342edo: Difference between revisions

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Subsets and supersets; notability in the 11-limit
Adopt template: EDO intro; cleanup; clarify the title row of the rank-2 temp table
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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.
{{EDO intro|342}}


== 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.


=== 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.  
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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
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| style="border-top: double;" | 2.3.5.7.11.13
| style="border-top: double;" | 2.3.5.7.11.13
| style="border-top: double;" | 676/675, 1001/1000, 1716/1715, 3025/3024, 19773/19712
| style="border-top: double;" | 676/675, 1001/1000, 1716/1715, 3025/3024, 19773/19712
| style="border-top: double;" | [{{val| 342 542 794 960 1183 1265 }}] (342f)
| style="border-top: double;" | {{mapping| 342 542 794 960 1183 1265 }} (342f)
| style="border-top: double;" | +0.178
| style="border-top: double;" | +0.178
| style="border-top: double;" | 0.1618
| style="border-top: double;" | 0.1618
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| style="border-top: double;" | 2.3.5.7.11.13
| style="border-top: double;" | 2.3.5.7.11.13
| style="border-top: double;" | 625/624, 729/728, 847/845, 1575/1573, 4096/4095
| style="border-top: double;" | 625/624, 729/728, 847/845, 1575/1573, 4096/4095
| style="border-top: double;" | [{{val| 342 542 794 960 1183 1266 }}] (342)
| style="border-top: double;" | {{mapping| 342 542 794 960 1183 1266 }} (342)
| style="border-top: double;" | +0.020
| style="border-top: double;" | +0.020
| style="border-top: double;" | 0.2061
| style="border-top: double;" | 0.2061
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|+Table of rank-2 temperaments by generator
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator<br>(Reduced)
! Generator*
! Cents<br>(Reduced)
! Cents*
! Associated<br>Ratio
! Associated<br>Ratio*
! Temperaments
! Temperaments
|-
|-
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| [[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

Revision as of 08:02, 15 November 2023

← 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

Template:EDO intro

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 and 32805/32768. 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.

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
6 142\342
(28\342) 		498.25
(98.25) 		4/3
(200/189) 		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 it is distinct

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