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 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 [[support]]s hemiennealimmal.

342edo is a very strong 11-limit system. It is, as one would expect, [[consistency|distinctly consistent]] through the [[11-odd-limit]], but goes no higher; nonetheless, it is a [[zeta peak edo]]. A [[comma basis|basis]] for the 11-limit [[comma]]s consists of [[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 ===

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=== Subset and supersets ===

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

342 factors as {{factorization|342}}, with subset edos {{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.

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| 2.3.5.7.11

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

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

| {{mapping| 342 542 794 960 1183 }}

| +0.110

| 0.0556

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| style="border-top: double;" | 2.3.5.7.11.13

| style="border-top: double;" | 676/675, 1001/1000, 1716/1715, 3025/3024, 19773/19712

| style="border-top: double;" | [{{~~val~~| 342 542 794 960 1183 1265 }}] (342f)

| style="border-top: double;" | {{mapping| 342 542 794 960 1183 1265 }} (342f)

| style="border-top: double;" | +0.178

| style="border-top: double;" | 0.1618

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| style="border-top: double;" | 2.3.5.7.11.13

| style="border-top: double;" | 625/624, 729/728, 847/845, 1575/1573, 4096/4095

| style="border-top: double;" | [{{~~val~~| 342 542 794 960 1183 1266 }}] (342)

| style="border-top: double;" | {{mapping| 342 542 794 960 1183 1266 }} (342)

| style="border-top: double;" | +0.020

| style="border-top: double;" | 0.2061

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|+Table of rank-2 temperaments by generator

! Periods per 8ve

! Generator~~ (Reduced)~~

! Generator*

! Cents~~ (Reduced)~~

! Cents*

! Associated Ratio

! Associated Ratio*

! Temperaments

|-

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| [[Hemienneadecal]]

|}

<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-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.
+{{EDO intro|342}}
 == 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 [[support]]s hemiennealimmal.
+edo is a very strong 11-limit system. It is, as one would expect, [[consistency|distinctly consistent]] through the [[11-odd-limit]], but goes no higher; nonetheless, it is a  [[zeta peak edo]]. A [[comma basis|basis]] for the 11-limit [[comma]]s consists of [[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 ===
@@ Line 9: / Line 9: @@
 === Subset and supersets ===
-factors as 2 × 3<sup>2</sup> × 19, with subset edos {{EDOs| 2, 3, 6, 9, 18, 19, 38, 57, 114, and 171 }}.
+factors as {{factorization|342}}, with subset edos {{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.
@@ Line 26: / Line 26: @@
 | 2.3.5.7.11
 | 2401/2400, 3025/3024, 4375/4374, 32805/32768
-| [{{val| 342 542 794 960 1183 }}]
+| {{mapping| 342 542 794 960 1183 }}
 | +0.110
 | 0.0556
@@ Line 33: / Line 33: @@
 | style="border-top: double;" | 2.3.5.7.11.13
 | style="border-top: double;" | 676/675, 1001/1000, 1716/1715, 3025/3024, 19773/19712
-| style="border-top: double;" | [{{val| 342 542 794 960 1183 1265 }}] (342f)
+| style="border-top: double;" | {{mapping| 342 542 794 960 1183 1265 }} (342f)
 | style="border-top: double;" | +0.178
 | style="border-top: double;" | 0.1618
@@ Line 40: / Line 40: @@
 | style="border-top: double;" | 2.3.5.7.11.13
 | style="border-top: double;" | 625/624, 729/728, 847/845, 1575/1573, 4096/4095
-| style="border-top: double;" | [{{val| 342 542 794 960 1183 1266 }}] (342)
+| style="border-top: double;" | {{mapping| 342 542 794 960 1183 1266 }} (342)
 | style="border-top: double;" | +0.020
 | style="border-top: double;" | 0.2061
@@ Line 51: / Line 51: @@
 |+Table of rank-2 temperaments by generator
 ! Periods<br>per 8ve
-! Generator<br>(Reduced)
+! Generator*
-! Cents<br>(Reduced)
+! Cents*
-! Associated<br>Ratio
+! Associated<br>Ratio*
 ! Temperaments
 |-
@@ Line 116: / Line 116: @@
 | [[Hemienneadecal]]
 |}
+<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct