342edo: Difference between revisions

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== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{{comma basis begin}}
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" | Tuning Error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
|-
| 2.3.5.7.11
| 2.3.5.7.11
Line 31: Line 23:
| 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
|}
{{comma basis end}}
* 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.


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{{rank-2 begin}}
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>Ratio*
! Temperaments
|-
|-
| 1
| 1
Line 75: Line 61:
|-
|-
| 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
| 97\342<br>(17\342)
| 97\342<br />(17\342)
| 340.35<br>(59.65)
| 340.35<br />(59.65)
| 162/133<br>(88/85)
| 162/133<br />(88/85)
| [[Semiseptichrome]]
| [[Semiseptichrome]]
|-
|-
| 6
| 6
| 142\342<br>(28\342)
| 142\342<br />(28\342)
| 498.25<br>(98.25)
| 498.25<br />(98.25)
| 4/3<br>(18/17)
| 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]]
|}
{{rank-2 end}}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
{{orf}}

Revision as of 04:58, 16 November 2024

← 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

Template:Comma basis begin |- | 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 |- style="border-top: double;" | 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 Template:Comma basis end

  • 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

Template:Rank-2 begin |- | 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 Template:Rank-2 end Template:Orf

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