342edo: Difference between revisions

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'''~~342EDO~~''' is the [[EDO|equal division of the octave]] into 342 parts of 3.50877 [[cent]]s each ~~(dividing the steps of [[171edo|171EDO]] in half)~~. It 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-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.

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.

== 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.

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

=== Prime harmonics ===

[[Category:Equal divisions of the octave]]

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-'''342EDO''' is the [[EDO|equal division of the octave]] into 342 parts of 3.50877 [[cent]]s each (dividing the steps of [[171edo|171EDO]] in half). It 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-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.
+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.
+== 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.
+factors as 2 × 3<sup>2</sup> × 19, with subset edos 2, 3, 6, 9, 18, 19, 38, 57, 114, and 171.
+=== Prime harmonics ===
+{{Primes in edo|342}}
 [[Category:Equal divisions of the octave]]