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.
Notation
Sagittal
224edo can be written in Sagittal using almost the entire Athenian extension (except for since it tempers 1240029/1239040), by virtue of its apotome being equal to 21 edosteps, which is the maximum equal division of the apotome (eda) supported by Athenian[1]. It is identical to 217edo's Sagittal notation, but it uses the 55C for the +6/-6 alteration instead of 11/7C.[2]
| 224edosteps | -21 | -20 | -19 | -18 | -17 | -16 | -15 | -14 | -13 | -12 | -11 | -10 | -9 | -8 | -7 | -6 | -5 | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Revo | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |
| Evo | | | | | | | | | | | | | | | | | | | | | | |
Ups-and-downs notation
The 4-up (quup) alteration maps to the pythagorean/syntonic comma.
| 224edosteps | -21 | -20 | -19 | -18 | -17 | -16 | -15 | -14 | -13 | -12 | -11 | -10 | -9 | -8 | -7 | -6 | -5 | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| b | <<<< | ^<<<< | vvv<<< | vv<<< | v<<< | <<< | ^<<< | vvv<< | vv<< | v<< | << | ^<< | vvv< | vv< | v< | < | ^< | vvv | vv | v | h | ^ | ^^ | ^^^ | v> | > | ^> | ^^> | ^^^> | v>> | >> | ^>> | ^^>> | ^^^>> | v>>> | >>> | ^>>> | ^^>>> | ^^^>>> | v>>>> | >>>> | # | |
| ^b | ^^b | ^^^b | v>b | >b | ^>b | ^^>b | ^^^>b | v>>b | >>b | ^>>b | ^^>>b | ^^^>>b | v>>>b | >>>b | ^>>>b | ^^>>>b | ^^^>>>b | v>>>>b | >>>>b | <<<<# | ^<<<<# | vvv<<<# | vv<<<# | v<<<# | <<<# | ^<<<# | vvv<<# | vv<<# | v<<# | <<# | ^<<# | vvv<# | vv<# | v<# | <# | ^<# | vvv# | vv# | v# |
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