2684edo: Difference between revisions

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2684edo is an extremely strong 13-limit system, with a lower 13-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] than any division until we reach [[5585edo]]. It is ~~distinctly~~ [[consistent]] through the [[17-odd-limit]], and is both a [[~~The Riemann~~ zeta ~~function and tuning #Zeta EDO lists~~|zeta peak and zeta integral edo]]. It is [[enfactoring|enfactored]] in the 2.3.5.13 subgroup, with the same tuning as [[1342edo]], tempering out kwazy, {{monzo| -53 10 16 }}, senior, {{monzo| -17 62 -35 }} and egads, {{monzo| -36 52 51 }}. A 13-limit [[comma basis]] is {9801/9800, 10648/10647, 140625/140608, 196625/196608, 823680/823543}; it also tempers out 123201/123200. It is less accurate, but still quite accurate in the 17-limit; a comma basis is {4914/4913, 5832/5831, 9801/9800, 10648/10647, 28561/28560, 140625/140608}.

== Theory ==

2684edo is an extremely strong 13-limit system, with a lower 13-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] than any division until we reach [[5585edo]]. It is [[consistency|distinctly consistent]] through the [[17-odd-limit]], and is both a [[zeta edo|zeta peak and zeta integral edo]]. It is [[enfactoring|enfactored]] in the 2.3.5.13 [[subgroup]], with the same tuning as [[1342edo]], [[tempering out]] kwazy, {{monzo| -53 10 16 }}, senior, {{monzo| -17 62 -35 }} and egads, {{monzo| -36 52 51 }}. A 13-limit [[comma basis]] is {[[9801/9800]], [[10648/10647]], 140625/140608, 196625/196608, 823680/823543}; it also tempers out [[123201/123200]]. It is less accurate, but still quite accurate in the 17-limit; a comma basis is {[[4914/4913]], [[5832/5831]], 9801/9800, 10648/10647, [[28561/28560]], 140625/140608}.

2684edo ~~sets the septimal comma, 64/63~~, ~~to an exact 1/44th~~ of ~~the octave~~ (~~61 steps~~)~~. As a corollary, it supports the period-44 [[ruthenium]] temperament~~.

In higher limits, 2684edo is a good no-19s 31-limit tuning, with errors of 25% or less on all harmonics (except 19).

=== Prime harmonics ===

=== ~~Divisors~~ ===

=== Subsets and supersets ===

Since 2684 factors ~~as 22 × 11 × 61~~, 2684edo has subset edos {{EDOs| 2, 4, 11, 22, 44, 61, 122, 244, 671, and 1342 }}.

Since 2684 factors into {{factorization|2684}}, 2684edo has subset edos {{EDOs| 2, 4, 11, 22, 44, 61, 122, 244, 671, and 1342 }}.

==Regular temperament properties==

2684edo tunes the septimal comma, 64/63, to an exact 1/44th of the octave (61 steps). As a corollary, it supports the period-44 [[ruthenium]] temperament.

~~2684edo~~ holds a record for the lowest relative error in the 13-limit~~. It~~ is only bettered by [[5585edo]].

== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

|-

! 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]] (%)

|-

| 2.3.5.7

| 78125000/78121827, {{monzo| -5 10 5 -8 }}, {{monzo| -48 0 11 8 }}

| {{mapping| 2684 4254 6232 7535 }}

| +0.0030

| 0.0085

| 1.90

|-

| 2.3.5.7.11

| 9801/9800, 1771561/1771470, 35156250/35153041, 67110351/67108864

| {{mapping| 2684 4254 6232 7535 9825 }}

| +0.0089

| 0.0089

| 1.99

|-

| 2.3.5.7.11.13

| 9801/9800, 10648/10647, 140625/140608, 196625/196608, 823680/823543

| {{mapping| 2684 4254 6232 7535 9825 9932 }}

| +0.0041

| 0.0086

| 1.93

|-

| 2.3.5.7.11.13.17

| 4914/4913, 5832/5831, 9801/9800, 10648/10647, 28561/28560, 140625/140608

| {{mapping| 2684 4254 6232 7535 9825 9932 10971 }}

| −0.0004

| 0.0136

| 3.04

|-

| 2.3.5.7.11.13.17.23

| 4761/4760, 4914/4913, 5832/5831, 8625/8624, 9801/9800, 10648/10647, 28561/28560

