2684edo: Difference between revisions

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The '''2684 equal divisions of the octave''' divides the octave into 2684 equal parts of 0.4471 [[cent]]s each. It is a very strong 13-limit tuning, 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 though the [[17-odd-limit]], and is both a [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak and zeta integral edo]]. A basis for its 13-limit commas is {9801/9800, 10648/10647, 140625/140608, 196625/196608, 823680/823543}; it also tempers out 123201/123200. It factors as 2<sup>2</sup> × 11 × 61, with divisors 2, 4, 11, 22, 44, 61, 122, 244, 671, and 1342.
{{Infobox ET}}
{{ED intro}}


{{Primes in edo|2684}}
== 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}.


[[Category:Equal divisions of the octave]]
In higher limits, 2684edo is a good no-19s 31-limit tuning, with errors of 25% or less on all harmonics (except 19).
[[Category:Zeta]]
 
=== 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 →
Prime factorization 22 × 11 × 61
Step size 0.447094 ¢ 
Fifth 1570\2684 (701.937 ¢) (→ 785\1342)
Semitones (A1:m2) 254:202 (113.6 ¢ : 90.31 ¢)
Consistency limit 17
Distinct consistency limit 17

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

Approximation of prime harmonics in 2684edo
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.

Table of rank-2 temperaments by generator
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

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