# 2684edo

 ← 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
Special properties

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
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 it is distinct