224edo
| ← 223edo | 224edo | 225edo → |
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
| 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 = 32 × 7, 224edo has subset edos 2, 4, 8, 16, 32, 7, 14, 28, 56, and 112.
Regular temperament properties
Template:Comma basis begin |- | 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 Template:Comma basis end
- 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
Template:Rank-2 begin
|-
| 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
Template:Rank-2 end
Template:Orf
Music
- Dreyfus (archived 2010) – SoundCloud | details | play – octoid[72] in 224edo tuning
- Kindness Is A Weakness (2023) – octant[24], hemigamera[26], oquatonic[56], bezique[64] in 224edo tuning