2684edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== 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 | 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 | 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 === | === Prime harmonics === | ||
{{Harmonics in equal|2684|columns=11}} | {{Harmonics in equal|2684|columns=11}} | ||
=== | === Subsets and supersets === | ||
Since 2684 factors | 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. | |||
== Regular temperament properties == | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
|- | |||
! rowspan="2" | [[Subgroup]] | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" | [[Comma list | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br>8ve stretch (¢) | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
! colspan="2" | Tuning error | ! colspan="2" | Tuning error | ||
|- | |- | ||
| Line 25: | Line 28: | ||
|- | |- | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 78125000/78121827, | | 78125000/78121827, {{monzo| -5 10 5 -8 }}, {{monzo| -48 0 11 8 }} | ||
| | | {{mapping| 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, | | 9801/9800, 1771561/1771470, 35156250/35153041, 67110351/67108864 | ||
| | | {{mapping| 2684 4254 6232 7535 9825 }} | ||
| +0.0089 | |||
| 0.0089 | | 0.0089 | ||
| 1.99 | | 1.99 | ||
|- | |- | ||
| 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 | ||
| | | {{mapping| 2684 4254 6232 7535 9825 9932 }} | ||
| 0.0041 | | +0.0041 | ||
| 0.0086 | | 0.0086 | ||
| 1.93 | | 1.93 | ||
| Line 47: | Line 50: | ||
| 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 | ||
| | | {{mapping| 2684 4254 6232 7535 9825 9932 10971 }} | ||
| | | −0.0004 | ||
| 0.0136 | | 0.0136 | ||
| 3.04 | | 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 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 === | === Rank-2 temperaments === | ||
| Line 58: | Line 69: | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
! Periods<br>per 8ve | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | ||
! Generator | |- | ||
! Cents | ! Periods<br />per 8ve | ||
! Associated<br> | ! Generator* | ||
! Cents* | |||
! Associated<br />ratio* | |||
! Temperaments | ! Temperaments | ||
|- | |||
| 1 | |||
| 353\2684 | |||
| 157.824 | |||
| 36756909/33554432 | |||
| [[Hemiegads]] | |||
|- | |- | ||
| 44 | | 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]] | | [[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 | |||