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Subsets and supersets; notability in the 11-limit
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{{Infobox ET}}
{{Infobox ET}}
The '''342 equal divisions of the octave''' ('''342edo'''), or the '''342(-tone) equal temperament''' ('''342tet''', '''342et''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 342 parts of about 3.51 [[cent]]s each.
{{ED intro}}


== Theory ==
== 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 [[The Riemann zeta function and tuning #Zeta EDO lists|zeta peak edo]]. A basis for the 11-limit commas is 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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=== Subset and supersets ===
=== Subset and supersets ===
342 factors as 2 × 3<sup>2</sup> × 19, with subset edos {{EDOs| 2, 3, 6, 9, 18, 19, 38, 57, 114, and 171 }}.  
342 factors as {{factorization|342}}, with subset edos {{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.  
[[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 ===
{{ZPI
| zpi = 2568
| steps = 341.974850913987
| step size = 3.50902996753355
| tempered height = 13.478611
| pure height = 12.437722
| integral = 1.890555
| gap = 20.767404
| octave = 1200.08824889647
| consistent = 12
| distinct = 12
}}


== 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
| [{{val| 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;" | [{{val| 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;" | [{{val| 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
|-
! Generator<br>(Reduced)
! Periods<br />per 8ve
! Cents<br>(Reduced)
! Generator*
! Associated<br>Ratio
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
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|-
|-
| 2
| 2
| 124\342<br>(47\342)
| 124\342<br />(47\342)
| 435.09<br>(164.91)
| 435.09<br />(164.91)
| 9/7<br>(11/10)
| 9/7<br />(11/10)
| [[Semisupermajor]]
| [[Semisupermajor]]
|-
|-
| 2
| 2
| 142\342<br>(29\342)
| 142\342<br />(29\342)
| 498.25<br>(101.75)
| 498.25<br />(101.75)
| 4/3<br>(35/33)
| 4/3<br />(35/33)
| [[Bipont]]
| [[Bipont]]
|-
|-
| 3
| 3
| 71\342<br>(43\342)
| 71\342<br />(43\342)
| 249.12<br>(150.88)
| 249.12<br />(150.88)
| 15/13<br>(12/11)
| 15/13<br />(12/11)
| [[Hemiterm]]
| [[Hemiterm]]
|-
|-
| 6
| 6
| 142\342<br>(28\342)
| 97\342<br />(17\342)
| 498.25<br>(98.25)
| 340.35<br />(59.65)
| 4/3<br>(200/189)
| 162/133<br />(88/85)
| [[Semiseptichrome]]
|-
| 6
| 142\342<br />(28\342)
| 498.25<br />(98.25)
| 4/3<br />(18/17)
| [[Semiterm]]
| [[Semiterm]]
|-
|-
| 9
| 9
| 63\342<br>(13\342)
| 63\342<br />(13\342)
| 221.05<br>(45.61)
| 221.05<br />(45.61)
| 25/22<br>(77/75)
| 25/22<br />(77/75)
| [[Quadraennealimmal]]
| [[Quadraennealimmal]]
|-
|-
| 18
| 18
| 71\342<br>(5\342)
| 71\342<br />(5\342)
| 249.12<br>(17.54)
| 249.12<br />(17.54)
| 15/13<br>(99/98)
| 15/13<br />(99/98)
| [[Hemiennealimmal]]
| [[Hemiennealimmal]]
|-
|-
| 38
| 38
| 142\342<br>(2\342)
| 142\342<br />(2\342)
| 498.25<br>(7.02)
| 498.25<br />(7.02)
| 4/3<br>(225/224)
| 4/3<br />(225/224)
| [[Hemienneadecal]]
| [[Hemienneadecal]]
|}
|}
<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 ==
* [[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
Retrieved from "https://en.xen.wiki/w/342edo"