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 ===

Line 30:

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

| {{mapping| 354 561 822 994 }}

| ~~−0~~.0319

| −0.0319

| 0.1432

| 4.23

Line 37:

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

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

| ~~−0~~.0963

| −0.0963

| 0.1817

| 5.36

Line 44:

| 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

Line 51:

| 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

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 }}

| ~~−0~~.0926

| −0.0926

| 0.1509

| 4.43

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| [[Oganesson]]

|}

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

<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 30: / Line 30: @@
 | 32805/32768, 118098/117649, 250047/250000
 | {{mapping| 354 561 822 994 }}
-| &minus;0.0319
+| −0.0319
 | 0.1432
 | 4.23
@@ Line 37: / 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 44: / 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 51: / 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 58: / 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
@@ Line 105: / Line 105: @@
 | [[Oganesson]]
 |}
-<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 [[Category:Stearnscape]]