342edo: Difference between revisions

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== 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, 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.


=== Prime harmonics ===
{{Harmonics in equal|342|columns=11}}
=== Miscellany ===
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 2 × 3<sup>2</sup> × 19, with subset edos {{EDOs| 2, 3, 6, 9, 18, 19, 38, 57, 114, and 171 }}.  
=== Prime harmonics ===
{{Primes in edo|342}}


== 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|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]] (¢)
Line 28: Line 29:
| 1.59
| 1.59
|-
|-
| 2.3.5.7.11.13
| style="border-top: double;" | 2.3.5.7.11.13
| 676/675, 1001/1000, 1716/1715, 3025/3024, 19773/19712
| style="border-top: double;" | 676/675, 1001/1000, 1716/1715, 3025/3024, 19773/19712
| [{{val| 342 542 794 960 1183 1265 }}] (342f)
| style="border-top: double;" | [{{val| 342 542 794 960 1183 1265 }}] (342f)
| +0.178
| style="border-top: double;" | +0.178
| 0.1618
| style="border-top: double;" | 0.1618
| 4.61
| style="border-top: double;" | 4.61
|-
|-
| 2.3.5.7.11.13
| style="border-top: double;" | 2.3.5.7.11.13
| 625/624, 729/728, 847/845, 1575/1573, 4096/4095
| style="border-top: double;" | 625/624, 729/728, 847/845, 1575/1573, 4096/4095
| [{{val| 342 542 794 960 1183 1266 }}] (342)
| style="border-top: double;" | [{{val| 342 542 794 960 1183 1266 }}] (342)
| +0.020
| style="border-top: double;" | +0.020
| 0.2061
| style="border-top: double;" | 0.2061
| 5.87
| style="border-top: double;" | 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. 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"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+Table of rank-2 temperaments by generator
! Periods<br>per octave
! Periods<br>per 8ve
! Generator<br>(reduced)
! Generator<br>(Reduced)
! Cents<br>(reduced)
! Cents<br>(Reduced)
! Associated<br>ratio
! Associated<br>Ratio
! Temperaments
! Temperaments
|-
|-

Revision as of 09:46, 26 November 2022

← 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

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 about 3.51 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.

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)

Miscellany

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

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. 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
(Reduced) 		Cents
(Reduced) 		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
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