342edo
342 equal divisions of the octave (abbreviated 342edo or 342ed2), also called 342-tone equal temperament (342tet) or 342 equal temperament (342et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 342 equal parts of about 3.51 ¢ each. Each step represents a frequency ratio of 21/342, or the 342nd root of 2.
Theory
342edo is a very strong 11-limit system. It is, as one would expect, distinctly consistent through the 11-odd-limit, but goes no higher; nonetheless, it is a zeta peak edo. A basis for the 11-limit commas consists of 2401/2400, 3025/3024, 4375/4374 and 32805/32768. It is the optimal patent val for 11-limit hemitert temperament, and supports hemiennealimmal.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.20 | -0.35 | -0.40 | -0.44 | +1.58 | +0.31 | +0.73 | -0.20 | -1.51 | -1.18 |
| Relative (%) | +0.0 | -5.7 | -9.9 | -11.5 | -12.6 | +45.0 | +8.8 | +20.9 | -5.8 | -43.0 | -33.5 | |
| Steps (reduced) |
342 (0) |
542 (200) |
794 (110) |
960 (276) |
1183 (157) |
1266 (240) |
1398 (30) |
1453 (85) |
1547 (179) |
1661 (293) |
1694 (326) | |
Subset and supersets
342 factors as 2 × 32 × 19, with subset edos 2, 3, 6, 9, 18, 19, 38, 57, 114, and 171.
684edo, which doubles 342edo, provides an approximation of harmonic 13 that works well with the flat tendency of its 11-limit mapping.
Approximation to JI
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 | |||||||||
| 2568zpi | 341.974851 | 3.50903 | 13.478611 | 12.437722 | 1.890555 | 20.767404 | 1200.088249 | 0.088249 | 12 | 12 |
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5.7.11 | 2401/2400, 3025/3024, 4375/4374, 32805/32768 | [⟨342 542 794 960 1183]] | +0.110 | 0.0556 | 1.59 |
| 2.3.5.7.11.13 | 676/675, 1001/1000, 1716/1715, 3025/3024, 19773/19712 | [⟨342 542 794 960 1183 1265]] (342f) | +0.178 | 0.1618 | 4.61 |
| 2.3.5.7.11.13 | 625/624, 729/728, 847/845, 1575/1573, 4096/4095 | [⟨342 542 794 960 1183 1266]] (342) | +0.020 | 0.2061 | 5.87 |
- 342et is lower in relative error than any previous equal temperaments in the 11-limit, being the first to beat 270. Not until 612 do we find a better equal temperament in terms of absolute error, and not until 1848 do we find one in terms of relative error.
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 11\342 | 38.60 | 45/44 | Hemitert |
| 2 | 5\342 | 17.54 | 99/98 | Poseidon |
| 2 | 50\342 | 175.44 | 448/405 | Bisesqui |
| 2 | 124\342 (47\342) |
435.09 (164.91) |
9/7 (11/10) |
Semisupermajor |
| 2 | 142\342 (29\342) |
498.25 (101.75) |
4/3 (35/33) |
Bipont |
| 3 | 71\342 (43\342) |
249.12 (150.88) |
15/13 (12/11) |
Hemiterm |
| 6 | 97\342 (17\342) |
340.35 (59.65) |
162/133 (88/85) |
Semiseptichrome |
| 6 | 142\342 (28\342) |
498.25 (98.25) |
4/3 (18/17) |
Semiterm |
| 9 | 63\342 (13\342) |
221.05 (45.61) |
25/22 (77/75) |
Quadraennealimmal |
| 18 | 71\342 (5\342) |
249.12 (17.54) |
15/13 (99/98) |
Hemiennealimmal |
| 38 | 142\342 (2\342) |
498.25 (7.02) |
4/3 (225/224) |
Hemienneadecal |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct
Scales
- Diamond11: 43 4 5 6 8 10 14 9 11 9 5 18 15 9 10 9 15 18 5 9 11 9 14 10 8 6 5 4 43
