449edo: Difference between revisions

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== Theory ==

~~449et tempers out 26873856/26796875 and~~ [[~~4375/4374~~]] in the 7-limit~~; 100663296~~/~~100656875~~, [[~~117440512~~/~~117406179~~]], ~~4302592~~/~~4296875~~, ~~825000/823543~~, ~~85937500~~/~~85766121, 160083~~/~~160000,~~ [[41503/41472]], 539055/537824 ~~and~~ 805255/802816 in the 11-limit.

449edo is [[consistent]] to the [[7-odd-limit]], but the errors of [[harmonic]]s [[3/1|3]], [[5/1|5]], and [[7/1|7]] are all quite large, giving us the option of treating it as a full 7-limit temperament, or a 2.9.15.21.11.13 [[subgroup]] temperament.

===Odd harmonics===

Using the [[patent val]], the equal temperament [[tempering out|tempers out]] [[4375/4374]] and 26873856/26796875 in the 7-limit; [[41503/41472]], 160083/160000, 539055/537824, 805255/802816, and 825000/823543 in the 11-limit.

=== Odd harmonics ===

===Subsets and supersets===

=== Subsets and supersets ===

449edo is the 87th [[prime edo]]. 898edo, which doubles it, gives a good correction to the harmonic 3, 5 and 7.

==Regular temperament properties==

== Regular temperament properties ==

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

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

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

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

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

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

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

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

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

! colspan="2" |Tuning Error

! colspan="2" | Tuning Error

|-

![[TE error|Absolute]] (¢)

! [[TE error|Absolute]] (¢)

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

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

|-

|2.9

| 2.9

|{{monzo|-1423 449}}

| {{monzo| -1423 449 }}

|{{~~val~~|449 1423}}

| {{mapping| 449 1423 }}

| 0.1249

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|+Table of rank-2 temperaments by generator

! Periods per 8ve

! Generator~~ (reduced)~~

! Generator*

! Cents~~ (reduced)~~

! Cents*

! Associated ~~ratio~~

! Associated Ratio*

! Temperaments

|-

|1

| 1

|127\449

| 127\449

|339.421

| 339.421

|243\200

| 243\200

|[[Amity]] (7-limit)

| [[Amity]] (7-limit)

|}

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

== Music ==

*[https://www.youtube.com/watch?v=vLZuStkREoE Little Victorious Dance] ~~by [[User:Francium|Francium]]~~

; [[User:Francium|Francium]]

* [https://www.youtube.com/watch?v=vLZuStkREoE ''Little Victorious Dance''] (2023)

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
 {{EDO intro|449}}
 == Theory ==
-et tempers out 26873856/26796875 and [[4375/4374]] in the 7-limit; 100663296/100656875, [[117440512/117406179]], 4302592/4296875, 825000/823543, 85937500/85766121, 160083/160000, [[41503/41472]], 539055/537824 and 805255/802816 in the 11-limit.
+edo is [[consistent]] to the [[7-odd-limit]], but the errors of [[harmonic]]s [[3/1|3]], [[5/1|5]], and [[7/1|7]] are all quite large, giving us the option of treating it as a full 7-limit temperament, or a 2.9.15.21.11.13 [[subgroup]] temperament.
-===Odd harmonics===
+Using the [[patent val]], the equal temperament [[tempering out|tempers out]] [[4375/4374]] and 26873856/26796875 in the 7-limit; [[41503/41472]], 160083/160000, 539055/537824, 805255/802816, and 825000/823543 in the 11-limit.
+=== Odd harmonics ===
 {{Harmonics in equal|449}}
-===Subsets and supersets===
+=== Subsets and supersets ===
 edo is the 87th [[prime edo]]. 898edo, which doubles it, gives a good correction to the harmonic 3, 5 and 7.
-==Regular temperament properties==
+== Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
-! rowspan="2" |[[Subgroup]]
+! rowspan="2" | [[Subgroup]]
-! rowspan="2" |[[Comma list|Comma List]]
+! rowspan="2" | [[Comma list|Comma List]]
-! rowspan="2" |[[Mapping]]
+! rowspan="2" | [[Mapping]]
-! rowspan="2" |Optimal<br>8ve Stretch (¢)
+! rowspan="2" | Optimal<br>8ve Stretch (¢)
-! colspan="2" |Tuning Error
+! colspan="2" | Tuning Error
 |-
-![[TE error|Absolute]] (¢)
+! [[TE error|Absolute]] (¢)
-![[TE simple badness|Relative]] (%)
+! [[TE simple badness|Relative]] (%)
 |-
-|2.9
+| 2.9
-|{{monzo|-1423 449}}
+| {{monzo| -1423 449 }}
-|{{val|449 1423}}
+| {{mapping| 449 1423 }}
 | 0.1249
 | 0.1249
@@ Line 30: / Line 36: @@
 |+Table of rank-2 temperaments by generator
 ! Periods<br>per 8ve
-! Generator<br>(reduced)
+! Generator*
-! Cents<br>(reduced)
+! Cents*
-! Associated<br>ratio
+! Associated<br>Ratio*
 ! Temperaments
 |-
-|1
+| 1
-|127\449
+| 127\449
-|339.421
+| 339.421
-|243\200
+| 243\200
-|[[Amity]] (7-limit)
+| [[Amity]] (7-limit)
 |}
+<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
 == Music ==
-*[https://www.youtube.com/watch?v=vLZuStkREoE Little Victorious Dance] by [[User:Francium|Francium]]
+; [[User:Francium|Francium]]
+* [https://www.youtube.com/watch?v=vLZuStkREoE ''Little Victorious Dance''] (2023)