1600edo

← 1599edo

1600edo

1601edo →

Prime factorization	26 × 52
Step size	0.75 ¢
Fifth	936\1600 (702 ¢) (→ 117\200)
Semitones (A1:m2)	152:120 (114 ¢ : 90 ¢)
Consistency limit	37
Distinct consistency limit	37

Theory

1600edo is a very strong 37-limit system, being distinctly consistent in the 37-odd-limit with a smaller relative error than anything else with this property until 4501. It is also the first division past 311 with a lower 43-limit relative error.

In the 5-limit, it supports kwazy. In the 11-limit, it supports the rank-3 temperament thor. In higher limits, it tempers out 12376/12375 in the 17-limit and due to being consistent higher than 33-odd-limit it enables the essentially tempered flashmic chords.

Odd harmonics

Approximation of prime harmonics in 1600edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.000	+0.045	-0.064	+0.174	-0.068	+0.222	+0.045	+0.237	+0.226	+0.173	+0.214
	Relative (%)	+0.0	+6.0	-8.5	+23.2	-9.1	+29.6	+5.9	+31.6	+30.1	+23.0	+28.6
Steps (reduced)		1600 (0)	2536 (936)	3715 (515)	4492 (1292)	5535 (735)	5921 (1121)	6540 (140)	6797 (397)	7238 (838)	7773 (1373)	7927 (1527)

Subsets and supersets

Since 1600 factors into 2⁶ × 5², 1600edo has subset edos 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 64, 80, 100, 160, 200, 320, 400, and 800.

One step of it is the relative cent for 16. Its high divisibility, high consistency limit, and compatibility with the decimal system make it a candidate for interval size measure. One step of 1600edo is already used as a measure called śata in the context of 16edo Armodue theory.

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3.5	[-53 10 16⟩, [26 -75 40⟩	[⟨1600 2536 3715]]	−0.0003	0.0228	3.04
2.3.5.7	4375/4374, [36 -5 0 -10⟩, [-17 5 16 -10⟩	[⟨1600 2536 3715 4492]]	−0.0157	0.0332	4.43
2.3.5.7.11	3025/3024, 4375/4374, [24 -1 -5 0 1⟩, [15 1 7 -8 -3⟩	[⟨1600 2536 3715 4492 5535]]	−0.0172	0.0329	4.39
2.3.5.7.11.13	3025/3024, 4096/4095, 4375/4374, 78125/78078, 823875/823543	[⟨1600 2536 3715 4492 5535 5921]]	−0.0087	0.0356	4.75
2.3.5.7.11.13.17	2500/2499, 3025/3024, 4096/4095, 4375/4374, 14875/14872, 63888/63869	[⟨1600 2536 3715 4492 5535 5921 6540]]	−0.0163	0.0331	4.41

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperaments
2	217\1600	162.75	1125/1024	Kwazy
32	23\1600	17.25	?	Dam / dike / polder
32	121\1600 (21/1600)	90.75 (15.75)	48828125/46294416 (?)	Windrose
32	357\1600 (7\1600)	267.75 (5.25)	245/143 (?)	Germanium
80	629\1600 (9\1600)	471.75 (6.75)	130/99 (?)	Tetraicosic

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct