224edo: Difference between revisions
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| 56/45 | | 56/45 | ||
| [[Quanharuk]] | | [[Quanharuk]] | ||
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| 1 | |||
| 87\224 | |||
| 466.07 | |||
| 55/42 | |||
| [[Hemiseptisix]] | |||
|- | |- | ||
| 1 | | 1 | ||
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| 11/8 | | 11/8 | ||
| [[Emkay]] | | [[Emkay]] | ||
|- | |||
| 2 | |||
| 93\224<br>(19\224) | |||
| 498.21<br>(101.79) | |||
| 4/3<br>(35/33) | |||
| [[Bipont]] | |||
|- | |- | ||
| 2 | | 2 | ||
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| 37\224<br>(19\224) | | 37\224<br>(19\224) | ||
| 198.21<br>(101.79) | | 198.21<br>(101.79) | ||
| 28/25<br> | | 28/25<br>(35/33) | ||
| [[Quadrant]] | | [[Quadrant]] | ||
|- | |- | ||
Revision as of 21:54, 19 July 2021
224EDO is the equal division of the octave into 224 parts of 5.3571 cents each. It 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 117440512/117406179 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 octoid temperament 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.
Script error: No such module "primes_in_edo".
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, 4096/4095 | [⟨224 355 520 629 775 829]] | -0.035 | 0.1805 | 3.37 |
Rank-2 temperaments
| 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 |
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 |