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== Theory ==
== Theory ==
342edo is a very strong 11-limit system. It is, as one would expect, [[consistency|distinctly consistent]] through the [[11-odd-limit]], but goes no higher; nonetheless, it is a [[zeta peak edo]]. A [[comma basis|basis]] for the 11-limit [[comma]]s consists of [[2401/2400]], [[3025/3024]], [[4375/4374]] and [[32805/32768]]. It is the [[optimal patent val]] for 11-limit [[Breedsmic temperaments #Hemitert|hemitert]] temperament, and [[support]]s hemiennealimmal.
342edo is a very strong 11-limit system. It is, as one would expect, [[consistency|distinctly consistent]] through the [[11-odd-limit]], but goes no higher; nonetheless, it is a [[zeta peak edo]]. A [[comma basis|basis]] for the 11-limit [[comma]]s consists of [[2401/2400]], [[3025/3024]], [[4375/4374]] and [[32805/32768]]. It is the [[optimal patent val]] for 11-limit [[Breedsmic temperaments #Hemitert|hemitert]] temperament, and [[support]]s hemiennealimmal.
If 3.5 cents is taken as the [[just-noticeable difference]], then 342edo may be regarded as the highest EDO whose step size remains individually discernible. However, the [[JND]] is not fixed and depends on the listener and musical context.


=== Prime harmonics ===
=== Prime harmonics ===
Line 139: Line 141:
| [[Hemienneadecal]]
| [[Hemienneadecal]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct


== Scales ==
== Scales ==


* [[11-odd-limit|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
* [[11-odd-limit|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:
{| class="wikitable mw-collapsible mw-collapsed sortable"
! [[Ratio]]
! [[Monzo]]
! [[Cent]]s
|-
| [[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
|}
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