2684edo: Difference between revisions

(8 intermediate revisions by 3 users not shown)

Line 1:

== 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 ~~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}.

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}.

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

Line 11:

=== 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 }}.

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.

Line 17:

== Regular temperament properties ==

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

|-

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

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

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

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

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

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

! colspan="2" | Tuning ~~Error~~

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

Line 28:

Line 29:

| 2.3.5.7

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

| [{{~~val~~| 2684 4254 6232 7535 }}]

| {{mapping| 2684 4254 6232 7535 }}

| +0.0030

| 0.0085

Line 35:

Line 36:

| 2.3.5.7.11

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

| [{{~~val~~| 2684 4254 6232 7535 9825 }}]

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

| +0.0089

| 0.0089

Line 42:

Line 43:

| 2.3.5.7.11.13

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

| [{{~~val~~| 2684 4254 6232 7535 9825 9932 }}]

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

| +0.0041

| 0.0086

Line 49:

Line 50:

| 2.3.5.7.11.13.17

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

| [{{~~val~~| 2684 4254 6232 7535 9825 9932 10971 }}]

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

| -0.0004

| −0.0004

| 0.0136

| 3.04

Line 56:

Line 57:

| 2.3.5.7.11.13.17.23

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

| [{{~~val~~| 2684 4254 6232 7535 9825 9932 10971 12141 }}]

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

| +0.0026

| 0.0150

Line 62:

Line 63:

|}

* 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 ===

Line 67:

Line 69:

{| 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

|-

Line 80:

Line 84:

|-

| 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)

| 557\2684 (29\2684)

| 249.031 (12.965)

| 249.031 (12.965)

| 11907/6875 (?)

| 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}}
 == 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 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}.
+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}.
 In higher limits, 2684edo is a good no-19s 31-limit tuning, with errors of 25% or less on all harmonics (except 19).
@@ Line 11: / Line 11: @@
 === 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 }}.
 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.
@@ Line 17: / Line 17: @@
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
-! rowspan="2" | [[Comma list|Comma List]]
+! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br>8ve Stretch (¢)
+! rowspan="2" | Optimal<br />8ve stretch (¢)
-! colspan="2" | Tuning Error
+! colspan="2" | Tuning error
 |-
 ! [[TE error|Absolute]] (¢)
@@ Line 28: / Line 29: @@
 | 2.3.5.7
 | 78125000/78121827, {{monzo| -5 10 5 -8 }}, {{monzo| -48 0 11 8 }}
-| [{{val| 2684 4254 6232 7535 }}]
+| {{mapping| 2684 4254 6232 7535 }}
 | +0.0030
 | 0.0085
@@ Line 35: / Line 36: @@
 | 2.3.5.7.11
 | 9801/9800, 1771561/1771470, 35156250/35153041, 67110351/67108864
-| [{{val| 2684 4254 6232 7535 9825 }}]
+| {{mapping| 2684 4254 6232 7535 9825 }}
 | +0.0089
 | 0.0089
@@ Line 42: / Line 43: @@
 | 2.3.5.7.11.13
 | 9801/9800, 10648/10647, 140625/140608, 196625/196608, 823680/823543
-| [{{val| 2684 4254 6232 7535 9825 9932 }}]
+| {{mapping| 2684 4254 6232 7535 9825 9932 }}
 | +0.0041
 | 0.0086
@@ Line 49: / Line 50: @@
 | 2.3.5.7.11.13.17
 | 4914/4913, 5832/5831, 9801/9800, 10648/10647, 28561/28560, 140625/140608
-| [{{val| 2684 4254 6232 7535 9825 9932 10971 }}]
+| {{mapping| 2684 4254 6232 7535 9825 9932 10971 }}
-| -0.0004
+| −0.0004
 | 0.0136
 | 3.04
@@ Line 56: / Line 57: @@
 | 2.3.5.7.11.13.17.23
 | 4761/4760, 4914/4913, 5832/5831, 8625/8624, 9801/9800, 10648/10647, 28561/28560
-| [{{val| 2684 4254 6232 7535 9825 9932 10971 12141 }}]
+| {{mapping| 2684 4254 6232 7535 9825 9932 10971 12141 }}
 | +0.0026
 | 0.0150
@@ Line 62: / Line 63: @@
 |}
 * 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 ===
@@ Line 67: / Line 69: @@
 {| 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
 |-
@@ Line 80: / Line 84: @@
 |-
 | 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)
+| 557\2684<br />(29\2684)
-| 249.031<br>(12.965)
+| 249.031<br />(12.965)
-| 11907/6875<br>(?)
+| 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