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3,302 edits
Tags: Reverted Visual edit
Line 154: Line 154:
! [[Cent]]s
! [[Cent]]s
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| [[2573571875/2569273344]]
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| 2.894
| 2.894
|-
|-
| [[9453125/9437184]]
| [[2573571875/2572306572]]
| [-20 -2 7 0 2⟩
| [-2 -1 5 7 -8⟩
| 2.922
| 0.851
|-
|-
| [[2100875/2097152]]
| [[2690514981/2684354560]]
| [-21 0 3 5⟩
| [-29 3 -1 7 2⟩
| 3.071
| 3.969
|-
|-
| [[411778125/411041792]]
| [[3487704605/3486784401]]
| [-23 2 5 -2 4⟩
| [0 -20 1 8 2⟩
| 3.099
| 0.457
|-
|-
| [[2542277241/2537553920]]
| [[3955078125/3954653486]]
| [-22 2 -1 10 -2⟩
| [-1 4 11 -11⟩
| 3.219
| 0.186
|-
| [[16808715/16777216]]
| [-24 4 1 3 2⟩
| 3.247
|-
|-
| [[4202539929/4194304000]]
| [[4202539929/4194304000]]
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| 3.396
| 3.396
|-
|-
| [[672417207/671088640]]
| [[4253517961/4246732800]]
| [-27 8 -1 1 4⟩
| [-21 -4 -2 4 6⟩
| 3.424
| 2.764
|-
|-
| [[907878125/905969664]]
| [[4253517961/4251528000]]
| [-25 -3 5 4 2⟩
| [-6 -12 -3 4 6⟩
| 3.643
| 0.810
|-
| [[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
|}
|}

Revision as of 23:47, 8 November 2025

← 341edo 342edo 343edo →
Prime factorization 2 × 32 × 19
Step size 3.50877 ¢ 
Fifth 200\342 (701.754 ¢) (→ 100\171)
Semitones (A1:m2) 32:26 (112.3 ¢ : 91.23 ¢)
Consistency limit 11
Distinct consistency limit 11

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, 9801/9800, 32805/32768, 41503/41472, 43923/43904, and 65625/65536. It is the optimal patent val for 11-limit hemitert temperament, and supports hemiennealimmal.

Prime harmonics

Approximation of prime harmonics in 342edo
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

