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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 3.50877 [[cent]]s each.
{{Infobox ET}}
{{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 supports 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.


342 factors as 2 × 3<sup>2</sup> × 19, with subset edos 2, 3, 6, 9, 18, 19, 38, 57, 114, and 171.  
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 ===
{{Primes in edo|342}}
{{Harmonics in equal|342|columns=11}}
 
=== Subset and supersets ===
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.
 
== 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]]
! 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
|-
|-
Line 22: Line 44:
| 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
Line 28: Line 50:
|-
|-
| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 676/675, 1001/1000, 1716/1715, 3025/3024, 32805/32768
| 676/675, 1001/1000, 1716/1715, 3025/3024, 19773/19712
| [{{val| 342 542 794 960 1183 1265 }}] (342f)
| {{mapping| 342 542 794 960 1183 1265 }} (342f)
| +0.178
| +0.178
| 0.1618
| 0.1618
| 4.61
| 4.61
|-
|- style="border-top: double;"
| 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
| [{{val| 342 542 794 960 1183 1266 }}] (342)
| {{mapping| 342 542 794 960 1183 1266 }} (342)
| +0.020
| +0.020
| 0.2061
| 0.2061
| 5.87
| 5.87
|}
|}
* 342et is lower in relative error than any previous ETs in the 11-limit. Not until 612 do we find a better ET in terms of absolute error, and not until 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.
 
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br />per 8ve
! Generator*
! Cents*
! Associated<br />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<br />(47\342)
| 435.09<br />(164.91)
| 9/7<br />(11/10)
| [[Semisupermajor]]
|-
| 2
| 142\342<br />(29\342)
| 498.25<br />(101.75)
| 4/3<br />(35/33)
| [[Bipont]]
|-
| 3
| 71\342<br />(43\342)
| 249.12<br />(150.88)
| 15/13<br />(12/11)
| [[Hemiterm]]
|-
| 6
| 97\342<br />(17\342)
| 340.35<br />(59.65)
| 162/133<br />(88/85)
| [[Semiseptichrome]]
|-
| 6
| 142\342<br />(28\342)
| 498.25<br />(98.25)
| 4/3<br />(18/17)
| [[Semiterm]]
|-
| 9
| 63\342<br />(13\342)
| 221.05<br />(45.61)
| 25/22<br />(77/75)
| [[Quadraennealimmal]]
|-
| 18
| 71\342<br />(5\342)
| 249.12<br />(17.54)
| 15/13<br />(99/98)
| [[Hemiennealimmal]]
|-
| 38
| 142\342<br />(2\342)
| 498.25<br />(7.02)
| 4/3<br />(225/224)
| [[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 ==


[[Category:Equal divisions of the octave]]
* [[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

Latest revision as of 21:53, 15 April 2026

← 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

342 equal divisions of the octave (abbreviated 342edo or 342ed2), also called 342-tone equal temperament (342tet) or 342 equal temperament (342et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 342 equal parts of about 3.51 ¢ each. Each step represents a frequency ratio of 21/342, or the 342nd root of 2.

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.

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

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.

Approximation to JI

Zeta peak index

Tuning Strength Octave (cents) Integer limit
ZPI Steps
per 8ve 		Step size
(cents) 		Height Integral Gap Size Stretch Consistent Distinct
Tempered Pure
2568zpi 341.974851 3.50903 13.478611 12.437722 1.890555 20.767404 1200.088249 0.088249 12 12

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 97\342
(17\342) 		340.35
(59.65) 		162/133
(88/85) 		Semiseptichrome
6 142\342
(28\342) 		498.25
(98.25) 		4/3
(18/17) 		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 distinct

Scales

  • 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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