224edo

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← 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
Special properties

224 equal divisions of the octave (abbreviated 224edo or 224ed2), also called 224-tone equal temperament (224tet) or 224 equal temperament (224et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 224 equal parts of about 5.36 ¢ each. Each step represents a frequency ratio of 21/224, or the 224th root of 2.

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

224edo tempers the syntonic comma to 1/56th of the octave (4 steps) and as a corollary supports the barium temperament. As a consequence of this, the 224bb val (flattening the fifth by one step) is a tuning for meantone and is very close (0.15 cents) to the quarter-comma meantone fifth. The generator however reduces to 112edo, being 65\112.

Prime harmonics

Approximation of prime harmonics in 224edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -0.17 -0.60 +0.82 +0.47 +0.54 +2.19 +2.49 -1.49 -1.01 +1.39
Relative (%) +0.0 -3.2 -11.2 +15.2 +8.7 +10.2 +40.8 +46.4 -27.8 -18.8 +26.0
Steps
(reduced)
224
(0)
355
(131)
520
(72)
629
(181)
775
(103)
829
(157)
916
(20)
952
(56)
1013
(117)
1088
(192)
1110
(214)

Subsets and supersets

Since 224 factors into 25 × 7, 224edo has subset edos 2, 4, 8, 16, 32, 7, 14, 28, 56, and 112.

Approximation to JI

Zeta peak index

Tuning Strength Closest edo Integer limit
ZPI Steps per octave Step size (cents) Height Integral Gap Edo Octave (cents) Consistent Distinct
1546zpi 224.002551156014 5.35708184485908 11.730463 1.700865 19.715639 224edo 1199.98633324843 16 16

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
2.3.5.7.11.13.17 375/374, 540/539, 625/624, 715/714, 729/728, 2200/2197 [224 355 520 629 775 829 916]] −0.106 0.2420 4.52
  • 224et has a lower relative error than any previous equal temperaments in the 13-limit, being the first to beat 72. The next equal temperament that does better in terms of either absolute or relative error is 270.
  • It is also notable in the 11- and 17-limit, with lower absolute errors than any previous equal temperaments. In the 11-limit it is the first to beat 152 and is superseded by 239. In the 17-limit it is the first to beat 217 and is superseded by 270.

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
ratio*
Temperament
1 43\224 230.36 8/7 Gamera
1 59\224 316.07 6/5 Counterkleismic / counterlytic
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 Pontiac / ponta
1 103\224 551.79 11/8 Emkay
1 111\224 594.64 55/39 Gaster
2 93\224
(19\224)
498.21
(101.79)
4/3
(35/33)
Bipont
2 31\224 166.07 11/10 Pogo
2 33\224 176.79 195/176 Quatracot
2 39\224 208.93 44/39 Abigail
2 43\224 230.36 8/7 Hemigamera
4 71\224
(15\224)
380.36
(80.36)
81/65
(22/21)
Quasithird
4 93\224
(19\224)
498.21
(101.79)
4/3
(35/33)
Quadrant
7 97\224
(1\224)
519.64
(5.36)
27/20
(325/324)
Brahmagupta
7 93\224
(3\224)
498.21
(16.07)
4/3
(99/98)
Septant
8 93\224
(9\224)
498.21
(48.21)
4/3
(36/35)
Octant
8 109\224
(3\224)
583.93
(16.07)
7/5
(100/99)
Octoid
14 93\224
(3\224)
498.21
(16.07)
4/3
(105/104)
Silicon
28 93\224
(3\224)
498.21
(16.07)
4/3
(126/125)
Oquatonic
32 50\224
(1\224)
267.86
(5.36)
245/143
(???)
Germanium
32 93\224
(2\224)
498.21
(10.71)
4/3
(???)
Bezique
56 93\224
(3\224)
498.21
(16.07)
4/3
(126/125)
Barium

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

Music

Gene Ward Smith
Mercury Amalgam