354edo: Difference between revisions

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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 [[enfactored]] in the 5-limit, with the same tuning as [[118edo]], defined by [[tempering out]] the [[schisma]] and the [[parakleisma]], but the approximation to higher [[harmonic]]s are much improved.

In the 7-limit, it tempers out 118098/117649 (stearnsma), 250047/250000 ([[landscape comma]]), 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]]. In the 13-limit, particularly 2.3.5.13 subgroup, one should consider [[peithoian]], as it preserves 5-limit tuning of 118edo while also improving the first harmonic 118edo tunes inconsistently.

354edo provides the [[optimal patent val]] for [[stearnscape]], the {{nowrap|72 & 282}} temperament, and 13- and 17-limit [[terminator]], the {{nowrap|171 & 183}} temperament.

=== Prime harmonics ===

=== Subsets and supersets ===

Since 354 factors into {{factorization|354}}, 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" | [[Mapping]]

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

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

! colspan="2" | Tuning error

|-

Line 21:

Line 29:

| 2.3.5.7

| 32805/32768, 118098/117649, 250047/250000

| [{{~~val~~| 354 561 822 994 }}]

| {{mapping| 354 561 822 994 }}

| -0.0319

| −0.0319

| 0.1432

| 4.23

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| 2.3.5.7.11

| 540/539, 4000/3993, 32805/32768, 137781/137500

| [{{~~val~~| 354 561 822 994 1225 }}]

| {{mapping| 354 561 822 994 1225 }}

| -0.0963

| −0.0963

| 0.1817

| 5.36

Line 35:

Line 43:

| 2.3.5.7.11.13

| 540/539, 729/728, 1575/1573, 4096/4095, 31250/31213

| [{{~~val~~| 354 561 822 994 1225 1310 }}]

| {{mapping| 354 561 822 994 1225 1310 }}

| -0.0871

| −0.0871

| 0.1671

| 4.93

Line 42:

Line 50:

| 2.3.5.7.11.13.17

| 540/539, 729/728, 936/935, 1156/1155, 1575/1573, 4096/4095

| [{{~~val~~| 354 561 822 994 1225 1310 1447 }}]

| {{mapping| 354 561 822 994 1225 1310 1447 }}

| -0.0791

| −0.0791

| 0.1559

| 4.60

Line 49:

Line 57:

| 2.3.5.7.11.13.17.19

| 540/539, 729/728, 936/935, 969/968, 1156/1155, 1445/1444, 1521/1520

| [{{~~val~~| 354 561 822 994 1225 1310 1447 1504 }}]

| {{mapping| 354 561 822 994 1225 1310 1447 1504 }}

| -0.0926

| −0.0926

| 0.1509

| 4.43

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{| class="wikitable center-all left-5"

|+Table of rank-2 temperaments by generator

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

! Periods per ~~octave~~

|-

! Generator~~ (reduced)~~

! Periods per 8ve

! Cents~~ (reduced)~~

! Generator*

! Associated ratio

! Cents*

! Associated ratio*

! Temperaments

|-

| 2

| 128\354 (49\354)

| 128\354 (49\354)

| 433.90 (166.10)

| 433.90 (166.10)

| 9/7 (11/10)

| 9/7 (11/10)

| [[Pogo]]

|-

| 3

| 147\354 (29\354)

| 147\354 (29\354)

| 498.31 (98.31)

| 498.31 (98.31)

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

| 4/3 (18/17)

| [[Term]] / terminator

| [[Term (temperament)|Term]] / terminator

|-

| 6

| 64\354 (5\354)

| 64\354 (5\354)

| 216.95 (16.95)

| 216.95 (16.95)

| ~~567~~/~~500~~ (245/243)

| 17/15 (245/243)

| [[Stearnscape]]

|-

| 6

| 147\354 (29\354)

| 147\354 (29\354)

| 498.31 (98.31)

| 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]]

|}

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

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

[[Category:Stearnscape]]

@@ 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.
+{{ED intro}}
 == 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 [[enfactored]] in the 5-limit, with the same tuning as [[118edo]], defined by [[tempering out]] the [[schisma]] and the [[parakleisma]], but the approximation to higher [[harmonic]]s are much improved.
+In the 7-limit, it tempers out 118098/117649 (stearnsma), 250047/250000 ([[landscape comma]]), 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]]. In the 13-limit, particularly 2.3.5.13 subgroup, one should consider [[peithoian]], as it preserves 5-limit tuning of 118edo while also improving the first harmonic 118edo tunes inconsistently.
+edo provides the [[optimal patent val]] for [[stearnscape]], the {{nowrap|72 &amp; 282}} temperament, and 13- and 17-limit [[terminator]], the {{nowrap|171 &amp; 183}} temperament.
 === Prime harmonics ===
-{{Primes in edo|354}}
+{{Harmonics in equal|354}}
+=== Subsets and supersets ===
+Since 354 factors into {{factorization|354}}, 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" | [[Mapping]]
-! rowspan="2" | Optimal<br>8ve stretch (¢)
+! rowspan="2" | Optimal<br />8ve stretch (¢)
 ! colspan="2" | Tuning error
 |-
@@ Line 21: / Line 29: @@
 | 2.3.5.7
 | 32805/32768, 118098/117649, 250047/250000
-| [{{val| 354 561 822 994 }}]
+| {{mapping| 354 561 822 994 }}
-| -0.0319
+| −0.0319
 | 0.1432
 | 4.23
@@ Line 28: / Line 36: @@
 | 2.3.5.7.11
 | 540/539, 4000/3993, 32805/32768, 137781/137500
-| [{{val| 354 561 822 994 1225 }}]
+| {{mapping| 354 561 822 994 1225 }}
-| -0.0963
+| −0.0963
 | 0.1817
 | 5.36
@@ Line 35: / Line 43: @@
 | 2.3.5.7.11.13
 | 540/539, 729/728, 1575/1573, 4096/4095, 31250/31213
-| [{{val| 354 561 822 994 1225 1310 }}]
+| {{mapping| 354 561 822 994 1225 1310 }}
-| -0.0871
+| −0.0871
 | 0.1671
 | 4.93
@@ Line 42: / Line 50: @@
 | 2.3.5.7.11.13.17
 | 540/539, 729/728, 936/935, 1156/1155, 1575/1573, 4096/4095
-| [{{val| 354 561 822 994 1225 1310 1447 }}]
+| {{mapping| 354 561 822 994 1225 1310 1447 }}
-| -0.0791
+| −0.0791
 | 0.1559
 | 4.60
@@ Line 49: / Line 57: @@
 | 2.3.5.7.11.13.17.19
 | 540/539, 729/728, 936/935, 969/968, 1156/1155, 1445/1444, 1521/1520
-| [{{val| 354 561 822 994 1225 1310 1447 1504 }}]
+| {{mapping| 354 561 822 994 1225 1310 1447 1504 }}
-| -0.0926
+| −0.0926
 | 0.1509
 | 4.43
@@ Line 59: / Line 67: @@
 {| class="wikitable center-all left-5"
-|+Table of rank-2 temperaments by generator
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
-! Periods<br>per octave
+|-
-! Generator<br>(reduced)
+! Periods<br />per 8ve
-! Cents<br>(reduced)
+! Generator*
-! Associated<br>ratio
+! Cents*
+! Associated<br />ratio*
 ! Temperaments
 |-
 | 2
-| 128\354<br>(49\354)
+| 128\354<br />(49\354)
-| 433.90<br>(166.10)
+| 433.90<br />(166.10)
-| 9/7<br>(11/10)
+| 9/7<br />(11/10)
 | [[Pogo]]
 |-
 | 3
-| 147\354<br>(29\354)
+| 147\354<br />(29\354)
-| 498.31<br>(98.31)
+| 498.31<br />(98.31)
-| 4/3<br>(200/189)
+| 4/3<br />(18/17)
-| [[Term]] / terminator
+| [[Term (temperament)|Term]] / terminator
 |-
 | 6
-| 64\354<br>(5\354)
+| 64\354<br />(5\354)
-| 216.95<br>(16.95)
+| 216.95<br />(16.95)
-| 567/500<br>(245/243)
+| 17/15<br />(245/243)
 | [[Stearnscape]]
 |-
 | 6
-| 147\354<br>(29\354)
+| 147\354<br />(29\354)
-| 498.31<br>(98.31)
+| 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]]
 |}
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
+[[Category:Stearnscape]]