2460edo: Difference between revisions

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== Theory ==

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 ~~below~~). It is also a [[~~zeta edo|~~zeta peak 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]].

=== Prime harmonics ===

{{Harmonics in equal|2460|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 2460edo (continued)}}

~~=== JI approximation~~ in ~~the 27-odd-limit~~ ===

~~{{15-odd-limit~~|~~2460|27~~}}

=== 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 equal division|~~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]].

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]].

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.

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

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.

== Approximation to JI ==

== Regular temperament properties ==

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! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve stretch (¢)

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning error

|-

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Line 34:

|-

| 2.3

| {{~~monzo~~| -3899 4320 }}

| {{Monzo| -3899 4320 }}

| {{~~mapping~~| 2460 3899 }}

| {{Mapping| 2460 3899 }}

| +0.001

| 0.001

Line 40:

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|-

| 2.3.5

| {{~~monzo~~| 91 -12 -31 }}, {{monzo| -70 72 -19 }}

| {{Monzo| 91 -12 -31 }}, {{monzo| -70 72 -19 }}

| {{~~mapping~~| 2460 3899 5712 }}

| {{Mapping| 2460 3899 5712 }}

| −0.003

| 0.006

Line 48:

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| 2.3.5.7

| 250047/250000, {{monzo| 3 -24 3 10 }}, {{monzo| -48 0 11 8 }}

| {{~~mapping~~| 2460 3899 5712 6096 }}

| {{Mapping| 2460 3899 5712 6096 }}

| +0.002

| 0.010

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| 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 }}

| {{Mapping| 2460 3899 5712 6096 8510 }}

| +0.007

| 0.014

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| 2.3.5.7.11.13

| 9801/9800, 10648/10647, 105644/105625, 196625/196608, 1063348/1063125

| {{~~mapping~~| 2460 3899 5712 6096 8510 9103 }}

| {{Mapping| 2460 3899 5712 6096 8510 9103 }}

| +0.008

| 0.013

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| 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 }}

| {{Mapping| 2460 3899 5712 6096 8510 9103 10055 }}

| +0.009

| 0.013

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|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

|-

! Periods per 8ve

! Periods per 8ve

! Generator*

! Cents*

! Associated ratio*

! Associated ratio*

! Temperaments

|-

Line 100:

Line 101:

|-

| 10

| 583\2460 (91\2460)

| 583\2460 (91\2460)

| 284.390 (44.390)

| 284.390 (44.390)

| {{Monzo| 10 29 -24 }} (?)

| {{Monzo| 10 29 -24 }} (?)

| [[Neon]]

|-

| 12

| 1021\2460 (4\2460)

| 1021\2460 (4\2460)

| 498.049 (1.951)

| 498.049 (1.951)

| 4/3 (32805/32768)

| 4/3 (32805/32768)

| [[Atomic]]

|-

| 20

| 353\2460 (16\2460)

| 353\2460 (16\2460)

| 172.195 (7.805)

| 172.195 (7.805)

| 169/153 (?)

| 169/153 (?)

| [[Calcium]]

|-

| 30

| 747\2460 (9\2460)

| 747\2460 (9\2460)

| 364.390 (4.390)

| 364.390 (4.390)

| 216/175 (385/384)

| 216/175 (385/384)

| [[Zinc]]

|-

| 41

| 1021\2460 (1\2460)

| 1021\2460 (1\2460)

| 498.049 (0.488)

| 498.049 (0.488)

| 4/3 ({{monzo| 215 -121 -10 }})

| [[Niobium]]

|-

| 60

| 747\2460 (9\2460)

| 747\2460 (9\2460)

| 364.390 (4.390)

| 364.390 (4.390)

| 216/175 (385/384)

| 216/175 (385/384)

| [[Neodymium]] / [[neodymium magnet]]

|-

| 60

| 1021\2460 (4\2460)

| 1021\2460 (4\2460)

| 498.049 (1.951)

| 498.049 (1.951)

| 4/3 (32805/32768)

| 4/3 (32805/32768)

| [[Minutes]]

|}

<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[~~Normal~~ lists|minimal form]] in parentheses if distinct

<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|minimal form]] in parentheses if distinct

