354edo: Difference between revisions

Line 1:

~~The '''354 equal divisions of the octave''' ('''354edo'''), or the '''354(-tone) equal temperament''' ('''354tet''', '''354et''') when viewed from a [[regular temperament]] perspective, is the [[~~EDO|~~equal division of the octave]] into~~ 354 ~~parts of about 3.39 [[cent]]s each.~~

== Theory ==

354edo is [[enfactoring|enfactored]] in the 5-limit, with the same tuning as [[118edo]], defined by tempering out the [[schisma]] and the [[parakleisma]]. In the 7-limit, it tempers out 118098/117649 (stearnsma), 250047/250000 ([[landscape comma|landscape]]), and 703125/702464 ([[~~meter comma|~~meter]]); in the 11-limit, [[540/539]], and [[4000/3993]]; in the 13-limit, [[729/728]], [[1575/1573]], [[1716/1715]], [[2080/2079]], [[4096/4095]], and [[4225/4224]]. It provides the [[optimal patent val]] for [[stearnscape]].

354edo is [[enfactoring|enfactored]] in the 5-limit, with the same tuning as [[118edo]], defined by tempering out the [[schisma]] and the [[parakleisma]]. In the 7-limit, it tempers out 118098/117649 (stearnsma), 250047/250000 ([[landscape comma|landscape]]), and 703125/702464 ([[meter]]); in the 11-limit, [[540/539]], and [[4000/3993]]; in the 13-limit, [[729/728]], [[1575/1573]], [[1716/1715]], [[2080/2079]], [[4096/4095]], and [[4225/4224]]. It provides the [[optimal patent val]] for [[stearnscape]].

=== Prime harmonics ===

=== Subsets and supersets ===

Since 354 factors into 2 × 3 × 59, 354edo has subset edos {{EDOs| 2, 3, 6, 59, 118, and 177 }}.

== Regular temperament properties ==

{| 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" | Optimal 8ve ~~stretch~~ (¢)

! rowspan="2" | Optimal 8ve Stretch (¢)

! colspan="2" | Tuning ~~error~~

! colspan="2" | Tuning Error

|-

! [[TE error|Absolute]] (¢)

Line 61:

Line 64:

|+Table of rank-2 temperaments by generator

! Periods per 8ve

! Generator (~~reduced~~)

! Generator (Reduced)

! Cents (~~reduced~~)

! Cents (Reduced)

! Associated ~~ratio~~

! Associated Ratio

! Temperaments

|-

Line 75:

Line 78:

| 147\354 (29\354)

| 498.31 (98.31)

| 4/3 (~~200~~/~~189~~)

| 4/3 (18/17)

| [[Term]] / terminator

|-

Line 87:

Line 90:

| 147\354 (29\354)

| 498.31 (98.31)

| 4/3 (~~200~~/~~189~~)

| 4/3 (18/17)

| [[Semiterm]]

|-

|118

| 118

|167\354 (2\354)

| 167\354 (2\354)

|566.101 (6.78)

| 566.101 (6.78)

|165/119 (?)

| 165/119 (?)

|[[Oganesson]]

| [[Oganesson]]

|}

~~[[Category:Equal divisions of the octave|###]] ~~

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-The '''354 equal divisions of the octave''' ('''354edo'''), or the '''354(-tone) equal temperament''' ('''354tet''', '''354et''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 354 parts of about 3.39 [[cent]]s each.
+{{EDO intro|354}}
 == Theory ==
-edo is [[enfactoring|enfactored]] in the 5-limit, with the same tuning as [[118edo]], defined by tempering out the [[schisma]] and the [[parakleisma]]. In the 7-limit, it tempers out 118098/117649 (stearnsma), 250047/250000 ([[landscape comma|landscape]]), and 703125/702464 ([[meter comma|meter]]); in the 11-limit, [[540/539]], and [[4000/3993]]; in the 13-limit, [[729/728]], [[1575/1573]], [[1716/1715]], [[2080/2079]], [[4096/4095]], and [[4225/4224]]. It provides the [[optimal patent val]] for [[stearnscape]].
+edo is [[enfactoring|enfactored]] in the 5-limit, with the same tuning as [[118edo]], defined by tempering out the [[schisma]] and the [[parakleisma]]. In the 7-limit, it tempers out 118098/117649 (stearnsma), 250047/250000 ([[landscape comma|landscape]]), and 703125/702464 ([[meter]]); in the 11-limit, [[540/539]], and [[4000/3993]]; in the 13-limit, [[729/728]], [[1575/1573]], [[1716/1715]], [[2080/2079]], [[4096/4095]], and [[4225/4224]]. It provides the [[optimal patent val]] for [[stearnscape]].
 === Prime harmonics ===
-{{Primes in edo|354}}
+{{Harmonics in equal|354}}
+=== Subsets and supersets ===
+Since 354 factors into 2 × 3 × 59, 354edo has subset edos {{EDOs| 2, 3, 6, 59, 118, and 177 }}.
 == Regular temperament properties ==
 {| 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" | Optimal<br>8ve stretch (¢)
+! rowspan="2" | Optimal<br>8ve Stretch (¢)
-! colspan="2" | Tuning error
+! colspan="2" | Tuning Error
 |-
 ! [[TE error|Absolute]] (¢)
@@ Line 61: / Line 64: @@
 |+Table of rank-2 temperaments by generator
 ! Periods<br>per 8ve
-! Generator<br>(reduced)
+! Generator<br>(Reduced)
-! Cents<br>(reduced)
+! Cents<br>(Reduced)
-! Associated<br>ratio
+! Associated<br>Ratio
 ! Temperaments
 |-
@@ Line 75: / Line 78: @@
 | 147\354<br>(29\354)
 | 498.31<br>(98.31)
-| 4/3<br>(200/189)
+| 4/3<br>(18/17)
 | [[Term]] / terminator
 |-
@@ Line 87: / Line 90: @@
 | 147\354<br>(29\354)
 | 498.31<br>(98.31)
-| 4/3<br>(200/189)
+| 4/3<br>(18/17)
 | [[Semiterm]]
 |-
-|118
+| 118
-|167\354<br>(2\354)
+| 167\354<br>(2\354)
-|566.101<br>(6.78)
+| 566.101<br>(6.78)
-|165/119<br>(?)
+| 165/119<br>(?)
-|[[Oganesson]]
+| [[Oganesson]]
 |}
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->