224edo

← 223edo

224edo

225edo →

Prime factorization	25 × 7
Step size	5.35714 ¢
Fifth	131\224 (701.786 ¢)
Semitones (A1:m2)	21:17 (112.5 ¢ : 91.07 ¢)
Consistency limit	15
Distinct consistency limit	15

The 224 equal divisions of the octave (224edo), or the 224(-tone) equal temperament (224tet, 224et) when viewed from a regular temperament perspective, is the equal division of the octave into 224 parts of about 5.36 cents each.

Theory

224edo is a very strong 13-limit system, tempering out 32805/32768 in the 5-limit; 4375/4374, 16875/16807 and 65625/65536 in the 7-limit; 540/539, 1375/1372, 4000/3993 and notably, the quartisma in the 11-limit; and 625/624, 729/728, 1575/1573 and 2200/2197 in the 13-limit, leading to an abundance of precisely-tuned essentially tempered chords. It defines the optimal patent val for the octoid in the 7-, 11- and 13-limit, and for mirkwai, the 7-limit planar temperament tempering out 16875/16807. It also provides an excellent tuning for indra and shibi temperaments. It is the twelfth zeta integral edo.

224 = 32 × 7, and has divisors 2, 4, 8, 16, 32, 7, 14, 28, 56, and 112.

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3	[-355 224⟩	[⟨224 355]]	+0.053	0.0534	1.00
2.3.5	32805/32768, [-5 -32 24⟩	[⟨224 355 520]]	+0.122	0.1059	1.98
2.3.5.7	4375/4374, 16875/16807, 32805/32768	[⟨224 355 520 629]]	+0.018	0.2009	3.75
2.3.5.7.11	540/539, 1375/1372, 4000/3993, 32805/32768	[⟨224 355 520 629 775]]	-0.012	0.1899	3.54
2.3.5.7.11.13	540/539, 625/624, 729/728, 1375/1372, 2200/2197	[⟨224 355 520 629 775 829]]	-0.035	0.1805	3.37

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per octave	Generator (reduced)	Cents (reduced)	Associated ratio	Temperaments
1	43\224	230.36	8/7	Gamera
1	59\224	316.07	6/5	Counterkleismic
1	65\224	348.21	11/9	Eris
1	71\224	380.36	56/45	Quanharuk
1	87\224	466.07	55/42	Hemiseptisix
1	93\224	498.21	4/3	Helmholtz / pontiac / ponta
1	103\224	551.79	11/8	Emkay
2	93\224 (19\224)	498.21 (101.79)	4/3 (35/33)	Bipont
2	31\224	166.07	11/10	Pogo
2	33\224	176.79	448/405, 195/176	Quatracot
2	39\224	208.93	44/39	Abigail
2	43\224	230.36	8/7	Hemigamera
4	15\224	80.36	22/21	Quasithird
4	37\224 (19\224)	198.21 (101.79)	28/25 (35/33)	Quadrant
7	97\224 (1\224)	519.64 (5.36)	27/20	Brahmagupta
7	93\224 (3\224)	498.21 (16.07)	4/3 (99/98)	Septant
8	3\224	16.07	100/99	Octoid
8	93\224 (9\224)	498.21 (48.21)	4/3 (36/35)	Octant
28	3\224	16.07	126/125	Oquatonic

Music

• Dreyfus play by Gene Ward Smith