2460 is divisible by {{EDOs|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 [http://www.h-pi.com/theory/measurement3.html 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]], and [[615edo]] has a step close in size to the [[schisma]].
=== Prime harmonics ===
{{Harmonics in equal|2460|columns=9}}
{{Harmonics in equal|2460|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 2460edo (continued)}}


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 [[mina]]s, and a mem, 1\205, is exactly 12 minas.  
=== Subsets and supersets ===
2460 is divisible by {{EDOs| 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 [http://www.h-pi.com/theory/measurement3.html 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]].


=== Prime harmonics ===
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 [[mina]]s, and a mem, 1\205, is exactly 12 minas.
{{Harmonics in equal|2460|columns=11}}
 
2460edo is also notable for being the smallest edo that is a multiple of 12 to be [[purely consistent]] in the 15-odd-limit (i.e. it is the smallest edo that is a multiple of 12 which maintains [[relative interval error|relative error]]s of less than 25% on all of the first 16 harmonics of the harmonic series). [[72edo]] comes close, but its approximations to [[13/8]] and [[15/8]] are somewhat inaccurate.


[[Category:Equal divisions of the octave|####]]
== Approximation to JI ==
{{15-odd-limit|2460|27}}


== Regular temperament properties ==
== Regular temperament properties ==
2460edo has lower 23-limit relative error than any EDO until [[8269edo|8269]]. Also it has a lower 23-limit [[TE logflat badness]] than any smaller edo and less than any until [[16808edo|16808]].
{| class="wikitable center-4 center-5 center-6"
 
|-
In addition, it has the lowest relative error in the 19-limit, being only bettered by [[3395edo]].
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{Monzo| -3899 4320 }}
| {{Mapping| 2460 3899 }}
| +0.001
| 0.001
| 0.24
|-
| 2.3.5
| {{Monzo| 91 -12 -31 }}, {{monzo| -70  72 -19 }}
| {{Mapping| 2460 3899 5712 }}
| −0.003
| 0.006
| 1.29
|-
| 2.3.5.7
| 250047/250000, {{monzo| 3 -24 3 10 }}, {{monzo| -48 0 11 8 }}
| {{Mapping| 2460 3899 5712 6096 }}
| +0.002
| 0.010
| 2.05
|-
| 2.3.5.7.11
| 9801/9800, 151263/151250, {{monzo| 24 -10 -5  0 1 }}, {{monzo| -3 -16 -1 6 4 }}
| {{Mapping| 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
| {{Mapping| 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
| {{Mapping| 2460 3899 5712 6096 8510 9103 10055 }}
| +0.009
| 0.013
| 2.56
|}
* 2460edo has lower 23-limit relative error than any edo until [[8269edo|8269]]. Also it has a lower 23-limit [[TE logflat badness]] than any smaller edo and less than any until [[16808edo|16808]].
* In addition, it has the lowest relative error in the 19-limit, being only bettered by [[3395edo]].


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
!Periods
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
per octave
|-
!Generator
! Periods<br>per 8ve
(reduced)
! Generator*
!Cents
! Cents*
(reduced)
! Associated<br>ratio*
!Associated
! Temperaments
ratio
|-
!Temperaments
| 1
| 271\2460
| 132.195
| {{Monzo| -38 5 13 }}
| [[Astro]]
|-
| 1
| 1219\2460
| 594.634
| {{Monzo| -70 72 -19 }}
| [[Gaster]]
|-
|-
|12
| 10
|1021\2460<br>(4\2460)
| 583\2460<br>(91\2460)
|498.049<br>(1.951)
| 284.390<br>(44.390)
|4/3<br>(32805/32768)
| {{Monzo| 10 29 -24 }}<br>(?)
|[[Atomic]]
| [[Neon]]
|}<!-- 4-digit number -->
|-
| 12
| 1021\2460<br>(4\2460)
| 498.049<br>(1.951)
| 4/3<br>(32805/32768)
| [[Atomic]]
|-
| 20
| 353\2460<br>(16\2460)
| 172.195<br>(7.805)
| 169/153<br>(?)
| [[Calcium]]
|-
| 30
| 747\2460<br>(9\2460)
| 364.390<br>(4.390)
| 216/175<br>(385/384)
| [[Zinc]]
|-
| 41
| 1021\2460<br>(1\2460)
| 498.049<br>(0.488)
| 4/3<br />({{monzo| 215 -121 -10 }})
| [[Niobium]]
|-
| 60
| 747\2460<br>(9\2460)
| 364.390<br>(4.390)
| 216/175<br>(385/384)
| [[Neodymium]] / [[neodymium magnet]]
|-
| 60
| 1021\2460<br>(4\2460)
| 498.049<br>(1.951)
| 4/3<br>(32805/32768)
| [[Minutes]]
|}
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|minimal form]] in parentheses if distinct
 
[[Category:Mina]]
[[Category:Mina]]
[[Category:Zeta]]

Latest revision as of 08:21, 7 March 2025

← 2459edo 2460edo 2461edo →
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

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 #Approximation to JI). 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, it tempers out 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
Error Absolute (¢) +0.000 -0.004 +0.028 -0.045 -0.098 -0.040 -0.077 +0.048 +0.018
Relative (%) +0.0 -0.8 +5.7 -9.3 -20.2 -8.2 -15.9 +9.8 +3.8
Steps
(reduced) 		2460
(0) 		3899
(1439) 		5712
(792) 		6906
(1986) 		8510
(1130) 		9103
(1723) 		10055
(215) 		10450
(610) 		11128
(1288)
Approximation of prime harmonics in 2460edo (continued)
Harmonic 29 31 37 41 43 47 53 59 61
Error Absolute (¢) +0.179 -0.158 -0.125 +0.206 +0.190 -0.141 +0.154 -0.147 +0.188
Relative (%) +36.7 -32.3 -25.5 +42.2 +38.9 -28.9 +31.6 -30.2 +38.6
Steps
(reduced) 		11951
(2111) 		12187
(2347) 		12815
(515) 		13180
(880) 		13349
(1049) 		13664
(1364) 		14091
(1791) 		14471
(2171) 		14590
(2290)

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 (i.e. it is the smallest edo that is a multiple of 12 which maintains relative errors of less than 25% on all of the first 16 harmonics of the harmonic series). 72edo comes close, but its approximations to 13/8 and 15/8 are somewhat inaccurate.

Approximation to JI

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

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

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