2684edo: Difference between revisions

(4 intermediate revisions by 2 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 50:

Line 51:

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

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

| -0.0004

| −0.0004

| 0.0136

| 3.04

Line 68:

Line 69:

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

! Periods per 8ve

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

|-

! Periods per 8ve

! Generator*

! Cents*

! Associated ~~Ratio~~

! Associated ratio*

! Temperaments

|-

Line 81:

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

<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 50: / Line 51: @@
 | 4914/4913, 5832/5831, 9801/9800, 10648/10647, 28561/28560, 140625/140608
 | {{mapping| 2684 4254 6232 7535 9825 9932 10971 }}
-| -0.0004
+| −0.0004
 | 0.0136
 | 3.04
@@ Line 68: / Line 69: @@
 {| class="wikitable center-all left-5"
-! Periods<br>per 8ve
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
+! Periods<br />per 8ve
 ! Generator*
 ! Cents*
-! Associated<br>Ratio
+! Associated<br />ratio*
 ! Temperaments
 |-
@@ Line 81: / 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>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct