2684edo

This page presents a novelty topic. It may contain ideas which are less likely to find practical applications in music, or numbers or structures that are arbitrary or exceedingly small, large, or complex. Novelty topics are often developed by a single person or a small group. As such, this page may also contain idiosyncratic terms, notation, or conceptual frameworks.

← 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

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 as 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.

Rank-2 temperaments

Note: 5-limit temperaments supported by 1342edo are not included.

Periods per 8ve	Generator (Reduced)	Cents (Reduced)	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