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342edo: Difference between revisions

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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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== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|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
|-
|-
! [[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
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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
| 1.59
| 1.59
|-
|-
| style="border-top: double;" | 2.3.5.7.11.13
| 2.3.5.7.11.13
| style="border-top: double;" | 676/675, 1001/1000, 1716/1715, 3025/3024, 19773/19712
| 676/675, 1001/1000, 1716/1715, 3025/3024, 19773/19712
| style="border-top: double;" | {{mapping| 342 542 794 960 1183 1265 }} (342f)
| {{Mapping| 342 542 794 960 1183 1265 }} (342f)
| style="border-top: double;" | +0.178
| +0.178
| style="border-top: double;" | 0.1618
| 0.1618
| style="border-top: double;" | 4.61
| 4.61
|-
|- style="border-top: double;"
| style="border-top: double;" | 2.3.5.7.11.13
| 2.3.5.7.11.13
| style="border-top: double;" | 625/624, 729/728, 847/845, 1575/1573, 4096/4095
| 625/624, 729/728, 847/845, 1575/1573, 4096/4095
| style="border-top: double;" | {{mapping| 342 542 794 960 1183 1266 }} (342)
| {{Mapping| 342 542 794 960 1183 1266 }} (342)
| style="border-top: double;" | +0.020
| +0.020
| style="border-top: double;" | 0.2061
| 0.2061
| style="border-top: double;" | 5.87
| 5.87
|}
|}
* 342et is lower in relative error than any previous equal temperaments in the 11-limit, being the first to beat [[270edo|270]]. Not until [[612edo|612]] do we find a better equal temperament in terms of absolute error, and not until [[1848edo|1848]] do we find one in terms of relative error.
* 342et is lower in relative error than any previous equal temperaments in the 11-limit, being the first to beat [[270edo|270]]. Not until [[612edo|612]] do we find a better equal temperament in terms of absolute error, and not until [[1848edo|1848]] do we find one in terms of relative error.
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=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+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>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is 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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