2684edo: Difference between revisions
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== Theory == | == 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 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}. | ||
=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|2684|columns=11}} | {{Harmonics in equal|2684|columns=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 as 2<sup>2</sup> × 11 × 61, 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. | |||
== Regular temperament properties == | == Regular temperament properties == | ||
| Line 26: | Line 26: | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 78125000/78121827, 184528125/184473632, {{monzo|-48 0 11 8}} | | 78125000/78121827, 184528125/184473632, {{monzo|-48 0 11 8}} | ||
| [{{val|2684 4254 6232 7535}}] | | [{{val| 2684 4254 6232 7535 }}] | ||
| 0.0030 | | 0.0030 | ||
| 0.0085 | | 0.0085 | ||
| | | 1.90 | ||
|- | |- | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 9801/9800, 47265625/47258883, 56953125/56942116, 369140625/369098752 | | 9801/9800, 47265625/47258883, 56953125/56942116, 369140625/369098752 | ||
| [{{val|2684 4254 6232 7535 9825}}] | | [{{val| 2684 4254 6232 7535 9825 }}] | ||
| 0.0089 | | 0.0089 | ||
| 0.0054 | | 0.0054 | ||
| Line 40: | Line 40: | ||
| 2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
| 9801/9800, 10648/10647, 140625/140608, 196625/196608, 823680/823543 | | 9801/9800, 10648/10647, 140625/140608, 196625/196608, 823680/823543 | ||
| [{{val|2684 4254 6232 7535 9825 9932}}] | | [{{val| 2684 4254 6232 7535 9825 9932 }}] | ||
| 0.0041 | | 0.0041 | ||
| 0.0086 | | 0.0086 | ||
| Line 47: | Line 47: | ||
| 2.3.5.7.11.13.17 | | 2.3.5.7.11.13.17 | ||
| 4914/4913, 5832/5831, 9801/9800, 10648/10647, 28561/28560, 140625/140608 | | 4914/4913, 5832/5831, 9801/9800, 10648/10647, 28561/28560, 140625/140608 | ||
| [{{val|2684 4254 6232 7535 9825 9932 10971}}] | | [{{val| 2684 4254 6232 7535 9825 9932 10971 }}] | ||
| -0.0004 | | -0.0004 | ||
| 0.0136 | | 0.0136 | ||