282edo

← 281edo

282edo

283edo →

Prime factorization	2 × 3 × 47
Step size	4.25532¢
Fifth	165\282 (702.128¢) (→55\94)
Semitones (A1:m2)	27:21 (114.9¢ : 89.36¢)
Consistency limit	29
Distinct consistency limit	23

282 equal divisions of the octave (abbreviated 282edo or 282ed2), also called 282-tone equal temperament (282tet) or 282 equal temperament (282et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 282 equal parts of about 4.26 ¢ each. Each step represents a frequency ratio of 2^1/282, or the 282nd root of 2.

Theory

282edo is the smallest edo distinctly consistent through to the 23-odd-limit, and also the smallest consistent to the 29-odd-limit. It shares the same 3rd, 7th, and 13th harmonics with 94edo (282 = 3 × 94), as well as 11/10 and 20/17 (supporting the garistearn temperament). It has a distinct sharp tendency for odd harmonics up to 29.

The equal temperament tempers out 6144/6125 (porwell), 118098/117649 (stearnsma), and 250047/250000 (landscape comma) in the 7-limit, and 540/539 and 5632/5625 in the 11-limit, so that it provides the optimal patent val for the jupiter temperament; it also tempers out 4000/3993 and 234375/234256, providing the optimal patent val for septisuperfourth temperament. In the 13-limit, it tempers out 729/728, 1575/1573, 1716/1715, 2080/2079, and 10648/10647.

It allows essentially tempered chords including swetismic chords, squbemic chords, and petrmic chords in the 13-odd-limit, in addition to nicolic chords in the 15-odd-limit.

Prime harmonics

Approximation of prime harmonics in 282edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.00	+0.17	+0.92	+1.39	+1.87	+2.03	+1.43	+0.36	+1.51	+0.21	-0.35
	Relative (%)	+0.0	+4.1	+21.6	+32.6	+44.0	+47.6	+33.5	+8.4	+35.6	+4.9	-8.3
Steps (reduced)		282 (0)	447 (165)	655 (91)	792 (228)	976 (130)	1044 (198)	1153 (25)	1198 (70)	1276 (148)	1370 (242)	1397 (269)

Subsets and supersets

Since 282 factors into 2 × 3 × 47, 282edo has subset edos 2, 3, 47, 94, and 141.

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3.5	[32 -7 -9⟩, [-7 22 -12⟩	[⟨282 447 655]]	−0.1684	0.1671	3.93
2.3.5.7	6144/6125, 118098/117649, 250047/250000	[⟨282 447 655 792]]	−0.2498	0.2020	4.75
2.3.5.7.11	540/539, 4000/3993, 5632/5625, 137781/137500	[⟨282 447 655 792 976]]	−0.3081	0.2151	5.06
2.3.5.7.11.13	540/539, 729/728, 1575/1573, 2200/2197, 3584/3575	[⟨282 447 655 792 976 1044]]	−0.3480	0.2156	5.07
2.3.5.7.11.13.17	540/539, 729/728, 936/935, 1156/1155, 1575/1573, 2200/2197	[⟨282 447 655 792 976 1044 1153]]	−0.3481	0.1996	4.69
2.3.5.7.11.13.17.19	456/455, 540/539, 729/728, 936/935, 969/968, 1156/1155, 1575/1573	[⟨282 447 655 792 976 1044 1153 1198]]	−0.3152	0.2061	4.84
2.3.5.7.11.13.17.19.23	456/455, 540/539, 729/728, 760/759, 936/935, 969/968, 1156/1155, 1288/1287	[⟨282 447 655 792 976 1044 1153 1198 1276]]	−0.3173	0.1944	4.57

• 282et has a lower relative error than any previous equal temperaments in the 23-limit, past 270 and before 311.

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperaments
1	13\282	55.32	33/32	Escapade
1	133\282	565.96	4096/2835	Trident (282ef)
2	13\282	55.32	33/32	Septisuperfourth
2	43\282	182.98	10/9	Unidecmic
3	33\282	140.43	243/224	Septichrome
3	37\282	157.45	35/32	Nessafof (7-limit)
6	51\282 (4\282)	217.02 (17.02)	17/15 (105/104)	Stearnscape
6	80\282 (14\282)	340.43 (59.57)	162/133 (88/85)	Semiseptichrome
6	117\282 (23\282)	497.87 (97.87)	4/3 (128/121)	Sextile

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