Commas of the 2.3.5.7.11 subgroup tempered out in 342edo patent val
Commas with numerator <= 2^32:
| Ratio | Monzo | Cents |
|---|---|---|
| 781258401/781250000 | [-4 2 -11 2 6⟩ | 0.019 |
| 3294225/3294172 | [-2 2 2 -7 4⟩ | 0.028 |
| 199297406/199290375 | [1 -13 -3 7 2⟩ | 0.061 |
| 78125000/78121827 | [3 -13 10 -2⟩ | 0.070 |
| 14348907/14348180 | [-2 15 -1 -2 -4⟩ | 0.088 |
| 1771561/1771470 | [-1 -11 -1 0 6⟩ | 0.089 |
| 645700815/645657712 | [-4 17 1 -9⟩ | 0.116 |
| 214375000/214358881 | [3 0 7 3 -8⟩ | 0.130 |
| 151263/151250 | [-1 2 -4 5 -2⟩ | 0.149 |
| 35156250/35153041 | [1 2 9 -4 -4⟩ | 0.158 |
| 9801/9800 | [-3 4 -2 -2 2⟩ | 0.177 |
| 3955078125/3954653486 | [-1 4 11 -11⟩ | 0.186 |
| 1291467969/1291315424 | [-5 6 0 -9 6⟩ | 0.205 |
| 21437500/21434787 | [2 -11 6 3 -2⟩ | 0.219 |
| 246071287/246037500 | [-2 -9 -5 5 4⟩ | 0.238 |
| 47265625/47258883 | [0 -9 8 -4 2⟩ | 0.247 |
| 1162261467/1162084000 | [-5 19 -3 -4 -2⟩ | 0.264 |
| 214358881/214326000 | [-4 -7 -3 -2 8⟩ | 0.266 |
| 1771875/1771561 | [0 4 5 1 -6⟩ | 0.307 |
| 250047/250000 | [-4 6 -6 3⟩ | 0.325 |
| 56953125/56942116 | [-2 6 7 -6 -2⟩ | 0.335 |
| 96059601/96040000 | [-6 8 -4 -4 4⟩ | 0.353 |
| 288240050/288178803 | [1 -9 2 8 -4⟩ | 0.368 |
| 4375/4374 | [-1 -7 4 1⟩ | 0.396 |
| 607645423/607500000 | [-5 -5 -7 3 6⟩ | 0.414 |
| 228765625/228709656 | [-3 -5 6 -6 4⟩ | 0.424 |
| 428830605/428717762 | [-1 6 1 6 -8⟩ | 0.456 |
| 3487704605/3486784401 | [0 -20 1 8 2⟩ | 0.457 |
| 820125/819896 | [-3 8 3 -1 -4⟩ | 0.483 |
| 1550115875/1549681956 | [-2 -18 3 1 6⟩ | 0.485 |
| 50014503/50000000 | [-7 10 -8 1 2⟩ | 0.502 |
| 184528125/184473632 | [-5 10 5 -8⟩ | 0.511 |
| 117649/117612 | [-2 -5 0 6 -2⟩ | 0.545 |
| 1220703125/1220312709 | [0 -5 13 -3 -4⟩ | 0.554 |
| 3025/3024 | [-4 -3 2 -1 2⟩ | 0.572 |
| 1500512167/1500000000 | [-8 -1 -9 1 8⟩ | 0.591 |
| 1107225625/1106841792 | [-6 -1 4 -8 6⟩ | 0.600 |
| 141776649/141724880 | [-4 10 -1 4 -6⟩ | 0.632 |
| 1722499009/1721868840 | [-3 -16 -1 6 4⟩ | 0.633 |
| 2657205/2656192 | [-6 12 1 -3 -2⟩ | 0.660 |
| 1071794405/1071385056 | [-5 -14 1 -1 8⟩ | 0.661 |
| 1977326743/1976535000 | [-3 -3 -4 11 -4⟩ | 0.693 |
| 95703125/95664294 | [-1 -3 9 2 -6⟩ | 0.703 |
| 2401/2400 | [-5 -1 -2 4⟩ | 0.721 |
| 48828125/48807528 | [-3 -1 11 -5 -2⟩ | 0.730 |
| 43923/43904 | [-7 1 0 -3 4⟩ | 0.749 |
| 19140625/19131876 | [-2 -14 8 2⟩ | 0.792 |
| 234365481/234256000 | [-7 14 -3 2 -4⟩ | 0.809 |
| 4253517961/4251528000 | [-6 -12 -3 4 6⟩ | 0.810 |
| 43046721/43025920 | [-9 16 -1 -5⟩ | 0.837 |
| 2573571875/2572306572 | [-2 -1 5 7 -8⟩ | 0.851 |
| 121060821/121000000 | [-6 1 -6 9 -2⟩ | 0.870 |
| 234375/234256 | [-4 1 7 0 -4⟩ | 0.879 |
| 160083/160000 | [-8 3 -4 2 2⟩ | 0.898 |
| 52734375/52706752 | [-6 3 9 -7⟩ | 0.907 |
| 430489323/430259200 | [-10 5 -2 -5 6⟩ | 0.926 |
| 514714375/514434888 | [-3 -12 4 7 -2⟩ | 0.940 |
| 1890625/1889568 | [-5 -10 6 0 2⟩ | 0.968 |
| 387420489/387200000 | [-10 18 -5 0 -2⟩ | 0.986 |
| 56723625/56689952 | [-5 3 3 5 -6⟩ | 1.028 |
| 200120949/200000000 | [-9 5 -8 7⟩ | 1.047 |
| 759375/758912 | [-7 5 5 -2 -2⟩ | 1.056 |
| 32019867/32000000 | [-11 7 -6 0 4⟩ | 1.074 |
