362edo: Difference between revisions

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== Theory ==

~~362et~~ is only consistent to the [[5-odd-limit]], with ~~three~~ mappings possible for the [[7-limit]]:

362edo is [[enfactoring|enfactored]] in the [[3-limit]] and is only [[consistent]] to the [[5-odd-limit]], with two mappings possible for the [[7-limit]]:

* {{val|362 574 841 1016}} (patent val),

* {{val| 362 574 841 1016 }} (patent val),

* {{val|362 574 '''840''' 1016}} (362c),

* {{val| 362 574 841 '''1017''' }} (362d).

* {{val|362 574 841 '''1017'''}} (362d).

Using the patent val, it tempers out [[393216/390625]] and {{monzo|25 -48 22}} in the 5-limit; [[4375/4374]], [[458752/455625]] and 11529602/11390625 in the 7-limit, [[support]]ing [[barbados]].

Using the patent val, it tempers out [[393216/390625]] and {{monzo| 25 -48 22 }} in the 5-limit; [[4375/4374]], [[458752/455625]] and 11529602/11390625 in the 7-limit, [[support]]ing [[barbados]].

~~Using the 362c val, it tempers out [[2109375/2097152]] and {{monzo|14 -22 9}} in the 5-limit; [[2401/2400]], [[10976/10935]] and 390625/387072 in the 7-limit.~~

Using the 362d val, it tempers out 393216/390625 and {{monzo| 25 -48 22 }} in the 5-limit; [[5120/5103]], 118098/117649 and 1959552/1953125 in the 7-limit.

Using the 362d val, it tempers out 393216/390625 and {{monzo|25 -48 22}} in the 5-limit; [[5120/5103]], 118098/117649 and 1959552/1953125 in the 7-limit.

=== Odd harmonics ===

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=== Subsets and supersets ===

362 factors into 2 × 181, ~~with~~ [[2edo]] and [[181edo]] as its ~~subset edos~~. [[1448edo]], which quadruples it, ~~gives~~ a ~~good correction to the harmonic 7~~.

Since 362 factors into 2 × 181, 372edo has [[2edo]] and [[181edo]] as its subsets. [[1448edo]], which quadruples it, is a strong full 13-limit system.

== Regular temperament properties ==

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

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

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

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

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

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

|-

~~![[TE error|Absolute]] (¢)~~

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

|-

|2.3

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

~~|{{monzo~~|~~287 -181}}~~

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

|~~{{mapping|362 574}}~~

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

| ~~-0.2547~~

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

| ~~0.2547~~

! colspan="2" | Tuning error

| ~~7.68~~

|-

|2.3.5

! [[TE error|Absolute]] (¢)

|393216/390625, {{monzo|25 -48 22}}

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

|{{mapping|362 574 841}}

|-

| -0.3896

| 2.3.5

| 393216/390625, {{monzo| 25 -48 22 }}

| {{mapping| 362 574 841 }}

| −0.3896

| 0.2822

| 8.51

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=== Rank-2 temperaments ===

{| 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 8ve

|-

! Periods per 8ve

! Generator*

! Cents*

! Associated ~~Ratio~~*

! Associated ratio*

! Temperaments

|-

|1

| 1

|117\362

| 117\362

|387.85

| 387.85

|5/4

| 5/4

|[[Würschmidt]]

| [[Würschmidt]]

|}

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

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|362}}
+{{ED intro}}
 == Theory ==
-et is only consistent to the [[5-odd-limit]], with three mappings possible for the [[7-limit]]:
+edo is [[enfactoring|enfactored]] in the [[3-limit]] and is only [[consistent]] to the [[5-odd-limit]], with two mappings possible for the [[7-limit]]:
-* {{val|362 574 841 1016}} (patent val),
+* {{val| 362 574 841 1016 }} (patent val),
-* {{val|362 574 '''840''' 1016}} (362c),
+* {{val| 362 574 841 '''1017''' }} (362d).
-* {{val|362 574 841 '''1017'''}} (362d).
-Using the patent val, it tempers out [[393216/390625]] and {{monzo|25 -48 22}} in the 5-limit; [[4375/4374]], [[458752/455625]] and 11529602/11390625 in the 7-limit, [[support]]ing [[barbados]].
+Using the patent val, it tempers out [[393216/390625]] and {{monzo| 25 -48 22 }} in the 5-limit; [[4375/4374]], [[458752/455625]] and 11529602/11390625 in the 7-limit, [[support]]ing [[barbados]].
-Using the 362c val, it tempers out [[2109375/2097152]] and {{monzo|14 -22 9}} in the 5-limit; [[2401/2400]], [[10976/10935]] and 390625/387072 in the 7-limit.
+Using the 362d val, it tempers out 393216/390625 and {{monzo| 25 -48 22 }} in the 5-limit; [[5120/5103]], 118098/117649 and 1959552/1953125 in the 7-limit.
-Using the 362d val, it tempers out 393216/390625 and {{monzo|25 -48 22}} in the 5-limit; [[5120/5103]], 118098/117649 and 1959552/1953125 in the 7-limit.
 === Odd harmonics ===
@@ Line 18: / Line 15: @@
 === Subsets and supersets ===
-factors into 2 × 181, with [[2edo]] and [[181edo]] as its subset edos. [[1448edo]], which quadruples it, gives a good correction to the harmonic 7.
+Since 362 factors into 2 × 181, 372edo has [[2edo]] and [[181edo]] as its subsets. [[1448edo]], which quadruples it, is a strong full 13-limit system.
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
-! rowspan="2" |[[Subgroup]]
-! rowspan="2" |[[Comma list|Comma List]]
-! rowspan="2" |[[Mapping]]
-! rowspan="2" |Optimal<br>8ve Stretch (¢)
-! colspan="2" |Tuning Error
-|-
-![[TE error|Absolute]] (¢)
-![[TE simple badness|Relative]] (%)
 |-
-|2.3
+! rowspan="2" | [[Subgroup]]
-|{{monzo|287 -181}}
+! rowspan="2" | [[Comma list]]
-|{{mapping|362 574}}
+! rowspan="2" | [[Mapping]]
-| -0.2547
+! rowspan="2" | Optimal<br />8ve stretch (¢)
-| 0.2547
+! colspan="2" | Tuning error
-| 7.68
 |-
-|2.3.5
+! [[TE error|Absolute]] (¢)
-|393216/390625, {{monzo|25 -48 22}}
+! [[TE simple badness|Relative]] (%)
-|{{mapping|362 574 841}}
+|-
-| -0.3896
+| 2.3.5
+| 393216/390625, {{monzo| 25 -48 22 }}
+| {{mapping| 362 574 841 }}
+| −0.3896
 | 0.2822
 | 8.51
@@ Line 48: / Line 39: @@
 === Rank-2 temperaments ===
 {| 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 8ve
+|-
+! Periods<br />per 8ve
 ! Generator*
 ! Cents*
-! Associated<br>Ratio*
+! Associated<br />ratio*
 ! Temperaments
 |-
-|1
+| 1
-|117\362
+| 117\362
-|387.85
+| 387.85
-|5/4
+| 5/4
-|[[Würschmidt]]
+| [[Würschmidt]]
 |}
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct