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{{Infobox ET}} | |||
{{ED intro}} | |||
== 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 [[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). | |||
=== Prime harmonics === | |||
{{Harmonics in equal|2684|columns=11}} | |||
=== Subsets and supersets === | |||
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 == | |||
{| class="wikitable center-4 center-5 center-6" | |||
|- | |||
! rowspan="2" | [[Subgroup]] | |||
! rowspan="2" | [[Comma list]] | |||
! rowspan="2" | [[Mapping]] | |||
! rowspan="2" | Optimal<br />8ve stretch (¢) | |||
! colspan="2" | Tuning error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |||
| 2.3.5.7 | |||
| 78125000/78121827, {{monzo| -5 10 5 -8 }}, {{monzo| -48 0 11 8 }} | |||
| {{mapping| 2684 4254 6232 7535 }} | |||
| +0.0030 | |||
| 0.0085 | |||
| 1.90 | |||
|- | |||
| 2.3.5.7.11 | |||
| 9801/9800, 1771561/1771470, 35156250/35153041, 67110351/67108864 | |||
| {{mapping| 2684 4254 6232 7535 9825 }} | |||
| +0.0089 | |||
| 0.0089 | |||
| 1.99 | |||
|- | |||
| 2.3.5.7.11.13 | |||
| 9801/9800, 10648/10647, 140625/140608, 196625/196608, 823680/823543 | |||
| {{mapping| 2684 4254 6232 7535 9825 9932 }} | |||
| +0.0041 | |||
| 0.0086 | |||
| 1.93 | |||
|- | |||
| 2.3.5.7.11.13.17 | |||
| 4914/4913, 5832/5831, 9801/9800, 10648/10647, 28561/28560, 140625/140608 | |||
| {{mapping| 2684 4254 6232 7535 9825 9932 10971 }} | |||
| −0.0004 | |||
| 0.0136 | |||
| 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 is also notable in the 11-limit, where it has the lowest absolute error, past [[1848edo|1848]] and before 3395. | |||
=== Rank-2 temperaments === | |||
Note: 5-limit temperaments supported by [[1342edo]] are not included. | |||
{| class="wikitable center-all left-5" | |||
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |||
|- | |||
! Periods<br />per 8ve | |||
! Generator* | |||
! Cents* | |||
! Associated<br />ratio* | |||
! Temperaments | |||
|- | |||
| 1 | |||
| 353\2684 | |||
| 157.824 | |||
| 36756909/33554432 | |||
| [[Hemiegads]] | |||
|- | |||
| 44 | |||
| 1114\2684<br />(16\2684) | |||
| 498.063<br />(7.154) | |||
| 4/3<br />(18375/18304) | |||
| [[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 | |||
Latest revision as of 14:26, 20 February 2025
| ← 2683edo | 2684edo | 2685edo → |
2684 equal divisions of the octave (abbreviated 2684edo or 2684ed2), also called 2684-tone equal temperament (2684tet) or 2684 equal temperament (2684et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 2684 equal parts of about 0.447 ¢ each. Each step represents a frequency ratio of 21/2684, or the 2684th root of 2.
Theory
2684edo is an extremely strong 13-limit system, with a lower 13-limit relative error than any division until we reach 5585edo. It is distinctly consistent through the 17-odd-limit, and is both a zeta peak and zeta integral edo. It is enfactored in the 2.3.5.13 subgroup, with the same tuning as 1342edo, tempering out kwazy, [-53 10 16⟩, senior, [-17 62 -35⟩ and egads, [-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).
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.000 | -0.018 | -0.025 | +0.027 | -0.051 | +0.009 | +0.112 | -0.196 | -0.107 | +0.080 | -0.028 |
| Relative (%) | +0.0 | -3.9 | -5.5 | +5.9 | -11.4 | +2.0 | +25.0 | -43.7 | -24.0 | +17.9 | -6.3 | |
| Steps (reduced) |
2684 (0) |
4254 (1570) |
6232 (864) |
7535 (2167) |
9285 (1233) |
9932 (1880) |
10971 (235) |
11401 (665) |
12141 (1405) |
13039 (2303) |
13297 (2561) | |
Subsets and supersets
Since 2684 factors into 22 × 11 × 61, 2684edo has subset 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
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5.7 | 78125000/78121827, [-5 10 5 -8⟩, [-48 0 11 8⟩ | [⟨2684 4254 6232 7535]] | +0.0030 | 0.0085 | 1.90 |
| 2.3.5.7.11 | 9801/9800, 1771561/1771470, 35156250/35153041, 67110351/67108864 | [⟨2684 4254 6232 7535 9825]] | +0.0089 | 0.0089 | 1.99 |
| 2.3.5.7.11.13 | 9801/9800, 10648/10647, 140625/140608, 196625/196608, 823680/823543 | [⟨2684 4254 6232 7535 9825 9932]] | +0.0041 | 0.0086 | 1.93 |
| 2.3.5.7.11.13.17 | 4914/4913, 5832/5831, 9801/9800, 10648/10647, 28561/28560, 140625/140608 | [⟨2684 4254 6232 7535 9825 9932 10971]] | −0.0004 | 0.0136 | 3.04 |
| 2.3.5.7.11.13.17.23 | 4761/4760, 4914/4913, 5832/5831, 8625/8624, 9801/9800, 10648/10647, 28561/28560 | [⟨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 2190 and is only bettered by 5585, which is more than twice its size. In terms of absolute error, it is narrowly beaten by 3395.
- 2684et is also notable in the 11-limit, where it has the lowest absolute error, past 1848 and before 3395.
Rank-2 temperaments
Note: 5-limit temperaments supported by 1342edo are not included.
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 353\2684 | 157.824 | 36756909/33554432 | Hemiegads |
| 44 | 1114\2684 (16\2684) |
498.063 (7.154) |
4/3 (18375/18304) |
Ruthenium |
| 61 | 557\2684 (29\2684) |
249.031 (12.965) |
11907/6875 (?) |
Promethium |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct