342edo: Difference between revisions

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== Theory ==

342edo is a very strong 11-limit system~~; not until [[1848edo|1848]] do we reach one with a lower 11-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]~~. 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, 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.

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

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=== Prime harmonics ===

== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

! rowspan="2" | Subgroup

! rowspan="2" | [[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

| 2401/2400, 3025/3024, 4375/4374, 32805/32768

| [{{val| 342 542 794 960 1183 }}]

| +0.110

| 0.0556

| 1.59

|}

* 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.

[[Category:Equal divisions of the octave]]

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 == Theory ==
-edo is a very strong 11-limit system; not until [[1848edo|1848]] do we reach one with a lower 11-limit [[Tenney-Euclidean temperament measures #TE simple badness|relative error]]. 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.
+edo 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.
 factors as 2 × 3<sup>2</sup> × 19, with subset edos 2, 3, 6, 9, 18, 19, 38, 57, 114, and 171.
@@ Line 8: / Line 8: @@
 === Prime harmonics ===
 {{Primes in edo|342}}
+== Regular temperament properties ==
+{| class="wikitable center-4 center-5 center-6"
+! rowspan="2" | Subgroup
+! rowspan="2" | [[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
+| 2401/2400, 3025/3024, 4375/4374, 32805/32768
+| [{{val| 342 542 794 960 1183 }}]
+| +0.110
+| 0.0556
+| 1.59
+|}
+* 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.
 [[Category:Equal divisions of the octave]]