354edo: Difference between revisions

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== Theory ==

354edo is [[~~enfactoring|~~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.

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, ~~the equal temperament~~ 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.

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 72 & 282 temperament, and 13- and 17-limit [[terminator]], the 171 & 183 temperament.

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 ===

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== Regular temperament properties ==

{~~{comma basis begin}}~~

{| class="wikitable center-4 center-5 center-6"

|-

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal<br />8ve stretch (¢)

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

! [[TE simple badness|Relative]] (%)

|-

| 2.3.5.7

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

| {{mapping| 354 561 822 994 }}

| ~~−0~~.0319

| −0.0319

| 0.1432

| 4.23

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

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

| ~~−0~~.0963

| −0.0963

| 0.1817

| 5.36

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

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

| ~~−0~~.0871

| −0.0871

| 0.1671

| 4.93

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

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

| ~~−0~~.0791

| −0.0791

| 0.1559

| 4.60

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

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

| ~~−0~~.0926

| −0.0926

| 0.1509

| 4.43

~~{{comma basis end}~~}

|}

=== Rank-2 temperaments ===

Note: 5-limit temperaments supported by [[118edo|118et]] are not included.

{{rank-2 ~~begin}}~~

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

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

|-

! Periods<br />per 8ve

! Generator*

! Cents*

! Associated<br />ratio*

! Temperaments

|-

| 2

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| 165/119<br />(?)

| [[Oganesson]]

~~{{rank-2 end}~~}

|}

~~{{orf}}~~

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

[[Category:Stearnscape]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|354}}
+{{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]], but the approximation to higher [[harmonic]]s are much improved.
+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, the equal temperament 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.
+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 72 & 282 temperament, and 13- and 17-limit [[terminator]], the 171 & 183 temperament.
+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 ===
@@ Line 16: / Line 16: @@
 == Regular temperament properties ==
-{{comma basis begin}}
+{| class="wikitable center-4 center-5 center-6"
+|-
+! rowspan="2" | [[Subgroup]]
+! rowspan="2" | [[Comma list]]
+! rowspan="2" | [[Mapping]]
+! rowspan="2" | Optimal<br />8ve stretch (¢)
+! colspan="2" | Tuning error
+|-
+! [[TE error|Absolute]] (¢)
+! [[TE simple badness|Relative]] (%)
 |-
 | 2.3.5.7
 | 32805/32768, 118098/117649, 250047/250000
 | {{mapping| 354 561 822 994 }}
-| &minus;0.0319
+| −0.0319
 | 0.1432
 | 4.23
@@ Line 28: / Line 37: @@
 | 540/539, 4000/3993, 32805/32768, 137781/137500
 | {{mapping| 354 561 822 994 1225 }}
-| &minus;0.0963
+| −0.0963
 | 0.1817
 | 5.36
@@ Line 35: / Line 44: @@
 | 540/539, 729/728, 1575/1573, 4096/4095, 31250/31213
 | {{mapping| 354 561 822 994 1225 1310 }}
-| &minus;0.0871
+| −0.0871
 | 0.1671
 | 4.93
@@ Line 42: / Line 51: @@
 | 540/539, 729/728, 936/935, 1156/1155, 1575/1573, 4096/4095
 | {{mapping| 354 561 822 994 1225 1310 1447 }}
-| &minus;0.0791
+| −0.0791
 | 0.1559
 | 4.60
@@ Line 49: / Line 58: @@
 | 540/539, 729/728, 936/935, 969/968, 1156/1155, 1445/1444, 1521/1520
 | {{mapping| 354 561 822 994 1225 1310 1447 1504 }}
-| &minus;0.0926
+| −0.0926
 | 0.1509
 | 4.43
-{{comma basis end}}
+|}
 === Rank-2 temperaments ===
 Note: 5-limit temperaments supported by [[118edo|118et]] are not included.
-{{rank-2 begin}}
+{| class="wikitable center-all left-5"
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
+! Periods<br />per 8ve
+! Generator*
+! Cents*
+! Associated<br />ratio*
+! Temperaments
 |-
 | 2
@@ Line 88: / Line 104: @@
 | 165/119<br />(?)
 | [[Oganesson]]
-{{rank-2 end}}
+|}
-{{orf}}
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 [[Category:Stearnscape]]