[[Category:Mina]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|2460}}
+{{ED intro}}
 == Theory ==
-edo 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 below). It is also a [[zeta edo|zeta peak edo]], and it has been used in [[Sagittal notation]] to define the ''olympian level'' of JI notation.
+edo 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]].
 === Prime harmonics ===
-{{Harmonics in equal|2460|columns=11}}
+{{Harmonics in equal|2460|columns=9}}
+{{Harmonics in equal|2460|columns=9|start=10|collapsed=true|title=Approximation of prime harmonics in 2460edo (continued)}}
-=== JI approximation in the 27-odd-limit ===
-{{15-odd-limit|2460|27}}
 === Subsets and supersets ===
-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 equal division|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]].
+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]].
 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.
-edo is also notable for being the smallest EDO that is a multiple of 12 to be [[Minimal consistent EDOs|purely consistent]] in the 15-odd-limit.
+edo 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.
+== Approximation to JI ==
+{{15-odd-limit|2460|27}}
 == Regular temperament properties ==
@@ Line 26: / Line 27: @@
 ! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br />8ve stretch (¢)
+! rowspan="2" | Optimal<br>8ve stretch (¢)
 ! colspan="2" | Tuning error
 |-
@@ Line 33: / Line 34: @@
 |-
 | 2.3
-| {{monzo| -3899 4320 }}
+| {{Monzo| -3899 4320 }}
-| {{mapping| 2460 3899 }}
+| {{Mapping| 2460 3899 }}
 | +0.001
 | 0.001
@@ Line 40: / Line 41: @@
 |-
 | 2.3.5
-| {{monzo| 91 -12 -31 }}, {{monzo| -70  72 -19 }}
+| {{Monzo| 91 -12 -31 }}, {{monzo| -70  72 -19 }}
-| {{mapping| 2460 3899 5712 }}
+| {{Mapping| 2460 3899 5712 }}
 | −0.003
 | 0.006
@@ Line 48: / Line 49: @@
 | 2.3.5.7
 | 250047/250000, {{monzo| 3 -24 3 10 }}, {{monzo| -48 0 11 8 }}
-| {{mapping| 2460 3899 5712 6096 }}
+| {{Mapping| 2460 3899 5712 6096 }}
 | +0.002
 | 0.010
@@ Line 55: / Line 56: @@
 | 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 }}
+| {{Mapping| 2460 3899 5712 6096 8510 }}
 | +0.007
 | 0.014
@@ Line 62: / Line 63: @@
 | 2.3.5.7.11.13
 | 9801/9800, 10648/10647, 105644/105625, 196625/196608, 1063348/1063125
-| {{mapping| 2460 3899 5712 6096 8510 9103 }}
+| {{Mapping| 2460 3899 5712 6096 8510 9103 }}
 | +0.008
 | 0.013
@@ Line 69: / Line 70: @@
 | 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 }}
+| {{Mapping| 2460 3899 5712 6096 8510 9103 10055 }}
 | +0.009
 | 0.013
@@ Line 81: / Line 82: @@
 |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
 |-
-! Periods<br />per 8ve
+! Periods<br>per 8ve
 ! Generator*
 ! Cents*
-! Associated<br />ratio*
+! Associated<br>ratio*
 ! Temperaments
 |-
@@ Line 100: / Line 101: @@
 |-
 | 10
-| 583\2460<br />(91\2460)
+| 583\2460<br>(91\2460)
-| 284.390<br />(44.390)
+| 284.390<br>(44.390)
-| {{Monzo| 10 29 -24 }}<br />(?)
+| {{Monzo| 10 29 -24 }}<br>(?)
 | [[Neon]]
 |-
 | 12
-| 1021\2460<br />(4\2460)
+| 1021\2460<br>(4\2460)
-| 498.049<br />(1.951)
+| 498.049<br>(1.951)
-| 4/3<br />(32805/32768)
+| 4/3<br>(32805/32768)
 | [[Atomic]]
 |-
 | 20
-| 353\2460<br />(16\2460)
+| 353\2460<br>(16\2460)
-| 172.195<br />(7.805)
+| 172.195<br>(7.805)
-| 169/153<br />(?)
+| 169/153<br>(?)
 | [[Calcium]]
 |-
 | 30
-| 747\2460<br />(9\2460)
+| 747\2460<br>(9\2460)
-| 364.390<br />(4.390)
+| 364.390<br>(4.390)
-| 216/175<br />(385/384)
+| 216/175<br>(385/384)
 | [[Zinc]]
 |-
 | 41
-| 1021\2460<br />(1\2460)
+| 1021\2460<br>(1\2460)
-| 498.049<br />(0.488)
+| 498.049<br>(0.488)
 | 4/3<br />({{monzo| 215 -121 -10 }})
 | [[Niobium]]
 |-
 | 60
-| 747\2460<br />(9\2460)
+| 747\2460<br>(9\2460)
-| 364.390<br />(4.390)
+| 364.390<br>(4.390)
-| 216/175<br />(385/384)
+| 216/175<br>(385/384)
 | [[Neodymium]] / [[neodymium magnet]]
 |-
 | 60
-| 1021\2460<br />(4\2460)
+| 1021\2460<br>(4\2460)
-| 498.049<br />(1.951)
+| 498.049<br>(1.951)
-| 4/3<br />(32805/32768)
+| 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
+<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[normal lists|minimal form]] in parentheses if distinct
 [[Category:Mina]]