2684edo: Difference between revisions

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== 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).

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=== 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.

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

|+Table of rank-2 temperaments by generator

|-

! Periods per 8ve

! Generator*

! Cents*

! Associated Ratio

! Associated Ratio*

! Temperaments

|-

@@ Line 3: / Line 3: @@
 == 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 68: / Line 68: @@
 {| class="wikitable center-all left-5"
+|+Table of rank-2 temperaments by generator
+|-
 ! Periods<br>per 8ve
 ! Generator*
 ! Cents*
-! Associated<br>Ratio
+! Associated<br>Ratio*
 ! Temperaments
 |-