| 420175/419904 | [-6 -8 2 5⟩ | 1.117 |
| 9150625/9144576 | [-8 -6 4 -2 4⟩ | 1.145 |
| 3750705/3748096 | [-8 7 1 3 -4⟩ | 1.205 |
| 2460375/2458624 | [-10 9 3 -4⟩ | 1.233 |
| 282475249/282268800 | [-7 -6 -2 10 -2⟩ | 1.266 |
| 341796875/341545248 | [-5 -6 11 1 -4⟩ | 1.275 |
| 41503/41472 | [-9 -4 0 3 2⟩ | 1.294 |
| 1220703125/1219784832 | [-7 -4 13 -6⟩ | 1.303 |
| 44289025/44255232 | [-11 -2 2 -4 6⟩ | 1.321 |
| 1240029/1239040 | [-11 11 -1 1 -2⟩ | 1.381 |
| 964565415/963780608 | [-13 13 1 -6 2⟩ | 1.409 |
| 5764801/5760000 | [-10 -2 -4 8⟩ | 1.442 |
| 1953125/1951488 | [-8 -2 9 -1 -2⟩ | 1.452 |
| 102487/102400 | [-12 0 -2 1 4⟩ | 1.470 |
| 1929229929/1927561216 | [-14 2 0 -6 8⟩ | 1.498 |
| 1838265625/1836660096 | [-7 -15 6 6⟩ | 1.513 |
| 14348907/14336000 | [-14 15 -3 -1⟩ | 1.558 |
| 7503125/7496192 | [-9 0 5 4 -4⟩ | 1.600 |
| 128119761/128000000 | [-13 2 -6 6 2⟩ | 1.619 |
| 703125/702464 | [-11 2 7 -3⟩ | 1.628 |
| 143496441/143360000 | [-15 4 -4 -1 6⟩ | 1.647 |
| 181575625/181398528 | [-10 -11 4 4 2⟩ | 1.689 |
| 1815912315/1814078464 | [-10 2 1 9 -6⟩ | 1.749 |
| 496125/495616 | [-12 4 3 2 -2⟩ | 1.777 |
| 275653125/275365888 | [-14 6 5 -5 2⟩ | 1.805 |
| 40353607/40310784 | [-11 -9 0 9⟩ | 1.838 |
| 1220703125/1219401216 | [-9 -9 13 0 -2⟩ | 1.847 |
| 17935225/17915904 | [-13 -7 2 2 4⟩ | 1.866 |
| 600362847/599695360 | [-13 6 -1 7 -4⟩ | 1.926 |
| 32805/32768 | [-15 8 1⟩ | 1.954 |
| 99648703/99532800 | [-14 -5 -2 7 2⟩ | 2.015 |
| 48828125/48771072 | [-12 -5 11 -2⟩ | 2.024 |
| 1771561/1769472 | [-16 -3 0 0 6⟩ | 2.043 |
| 992436543/991232000 | [-16 10 -3 5 -2⟩ | 2.103 |
| 64304361/64225280 | [-18 12 -1 -2 2⟩ | 2.130 |
| 26796875/26763264 | [-13 -3 7 3 -2⟩ | 2.173 |
| 246071287/245760000 | [-17 -1 -4 5 4⟩ | 2.191 |
| 236328125/236027904 | [-15 -1 9 -4 2⟩ | 2.201 |
| 643076643/642252800 | [-19 1 -2 -2 8⟩ | 2.219 |
| 1640558367/1638400000 | [-19 14 -5 3⟩ | 2.279 |
| 720600125/719634432 | [-14 -1 3 8 -4⟩ | 2.322 |
| 65625/65536 | [-16 1 5 1⟩ | 2.349 |
| 15882615/15859712 | [-17 3 1 6 -2⟩ | 2.498 |
| 3675375/3670016 | [-19 5 3 -1 2⟩ | 2.526 |
| 1722499009/1719926784 | [-18 -8 0 6 4⟩ | 2.587 |
| 5250987/5242880 | [-20 7 -1 4⟩ | 2.675 |
| 1440894015/1438646272 | [-22 9 1 -3 4⟩ | 2.703 |
| 95703125/95551488 | [-17 -6 9 2⟩ | 2.745 |
| 4253517961/4246732800 | [-21 -4 -2 4 6⟩ | 2.764 |
| 1050304563/1048576000 | [-23 11 -3 2 2⟩ | 2.852 |
| 2573571875/2569273344 | [-18 -4 5 7 -2⟩ | 2.894 |
| 9453125/9437184 | [-20 -2 7 0 2⟩ | 2.922 |
| 2100875/2097152 | [-21 0 3 5⟩ | 3.071 |
| 411778125/411041792 | [-23 2 5 -2 4⟩ | 3.099 |
| 2542277241/2537553920 | [-22 2 -1 10 -2⟩ | 3.219 |
| 16808715/16777216 | [-24 4 1 3 2⟩ | 3.247 |
| 4202539929/4194304000 | [-25 6 -3 8⟩ | 3.396 |
| 672417207/671088640 | [-27 8 -1 1 4⟩ | 3.424 |
| 907878125/905969664 | [-25 -3 5 4 2⟩ | 3.643 |
| 201768035/201326592 | [-26 -1 1 9⟩ | 3.792 |
| 269028375/268435456 | [-28 1 3 2 4⟩ | 3.820 |
| 1076168025/1073741824 | [-30 16 2⟩ | 3.907 |
| 2690514981/2684354560 | [-29 3 -1 7 2⟩ | 3.969 |
| 2152446615/2147483648 | [-31 5 1 0 6⟩ | 3.996 |
| 2152828125/2147483648 | [-31 9 6 1⟩ | 4.303 |