| {{mapping| 2684 4254 6232 7535 9825 9932 10971 12141 }}

| +0.0026

| 0.0150

| 3.36

|}

* 2684et holds a record for the lowest relative error in the 13-limit, past [[2190edo|2190]] and is only bettered by [[5585edo|5585]], which is more than twice its size. In terms of absolute error, it is narrowly beaten by [[3395edo|3395]].

* 2684et is also notable in the 11-limit, where it has the lowest absolute error, past [[1848edo|1848]] and before 3395.

=== Rank-2 temperaments ===

Note: 5-limit temperaments ~~represented~~ by [[1342edo]] are not included.

Note: 5-limit temperaments supported by [[1342edo]] are not included.

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

! Periods per 8ve

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

! Generator~~ (Reduced)~~

|-

! Cents~~ (Reduced)~~

! Periods per 8ve

! Associated ~~Ratio~~

! Generator*

! Cents*

! Associated ratio*

! Temperaments

|-

| 1

| 353\2684

| 157.824

| 36756909/33554432

| [[Hemiegads]]

|-

| 44

| 1114\2684 (16\2684)

| 1114\2684 (16\2684)

| 498.063 (7.154)

| 498.063 (7.154)

| 4/3 (18375/18304)

| 4/3 (18375/18304)

| [[Ruthenium]]

|-

| 61

| 557\2684 (29\2684)

| 249.031 (12.965)

| 11907/6875 (?)

| [[Promethium]]

