342edo: Difference between revisions

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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

| Prime factorization = 2 × 32 × 19

| Step size = 3.50877¢

| Fifth = 200\342 (701.75¢) (→ [[171edo|100\171]])

| Semitones = 32:26 (112.28¢ : 91.23¢)

| Consistency = 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 [[EDO|equal division of the octave]] into 342 parts of about 3.51 [[cent]]s 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 [[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 × 32 × 19, with subset edos 2, 3, 6, 9, 18, 19, 38, 57, 114, and 171.

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

=== Prime harmonics ===

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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
+| Prime factorization = 2 × 3<sup>2</sup> × 19
+| Step size = 3.50877¢
+| Fifth = 200\342 (701.75¢) (→ [[171edo|100\171]])
+| Semitones = 32:26 (112.28¢ : 91.23¢)
+| Consistency = 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 [[EDO|equal division of the octave]] into 342 parts of about 3.51 [[cent]]s each.
 == Theory ==
 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.
+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 ===