342edo: Difference between revisions

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* 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 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.
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods<br>per octave
! Generator<br>(reduced)
! Cents<br>(reduced)
! Associated<br>ratio
! Temperaments
|-
| 1
| 11\342
| 38.60
| 45/44
| [[Hemitert]]
|-
| 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
| 142\342<br>(28\342)
| 498.25<br>(98.25)
| 4/3<br>(200/189)
| [[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]]
|}


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

Revision as of 17:10, 29 December 2021

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 equal division of the octave into 342 parts of 3.50877 cents each.

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 is 2401/2400, 3025/3024, 4375/4374 and 32805/32768. It is the optimal patent val for 11-limit hemitert temperament, and supports hemiennealimmal.

342 factors as 2 × 32 × 19, with subset edos 2, 3, 6, 9, 18, 19, 38, 57, 114, and 171.

Prime harmonics

Script error: No such module "primes_in_edo".

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, 32805/32768 [⟨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 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.

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per octave 		Generator
(reduced) 		Cents
(reduced) 		Associated
ratio 		Temperaments
1 11\342 38.60 45/44 Hemitert
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
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