Table of rank-2 temperaments by generator
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
2401/2400 [-5 -1 -2 4⟩ 0.721
3025/3024 [-4 -3 2 -1 2⟩ 0.572
4375/4374 [-1 -7 4 1⟩ 0.396
9801/9800 [-3 4 -2 -2 2⟩ 0.177
32805/32768 [-15 8 1⟩ 1.954
41503/41472 [-9 -4 0 3 2⟩ 1.294
43923/43904 [-7 1 0 -3 4⟩ 0.749
65625/65536 [-16 1 5 1⟩ 2.349
102487/102400 [-12 0 -2 1 4⟩ 1.470
117649/117612 [-2 -5 0 6 -2⟩ 0.545
151263/151250 [-1 2 -4 5 -2⟩ 0.149
160083/160000 [-8 3 -4 2 2⟩ 0.898
234375/234256 [-4 1 7 0 -4⟩ 0.879
250047/250000 [-4 6 -6 3⟩ 0.325
420175/419904 [-6 -8 2 5⟩ 1.117
496125/495616 [-12 4 3 2 -2⟩ 1.777
703125/702464 [-11 2 7 -3⟩ 1.628
759375/758912 [-7 5 5 -2 -2⟩ 1.056
820125/819896 [-3 8 3 -1 -4⟩ 0.483
1240029/1239040 [-11 11 -1 1 -2⟩ 1.381
1771561/1769472 [-16 -3 0 0 6⟩ 2.043
1771561/1771470 [-1 -11 -1 0 6⟩ 0.089
1771875/1771561 [0 4 5 1 -6⟩ 0.307
1890625/1889568 [-5 -10 6 0 2⟩ 0.968
1953125/1951488 [-8 -2 9 -1 -2⟩ 1.452
2100875/2097152 [-21 0 3 5⟩ 3.071
2460375/2458624 [-10 9 3 -4⟩ 1.233
2657205/2656192 [-6 12 1 -3 -2⟩ 0.660
3294225/3294172 [-2 2 2 -7 4⟩ 0.028
3675375/3670016 [-19 5 3 -1 2⟩ 2.526
3750705/3748096 [-8 7 1 3 -4⟩ 1.205
5250987/5242880 [-20 7 -1 4⟩ 2.675
5764801/5760000 [-10 -2 -4 8⟩ 1.442
7503125/7496192 [-9 0 5 4 -4⟩ 1.600
9150625/9144576 [-8 -6 4 -2 4⟩ 1.145
9453125/9437184 [-20 -2 7 0 2⟩ 2.922
14348907/14336000 [-14 15 -3 -1⟩ 1.558
14348907/14348180 [-2 15 -1 -2 -4⟩ 0.088
15882615/15859712 [-17 3 1 6 -2⟩ 2.498
16808715/16777216 [-24 4 1 3 2⟩ 3.247
17935225/17915904 [-13 -7 2 2 4⟩ 1.866
19140625/19131876 [-2 -14 8 2⟩ 0.792
21437500/21434787 [2 -11 6 3 -2⟩ 0.219
26796875/26763264 [-13 -3 7 3 -2⟩ 2.173
32019867/32000000 [-11 7 -6 0 4⟩ 1.074
35156250/35153041 [1 2 9 -4 -4⟩ 0.158
40353607/40310784 [-11 -9 0 9⟩ 1.838
43046721/43025920 [-9 16 -1 -5⟩ 0.837
44289025/44255232 [-11 -2 2 -4 6⟩ 1.321
47265625/47258883 [0 -9 8 -4 2⟩ 0.247
48828125/48771072 [-12 -5 11 -2⟩ 2.024
48828125/48807528 [-3 -1 11 -5 -2⟩ 0.730
50014503/50000000 [-7 10 -8 1 2⟩ 0.502
52734375/52706752 [-6 3 9 -7⟩ 0.907
56723625/56689952 [-5 3 3 5 -6⟩ 1.028
56953125/56942116 [-2 6 7 -6 -2⟩ 0.335
64304361/64225280 [-18 12 -1 -2 2⟩ 2.130
78125000/78121827 [3 -13 10 -2⟩ 0.070
95703125/95551488 [-17 -6 9 2⟩ 2.745
95703125/95664294 [-1 -3 9 2 -6⟩ 0.703
96059601/96040000 [-6 8 -4 -4 4⟩ 0.353
99648703/99532800 [-14 -5 -2 7 2⟩ 2.015
121060821/121000000 [-6 1 -6 9 -2⟩ 0.870
128119761/128000000 [-13 2 -6 6 2⟩ 1.619
141776649/141724880 [-4 10 -1 4 -6⟩ 0.632
143496441/143360000 [-15 4 -4 -1 6⟩ 1.647
181575625/181398528 [-10 -11 4 4 2⟩ 1.689
184528125/184473632 [-5 10 5 -8⟩ 0.511
