444edo: Difference between revisions

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== Theory ==

~~444et~~ is only consistent to the 5-limit. Using the patent val, it tempers out ~~67108864/66976875, 29360128/29296875 and~~ [[250047/250000]] in the 7-limit; ~~100663296/100656875, 2097152/2096325,~~ [[~~131072~~/~~130977~~]]~~, 172032/171875~~, [[5632/5625]], ~~47265625~~/~~47258883~~, [[~~3025~~/~~3024~~]], 160083/160000, ~~42592~~/~~42525~~, 391314/390625~~, 102487/102400, 322102/321489~~ and [[1771561/1769472]] in the 11-limit. It ~~provides the~~ [[~~optimal patent val~~]] ~~for~~ the [[magnesium]] temperament.

444edo is only [[consistent]] to the [[5-odd-limit]] since [[harmonic]] [[7/1|7]] is about halfway between its steps. Since {{nowrap|444 {{=}} 4 × 111}}, its harmonic [[3/1|3]] derives from [[111edo]]. Using the [[patent val]], the equal temperament [[tempering out|tempers out]] [[250047/250000]], 29360128/29296875, 67108864/66976875 and in the 7-limit; [[3025/3024]], [[5632/5625]], 42592/42525, 102487/102400, [[131072/130977]], 160083/160000, 172032/171875, 322102/321489, 391314/390625 and [[1771561/1769472]] in the 11-limit. It [[support]]s the [[magnesium]] temperament.

=== Odd harmonics ===

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=== Subsets and supersets ===

444 factors into ~~22 × 3 × 37~~, ~~with~~ subset edos {{EDOs|2, 3, 4, 6, 12, 37, 74, 111, 148, and 222}}. [[1332edo]], which triples it, gives a good correction to the harmonic 7.

Since 444 factors into {{factorization|444}}, 444edo has subset edos {{EDOs| 2, 3, 4, 6, 12, 37, 74, 111, 148, and 222 }}. [[1332edo]], which triples it, gives a good correction to the harmonic 7.

== 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~~|~~176 -111}}~~

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

|~~{{mapping|444 704}}~~

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

| ~~-0.2359~~

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

| ~~0.2358~~

! colspan="2" | Tuning error

| ~~8.72~~

|-

|2.3.5

! [[TE error|Absolute]] (¢)

|{{monzo|41 -20 -4}}, {{monzo|-29 -11 20}}

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

|{{mapping|444 704 1031}}

|-

| -0.1821

| 2.3.5

| {{monzo| 41 -20 -4 }}, {{monzo| -29 -11 20 }}

| {{mapping| 444 704 1031 }}

| −0.1821

| 0.2071

| 7.66

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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

|-

! Generator~~ (reduced)~~*

! Periods per 8ve

! Cents~~ (reduced)~~*

! Generator*

! Associated ~~Ratio~~*

! Cents*

! Associated ratio*

! Temperaments

|-

|1

| 1

|13\444

| 13\444

|35.14

| 35.14

|1990656/1953125

| 1990656/1953125

|[[Gammic]]

| [[Gammic]] (5-limit)

|-

|4

| 4

|184\444 (38\444)

| 184\444 (38\444)

|497.30 (102.70)

| 497.30 (102.70)

|4/3 (35/33)

| 4/3 (35/33)

|[[Undim]]

| [[Undim]] (444d)

|}

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

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

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|444}}
+{{ED intro}}
 == Theory ==
-et is only consistent to the 5-limit. Using the patent val, it tempers out 67108864/66976875, 29360128/29296875 and [[250047/250000]] in the 7-limit; 100663296/100656875, 2097152/2096325, [[131072/130977]], 172032/171875, [[5632/5625]], 47265625/47258883, [[3025/3024]], 160083/160000, 42592/42525, 391314/390625, 102487/102400, 322102/321489 and [[1771561/1769472]] in the 11-limit. It provides the [[optimal patent val]] for the [[magnesium]] temperament.
+edo is only [[consistent]] to the [[5-odd-limit]] since [[harmonic]] [[7/1|7]] is about halfway between its steps. Since {{nowrap|444 {{=}} 4 × 111}}, its harmonic [[3/1|3]] derives from [[111edo]]. Using the [[patent val]], the equal temperament [[tempering out|tempers out]] [[250047/250000]], 29360128/29296875, 67108864/66976875 and in the 7-limit; [[3025/3024]], [[5632/5625]], 42592/42525, 102487/102400, [[131072/130977]], 160083/160000, 172032/171875, 322102/321489, 391314/390625 and [[1771561/1769472]] in the 11-limit. It [[support]]s the [[magnesium]] temperament.
 === Odd harmonics ===
@@ Line 9: / Line 9: @@
 === Subsets and supersets ===
-factors into 2<sub>2</sub> × 3 × 37, with subset edos {{EDOs|2, 3, 4, 6, 12, 37, 74, 111, 148, and 222}}. [[1332edo]], which triples it, gives a good correction to the harmonic 7.
+Since 444 factors into {{factorization|444}}, 444edo has subset edos {{EDOs| 2, 3, 4, 6, 12, 37, 74, 111, 148, and 222 }}. [[1332edo]], which triples it, gives a good correction to the harmonic 7.
 == 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|176 -111}}
+! rowspan="2" | [[Comma list]]
-|{{mapping|444 704}}
+! rowspan="2" | [[Mapping]]
-| -0.2359
+! rowspan="2" | Optimal<br />8ve stretch (¢)
-| 0.2358
+! colspan="2" | Tuning error
-| 8.72
 |-
-|2.3.5
+! [[TE error|Absolute]] (¢)
-|{{monzo|41 -20 -4}}, {{monzo|-29 -11 20}}
+! [[TE simple badness|Relative]] (%)
-|{{mapping|444 704 1031}}
+|-
-| -0.1821
+| 2.3.5
+| {{monzo| 41 -20 -4 }}, {{monzo| -29 -11 20 }}
+| {{mapping| 444 704 1031 }}
+| −0.1821
 | 0.2071
 | 7.66
@@ Line 39: / Line 33: @@
 === 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
+|-
-! Generator<br>(reduced)*
+! Periods<br />per 8ve
-! Cents<br>(reduced)*
+! Generator*
-! Associated<br>Ratio*
+! Cents*
+! Associated<br />ratio*
 ! Temperaments
 |-
-|1
+| 1
-|13\444
+| 13\444
-|35.14
+| 35.14
-|1990656/1953125
+| 1990656/1953125
-|[[Gammic]]
+| [[Gammic]] (5-limit)
 |-
-|4
+| 4
-|184\444<br>(38\444)
+| 184\444<br>(38\444)
-|497.30<br>(102.70)
+| 497.30<br>(102.70)
-|4/3<br>(35/33)
+| 4/3<br>(35/33)
-|[[Undim]]
+| [[Undim]] (444d)
 |}
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
-<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct