2684edo: Difference between revisions

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== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{{comma basis begin}}
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|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
| 2.3.5.7
Line 60: Line 52:
| 0.0150
| 0.0150
| 3.36
| 3.36
|}
{{comma basis end}}
* 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.  
* 2684et is also notable in the 11-limit, where it has the lowest absolute error, past [[1848edo|1848]] and before 3395.  
Line 67: Line 59:
Note: 5-limit temperaments supported by [[1342edo]] are not included.
Note: 5-limit temperaments supported by [[1342edo]] are not included.


{| class="wikitable center-all left-5"
{{rank-2 begin}}
|+Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>Ratio*
! Temperaments
|-
|-
| 1
| 1
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|-
|-
| 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
| 61
| 557\2684<br>(29\2684)
| 557\2684<br />(29\2684)
| 249.031<br>(12.965)
| 249.031<br />(12.965)
| 11907/6875<br>(?)
| 11907/6875<br />(?)
| [[Promethium]]
| [[Promethium]]
|}
{{rank-2 end}}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
{{orf}}

Revision as of 04:59, 16 November 2024

← 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

Template:EDO intro

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

Template:Comma basis begin |- | 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 Template:Comma basis end

  • 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.

Template:Rank-2 begin |- | 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 Template:Rank-2 end Template:Orf

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