199297406/199290375 [1 -13 -3 7 2⟩ 0.061
200120949/200000000 [-9 5 -8 7⟩ 1.047
201768035/201326592 [-26 -1 1 9⟩ 3.792
214358881/214326000 [-4 -7 -3 -2 8⟩ 0.266
214375000/214358881 [3 0 7 3 -8⟩ 0.130
228765625/228709656 [-3 -5 6 -6 4⟩ 0.424
234365481/234256000 [-7 14 -3 2 -4⟩ 0.809
236328125/236027904 [-15 -1 9 -4 2⟩ 2.201
246071287/245760000 [-17 -1 -4 5 4⟩ 2.191
246071287/246037500 [-2 -9 -5 5 4⟩ 0.238
269028375/268435456 [-28 1 3 2 4⟩ 3.820
275653125/275365888 [-14 6 5 -5 2⟩ 1.805
282475249/282268800 [-7 -6 -2 10 -2⟩ 1.266
288240050/288178803 [1 -9 2 8 -4⟩ 0.368
341796875/341545248 [-5 -6 11 1 -4⟩ 1.275
387420489/387200000 [-10 18 -5 0 -2⟩ 0.986
411778125/411041792 [-23 2 5 -2 4⟩ 3.099
428830605/428717762 [-1 6 1 6 -8⟩ 0.456
430489323/430259200 [-10 5 -2 -5 6⟩ 0.926
514714375/514434888 [-3 -12 4 7 -2⟩ 0.940
600362847/599695360 [-13 6 -1 7 -4⟩ 1.926
607645423/607500000 [-5 -5 -7 3 6⟩ 0.414
643076643/642252800 [-19 1 -2 -2 8⟩ 2.219
645700815/645657712 [-4 17 1 -9⟩ 0.116
672417207/671088640 [-27 8 -1 1 4⟩ 3.424
720600125/719634432 [-14 -1 3 8 -4⟩ 2.322
781258401/781250000 [-4 2 -11 2 6⟩ 0.019
907878125/905969664 [-25 -3 5 4 2⟩ 3.643
964565415/963780608 [-13 13 1 -6 2⟩ 1.409
992436543/991232000 [-16 10 -3 5 -2⟩ 2.103
1050304563/1048576000 [-23 11 -3 2 2⟩ 2.852
1071794405/1071385056 [-5 -14 1 -1 8⟩ 0.661
1076168025/1073741824 [-30 16 2⟩ 3.907
1107225625/1106841792 [-6 -1 4 -8 6⟩ 0.600
1162261467/1162084000 [-5 19 -3 -4 -2⟩ 0.264
1220703125/1219401216 [-9 -9 13 0 -2⟩ 1.847
1220703125/1219784832 [-7 -4 13 -6⟩ 1.303
1220703125/1220312709 [0 -5 13 -3 -4⟩ 0.554
1291467969/1291315424 [-5 6 0 -9 6⟩ 0.205
1440894015/1438646272 [-22 9 1 -3 4⟩ 2.703
1500512167/1500000000 [-8 -1 -9 1 8⟩ 0.591
1550115875/1549681956 [-2 -18 3 1 6⟩ 0.485
1640558367/1638400000 [-19 14 -5 3⟩ 2.279
1722499009/1719926784 [-18 -8 0 6 4⟩ 2.587
1722499009/1721868840 [-3 -16 -1 6 4⟩ 0.633
1815912315/1814078464 [-10 2 1 9 -6⟩ 1.749
1838265625/1836660096 [-7 -15 6 6⟩ 1.513
1929229929/1927561216 [-14 2 0 -6 8⟩ 1.498
1977326743/1976535000 [-3 -3 -4 11 -4⟩ 0.693
2152446615/2147483648 [-31 5 1 0 6⟩ 3.996
2152828125/2147483648 [-31 9 6 1⟩ 4.303
2542277241/2537553920 [-22 2 -1 10 -2⟩ 3.219
2573571875/2569273344 [-18 -4 5 7 -2⟩ 2.894
2573571875/2572306572 [-2 -1 5 7 -8⟩ 0.851
2690514981/2684354560 [-29 3 -1 7 2⟩ 3.969
3487704605/3486784401 [0 -20 1 8 2⟩ 0.457
3955078125/3954653486 [-1 4 11 -11⟩ 0.186
4202539929/4194304000 [-25 6 -3 8⟩ 3.396
4253517961/4246732800 [-21 -4 -2 4 6⟩ 2.764
4253517961/4251528000 [-6 -12 -3 4 6⟩ 0.810
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