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

1600 equal divisions of the octave (abbreviated 1600edo or 1600ed2), also called 1600-tone equal temperament (1600tet) or 1600 equal temperament (1600et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1600 equal parts of exactly 0.75 ¢ each. Each step represents a frequency ratio of 2^1/1600, or the 1600th root of 2.

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, being almost consistent in the 45-odd-limit, missing 50/39 and 39/25, both of which being off by 52,6% by patent val mapping, which is still just an error of 0.3945 cents.

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	37
Error	Absolute (¢)	+0.000	+0.045	-0.064	+0.174	-0.068	+0.222	+0.045	+0.237	+0.226	+0.173	+0.214	-0.094
	Relative (%)	+0.0	+6.0	-8.5	+23.2	-9.1	+29.6	+5.9	+31.6	+30.1	+23.0	+28.6	-12.5
Steps (reduced)		1600 (0)	2536 (936)	3715 (515)	4492 (1292)	5535 (735)	5921 (1121)	6540 (140)	6797 (397)	7238 (838)	7773 (1373)	7927 (1527)	8335 (335)

Approximation of prime harmonics in 1600edo (continued)

Harmonic		41	43	47	53	59	61	67	71	73	79	83	89
Error	Absolute (¢)	-0.062	-0.018	-0.257	+0.245	-0.172	-0.135	+0.193	+0.303	+0.211	-0.037	-0.047	-0.130
	Relative (%)	-8.3	-2.4	-34.2	+32.7	-22.9	-18.0	+25.7	+40.5	+28.1	-4.9	-6.3	-17.3
Steps (reduced)		8572 (572)	8682 (682)	8887 (887)	9165 (1165)	9412 (1412)	9489 (1489)	9706 (106)	9840 (240)	9904 (304)	10086 (486)	10200 (600)	10361 (761)

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. Similar to the Mina in the 27-odd-limit, All 45-odd limit intervals can be written using integer values of śata, being more in tune than out of tune.

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	Crazy
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