2016edo

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2016edo

2017edo →

Prime factorization	25 × 32 × 7
Step size	0.595238 ¢
Fifth	1179\2016 (701.786 ¢) (→ 131\224)
Semitones (A1:m2)	189:153 (112.5 ¢ : 91.07 ¢)
Consistency limit	5
Distinct consistency limit	5

The 2016 equal divisions of the octave (2016edo), or the 2016-tone equal temperament (2016tet), 2016 equal temperament (2016et) when viewed from a regular temperament perspective, divides the octave into 2016 equal parts of about 0.595 cents each.

Theory

Approximation of odd harmonics in 2016edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	-0.169	-0.004	+0.222	+0.257	-0.127	-0.051	-0.173	-0.194	+0.106	+0.052	+0.297
	Relative (%)	-28.4	-0.7	+37.2	+43.1	-21.4	-8.6	-29.1	-32.5	+17.8	+8.8	+49.9
Steps (reduced)		3195 (1179)	4681 (649)	5660 (1628)	6391 (343)	6974 (926)	7460 (1412)	7876 (1828)	8240 (176)	8564 (500)	8855 (791)	9120 (1056)

2016 is a significantly composite number, with its divisors being 1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 16, 18, 21, 24, 28, 32, 36, 42, 48, 56, 63, 72, 84, 96, 112, 126, 144, 168, 224, 252, 288, 336, 504, 672, 1008. It's abundancy index is 2.25.

Prime harmonics (below 61) with less than 22% error in 2016edo are: 2, 5, 11, 13, 19, 41, 47. With next error being 26% on the 37th harmonic, it is reasonable to make cutoff here.

2016 shares the mapping for 3 with 224edo, albeit with a 28 relative cent error.

2016edo has two reasonable mappings for 7. The 2016d val, ⟨2016 3195 4681 5659], tempers out 5250987/5242880, 40353607/40310784 (tritrizo), and [14, 11, -22, 7⟩. As such, its circle of the interval 7/6 is the same as in 9edo.

The patent val, ⟨2016 3195 4681 5658] tempers out 250047/250000, along with [7, 18, -2, -11⟩ and [43, -1, -13, -4⟩. This means that the symmetrical major third (400 cents, 1/3 of the octave) in 2016edo corresponds to 63/50.

In the 11-limit, 2016edo tempers out the [0 0 -22 0 3 11⟩ comma, which equates a stack of 11 25/13s with 3 hendecataves.

Fractional octave temperaments

The patent val 7-limit in 2016edo gives rise to the to rank two temperaments of 72 & 624 with period 24 and 441 & 1407, period 21. The 2016d val gives rise to 171 & 306, period 9 and 270 & 936bd, period 18. Using the 441 & 1407 temperament, 2016edo also tempers out the akjaysma.

If we assume that 2016edo is a dual-seventh system, where 5659th and 5660th steps represent 7- and 7+, two distinct dimensions amounting to 49/1, then this allows for mixing of these two temperaments. 236 & 600 is the temperament which best represents that.

Regular temperament properties

Subgroup	Comma list (zeroes skipped for clarity)	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3.5	[-83, 26, 18⟩, [30, 47, -45⟩	[⟨2016 3195 4681]]	0.036	0.050	8.4
2.3.5.7	250047/250000, [7, 18, -2, -11⟩, [43, -1, -13, -4⟩	[⟨2016 3195 4681 5660]]	0.007	0.066	11.1
2.3.5.7	5250987/5242880, 40353607/40310784, [14, 11, -22, 7⟩	[⟨2016 3195 4681 5659]] (2016d)	0.060	0.060	10.1
2.5.11.13	[5 -6 9 6⟩, [-38 12 4 -1⟩, [0 -22 3 11⟩	[⟨2016 4681 6974 7460]]	0.013	0.015	2.5
2.5.11.13.19.41.47	7943/7942, 322465/322373, 415292/415207, 511225/511024, 5078491/5078125, 22151168/22150865	⟨2016 4681 6974 7460 8564 10801 11198]	0.002	0.019	3.2

Rank two temperaments by generator

Periods per octave	Generator (reduced)	Cents (reduced)	Temperaments
21	23\2016	13.690	441 & 1407
24	29\2016	17.262	72 & 624