2460edo: Difference between revisions

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== 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]].

=== Prime harmonics ===

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

=== ~~Divisors~~ ===

=== 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 [[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 ==

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 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 ===

{| class="wikitable center-all left-5"

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

|-

! Periods per 8ve

! Generator~~ (Reduced)~~

! Generator*

! Cents~~ (Reduced)~~

! Cents*

! Associated ~~Ratio~~

! Associated ratio*

! Temperaments

|-

Line 33:

Line 93:

| {{Monzo| -38 5 13 }}

| [[Astro]]

|-

| 1

| 1219\2460

| 594.634

| {{Monzo| -70 72 -19 }}

| [[Gaster]]

|-

| 10

| 583\2460 (91\2460)

| 284.390 (44.390)

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

| [[Neon]]

|-

| 12

Line 40:

Line 112:

| [[Atomic]]

|-

|20

| 20

|353\2460 (~~107~~\2460)

| 353\2460 (16\2460)

|172.195 (52.~~195~~)

| 172.195 (7.805)

|169/153 (?)

| 169/153 (?)

|[[Calcium]]

| [[Calcium]]

|-

| 30

| 747\2460 (9\2460)

| 364.390 (4.390)

| 216/175 (385/384)

| [[Zinc]]

|-

|60

| 41

|~~747~~\2460 (9\2460)

| 1021\2460 (1\2460)

|~~364~~.~~390~~ (4.~~390~~)

| 498.049 (0.488)

|~~216~~/~~175~~ (?)

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

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

| [[Niobium]]

|-

|60

| 60

|1021\2460 (4\2460)

| 747\2460 (9\2460)

|498.049 (1.951)

| 364.390 (4.390)

|4/3 (32805/32768)

| 216/175 (385/384)

|[[Minutes]]

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

|-

| 60

| 1021\2460 (4\2460)

| 498.049 (1.951)

| 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

~~[[Category:Equal divisions of the octave|####]] ~~

[[Category:Mina]]

~~[[Category:Zeta]]~~

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|2460}}
+{{ED intro}}
 == Theory ==
-edo 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.
+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)}}
-=== Divisors ===
+=== 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 [[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 ==
-edo 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 ===
 {| class="wikitable center-all left-5"
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
 ! Periods<br>per 8ve
-! Generator<br>(Reduced)
+! Generator*
-! Cents<br>(Reduced)
+! Cents*
-! Associated<br>Ratio
+! Associated<br>ratio*
 ! Temperaments
 |-
@@ Line 33: / Line 93: @@
 | {{Monzo| -38 5 13 }}
 | [[Astro]]
+|-
+| 1
+| 1219\2460
+| 594.634
+| {{Monzo| -70 72 -19 }}
+| [[Gaster]]
+|-
+| 10
+| 583\2460<br>(91\2460)
+| 284.390<br>(44.390)
+| {{Monzo| 10 29 -24 }}<br>(?)
+| [[Neon]]
 |-
 | 12
@@ Line 40: / Line 112: @@
 | [[Atomic]]
 |-
-|20
+| 20
-|353\2460<br>(107\2460)
+| 353\2460<br>(16\2460)
-|172.195<br>(52.195)
+| 172.195<br>(7.805)
-|169/153<br>(?)
+| 169/153<br>(?)
-|[[Calcium]]
+| [[Calcium]]
+|-
+| 30
+| 747\2460<br>(9\2460)
+| 364.390<br>(4.390)
+| 216/175<br>(385/384)
+| [[Zinc]]
 |-
-|60
+| 41
-|747\2460<br>(9\2460)
+| 1021\2460<br>(1\2460)
-|364.390<br>(4.390)
+| 498.049<br>(0.488)
-|216/175<br>(?)
+| 4/3<br />({{monzo| 215 -121 -10 }})
-|[[Neodymium]] / [[neodymium magnet]]
+| [[Niobium]]
 |-
-|60
+| 60
-|1021\2460<br>(4\2460)
+| 747\2460<br>(9\2460)
-|498.049<br>(1.951)
+| 364.390<br>(4.390)
-|4/3<br>(32805/32768)
+| 216/175<br>(385/384)
-|[[Minutes]]
+| [[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:Equal divisions of the octave|####]] <!-- 4-digit number -->
 [[Category:Mina]]
-[[Category:Zeta]]