|}

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

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|2684}}
+{{ED intro}}
-edo is an extremely strong 13-limit system, with a lower 13-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] than any division until we reach [[5585edo]]. It is distinctly [[consistent]] through the [[17-odd-limit]], and is both a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak and zeta integral edo]]. It is [[enfactoring|enfactored]] in the 2.3.5.13 subgroup, with the same tuning as [[1342edo]], tempering out kwazy, {{monzo| -53 10 16 }}, senior, {{monzo| -17 62 -35 }} and egads, {{monzo| -36 52 51 }}. A 13-limit [[comma basis]] is {9801/9800, 10648/10647, 140625/140608, 196625/196608, 823680/823543}; it also tempers out 123201/123200. It is less accurate, but still quite accurate in the 17-limit; a comma basis is {4914/4913, 5832/5831, 9801/9800, 10648/10647, 28561/28560, 140625/140608}.
+== Theory ==
+edo is an extremely strong 13-limit system, with a lower 13-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]] than any division until we reach [[5585edo]]. It is [[consistency|distinctly consistent]] through the [[17-odd-limit]], and is both a [[zeta edo|zeta peak and zeta integral edo]]. It is [[enfactoring|enfactored]] in the 2.3.5.13 [[subgroup]], with the same tuning as [[1342edo]], [[tempering out]] kwazy, {{monzo| -53 10 16 }}, senior, {{monzo| -17 62 -35 }} and egads, {{monzo| -36 52 51 }}. A 13-limit [[comma basis]] is {[[9801/9800]], [[10648/10647]], 140625/140608, 196625/196608, 823680/823543}; it also tempers out [[123201/123200]]. It is less accurate, but still quite accurate in the 17-limit; a comma basis is {[[4914/4913]], [[5832/5831]], 9801/9800, 10648/10647, [[28561/28560]], 140625/140608}.
-edo sets the septimal comma, 64/63, to an exact 1/44th of the octave (61 steps). As a corollary, it supports the period-44 [[ruthenium]] temperament.
+In higher limits, 2684edo is a good no-19s 31-limit tuning, with errors of 25% or less on all harmonics (except 19).
 === Prime harmonics ===
 {{Harmonics in equal|2684|columns=11}}
-=== Divisors ===
+=== Subsets and supersets ===
-Since 2684 factors as 2<sup>2</sup> × 11 × 61, 2684edo has subset edos {{EDOs| 2, 4, 11, 22, 44, 61, 122, 244, 671, and 1342 }}.
+Since 2684 factors into {{factorization|2684}}, 2684edo has subset edos {{EDOs| 2, 4, 11, 22, 44, 61, 122, 244, 671, and 1342 }}.
-==Regular temperament properties==
+edo tunes the septimal comma, 64/63, to an exact 1/44th of the octave (61 steps). As a corollary, it supports the period-44 [[ruthenium]] temperament.
-edo holds a record for the lowest relative error in the 13-limit. It is only bettered by [[5585edo]].
+== Regular temperament properties ==
+{| class="wikitable center-4 center-5 center-6"
+|-
+! 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]] (%)
+|-
+| 2.3.5.7
+| 78125000/78121827, {{monzo| -5 10 5 -8 }}, {{monzo| -48 0 11 8 }}
+| {{mapping| 2684 4254 6232 7535 }}
+| +0.0030
+| 0.0085
+| 1.90
+|-
+| 2.3.5.7.11
+| 9801/9800, 1771561/1771470, 35156250/35153041, 67110351/67108864
+| {{mapping| 2684 4254 6232 7535 9825 }}
+| +0.0089
+| 0.0089
+| 1.99
+|-
+| 2.3.5.7.11.13
+| 9801/9800, 10648/10647, 140625/140608, 196625/196608, 823680/823543
+| {{mapping| 2684 4254 6232 7535 9825 9932 }}
+| +0.0041
+| 0.0086
+| 1.93
+|-
+| 2.3.5.7.11.13.17
+| 4914/4913, 5832/5831, 9801/9800, 10648/10647, 28561/28560, 140625/140608
+| {{mapping| 2684 4254 6232 7535 9825 9932 10971 }}
+| −0.0004
+| 0.0136
+| 3.04
+|-
+| 2.3.5.7.11.13.17.23
+| 4761/4760, 4914/4913, 5832/5831, 8625/8624, 9801/9800, 10648/10647, 28561/28560
+| {{mapping| 2684 4254 6232 7535 9825 9932 10971 12141 }}
+| +0.0026
+| 0.0150
+| 3.36
+|}
+* 2684et holds a record for the lowest relative error in the 13-limit, past [[2190edo|2190]] and is only bettered by [[5585edo|5585]], which is more than twice its size. In terms of absolute error, it is narrowly beaten by [[3395edo|3395]].
+* 2684et is also notable in the 11-limit, where it has the lowest absolute error, past [[1848edo|1848]] and before 3395.
 === Rank-2 temperaments ===
-Note: 5-limit temperaments represented by [[1342edo]] are not included.
+Note: 5-limit temperaments supported by [[1342edo]] are not included.
 {| class="wikitable center-all left-5"
-! Periods<br>per 8ve
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
-! Generator<br>(Reduced)
+|-
-! Cents<br>(Reduced)
+! Periods<br />per 8ve
-! Associated<br>Ratio
+! Generator*
+! Cents*
+! Associated<br />ratio*
 ! Temperaments
+|-
+| 1
+| 353\2684
+| 157.824
+| 36756909/33554432
+| [[Hemiegads]]
 |-
 | 44
-| 1114\2684<br>(16\2684)
+| 1114\2684<br />(16\2684)
-| 498.063<br>(7.154)
+| 498.063<br />(7.154)
-| 4/3<br>(18375/18304)
+| 4/3<br />(18375/18304)
 | [[Ruthenium]]
+|-
+| 61
+| 557\2684<br />(29\2684)
+| 249.031<br />(12.965)
+| 11907/6875<br />(?)
+| [[Promethium]]
 |}
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct