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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|342}}
{{ED intro}}


== 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 ===
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! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
! colspan="2" | Tuning error
|-
|-
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| 2.3.5.7.11
| 2.3.5.7.11
| 2401/2400, 3025/3024, 4375/4374, 32805/32768
| 2401/2400, 3025/3024, 4375/4374, 32805/32768
| {{mapping| 342 542 794 960 1183 }}
| {{Mapping| 342 542 794 960 1183 }}
| +0.110
| +0.110
| 0.0556
| 0.0556
Line 34: Line 36:
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 676/675, 1001/1000, 1716/1715, 3025/3024, 19773/19712
| 676/675, 1001/1000, 1716/1715, 3025/3024, 19773/19712
| {{mapping| 342 542 794 960 1183 1265 }} (342f)
| {{Mapping| 342 542 794 960 1183 1265 }} (342f)
| +0.178
| +0.178
| 0.1618
| 0.1618
Line 41: Line 43:
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 625/624, 729/728, 847/845, 1575/1573, 4096/4095
| 625/624, 729/728, 847/845, 1575/1573, 4096/4095
| {{mapping| 342 542 794 960 1183 1266 }} (342)
| {{Mapping| 342 542 794 960 1183 1266 }} (342)
| +0.020
| +0.020
| 0.2061
| 0.2061
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|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
|-
! Periods<br />per 8ve
! Periods<br>per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br />ratio*
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
|-
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|-
|-
| 2
| 2
| 124\342<br />(47\342)
| 47\342
| 435.09<br />(164.91)
| 164.91
| 9/7<br />(11/10)
| 11/10
| [[Semisupermajor]]
| [[Semisupermajor]]
|-
|-
| 2
| 2
| 142\342<br />(29\342)
| 29\342
| 498.25<br />(101.75)
| 101.75
| 4/3<br />(35/33)
| 35/33
| [[Bipont]]
| [[Bipont]]
|-
| 2
| 49\342
| 171.93
| 243/220
| [[Semiosiris]]
|-
|-
| 3
| 3
| 71\342<br />(43\342)
| 43\342
| 249.12<br />(150.88)
| 150.88
| 15/13<br />(12/11)
| 12/11
| [[Hemiterm]]
| [[Hemiterm]]
|-
|-
| 6
| 6
| 97\342<br />(17\342)
| 17\342
| 340.35<br />(59.65)
| 59.65
| 162/133<br />(88/85)
| 88/85
| [[Semiseptichrome]]
| [[Semiseptichrome]]
|-
|-
| 6
| 6
| 142\342<br />(28\342)
| 28\342
| 498.25<br />(98.25)
| 98.25
| 4/3<br />(18/17)
| 18/17
| [[Semiterm]]
| [[Semiterm]]
|-
|-
| 9
| 9
| 63\342<br />(13\342)
| 13\342
| 221.05<br />(45.61)
| 45.61
| 25/22<br />(77/75)
| 77/75
| [[Quadraennealimmal]]
| [[Quadraennealimmal]]
|-
|-
| 18
| 18
| 71\342<br />(5\342)
| 5\342
| 249.12<br />(17.54)
| 17.54
| 15/13<br />(99/98)
| 99/98
| [[Hemiennealimmal]]
| [[Hemiennealimmal]]
|-
|-
| 38
| 38
| 142\342<br />(2\342)
| 2\342
| 498.25<br />(7.02)
| 7.02
| 4/3<br />(225/224)
| 225/224
| [[Hemienneadecal]]
| [[Hemienneadecal]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]]
 
== 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
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