224edo
| ← 223edo | 224edo | 225edo → |
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 and is the second-smallest EDO after 87 to approximate all of the first 16 harmonics of the harmonic series with no greater than 25% relative error.
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 | 37 | 41 | 43 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.17 | -0.60 | +0.82 | +0.47 | +0.54 | +2.19 | +2.49 | -1.49 | -1.01 | +1.39 | +0.44 | -0.49 | -2.59 |
| Relative (%) | +0.0 | -3.2 | -11.2 | +15.2 | +8.7 | +10.2 | +40.8 | +46.4 | -27.8 | -18.8 | +26.0 | +8.2 | -9.2 | -48.3 | |
| Steps (reduced) |
224 (0) |
355 (131) |
520 (72) |
629 (181) |
775 (103) |
829 (157) |
916 (20) |
952 (56) |
1013 (117) |
1088 (192) |
1110 (214) |
1167 (47) |
1200 (80) |
1215 (95) | |
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
Interval mappings
The following table shows how 15-odd-limit intervals are represented in 224edo. Prime harmonics are in bold.
As 224edo is consistent in the 15-odd-limit, the mappings by direct approximation and through the patent val are identical.
| Interval and complement | Error (abs, ¢) | Error (rel, %) |
|---|---|---|
| 1/1, 2/1 | 0.000 | 0.0 |
| 13/11, 22/13 | 0.076 | 1.4 |
| 3/2, 4/3 | 0.169 | 3.2 |
| 9/5, 10/9 | 0.261 | 4.9 |
| 13/7, 14/13 | 0.273 | 5.1 |
| 9/8, 16/9 | 0.339 | 6.3 |
| 11/7, 14/11 | 0.349 | 6.5 |
| 5/3, 6/5 | 0.430 | 8.0 |
| 11/8, 16/11 | 0.468 | 8.7 |
| 13/8, 16/13 | 0.544 | 10.2 |
| 5/4, 8/5 | 0.599 | 11.2 |
| 11/6, 12/11 | 0.637 | 11.9 |
| 13/12, 24/13 | 0.713 | 13.3 |
| 15/8, 16/15 | 0.769 | 14.3 |
| 11/9, 18/11 | 0.806 | 15.1 |
| 7/4, 8/7 | 0.817 | 15.2 |
| 13/9, 18/13 | 0.882 | 16.5 |
| 7/6, 12/7 | 0.986 | 18.4 |
| 11/10, 20/11 | 1.067 | 19.9 |
| 13/10, 20/13 | 1.143 | 21.3 |
| 9/7, 14/9 | 1.156 | 21.6 |
| 15/11, 22/15 | 1.236 | 23.1 |
| 15/13, 26/15 | 1.312 | 24.5 |
| 7/5, 10/7 | 1.416 | 26.4 |
| 15/14, 28/15 | 1.586 | 29.6 |
Zeta peak index
| Tuning | Strength | Octave (cents) | Integer limit | |||||||
|---|---|---|---|---|---|---|---|---|---|---|
| ZPI | Steps per 8ve |
Step size (cents) |
Height | Integral | Gap | Size | Stretch | Consistent | Distinct | |
| Tempered | Pure | |||||||||
| 1546zpi | 224.002551 | 5.357082 | 11.730463 | 11.721612 | 1.700865 | 19.715639 | 1199.986333 | −0.013667 | 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
| 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
- 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