416edo: Difference between revisions

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== Theory ==

416et is consistent to the [[7-odd-limit]] ~~and~~ the [[harmonic]] [[3/1|3]] is ~~about halfway its steps~~. ~~It is~~ suitable for ~~the~~ 2.9.5.7.11.19.23~~.29.31.37~~ [[subgroup]], tempering out [[1540/1539]], [[5632/5625]], [[9801/9800]], 10241/10240, ~~1045~~/~~1044~~, ~~26125~~/~~26082~~, ~~46000~~/~~45927~~, ~~17600/17577 and 1036/1035. It~~ [[support]]s [[tridecatonic]], [[fermionic~~]], [[embankment]] and [[polder~~]].

416et is [[consistent]] to the [[7-odd-limit]], but the error of [[harmonic]] [[3/1|3]] is quite large. Nonetheless, it gives reasonable approximations to harmonics [[5/1|5]], [[7/1|7]], [[9/1|9]], [[11/1|11]], [[19/1|19]], and [[23/1|23]], making it suitable for a 2.9.5.7.11.19.23 [[subgroup]] interpretation, where it notably [[tempering out|tempers out]] [[1331/1330]], [[1540/1539]], [[5632/5625]], [[9801/9800]], and 10241/10240.

Using the patent val regardless, it tempers out 321489/320000, 589824/588245, and 703125/702464, [[support]]ing [[tridecatonic]] and [[fermionic]].

=== Odd harmonics ===

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=== Subsets and supersets ===

416 factors into 25 × 13, ~~with~~ subset edos {{EDOs|2, 4, 8, 13, 16, 26, 32, 52, 104, and 208}}. [[832edo]], which doubles it, gives a good correction to the harmonic 3.

Since 416 factors into 25 × 13, 416edo subset edos {{EDOs| 2, 4, 8, 13, 16, 26, 32, 52, 104, and 208 }}. [[832edo]], which doubles it, gives a good correction to the harmonic 3.

== Regular temperament properties ==

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

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

|-

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

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

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

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

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

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

! colspan="2" |Tuning Error

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

! colspan="2" | Tuning Error

|-

![[TE error|Absolute]] (¢)

! [[TE error|Absolute]] (¢)

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

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

|-

|2.9

| 2.9

|{{monzo|1319 -416}}

| {{monzo| 1319 -416 }}

|{{mapping|416 1319}}

| {{mapping| 416 1319 }}

| -0.1416

| −0.1416

| 0.1416

| 4.91

|-

|2.9.5

| 2.9.5

|{{monzo|56 -14 -5}}, {{monzo|-5 -16 24}}

| {{monzo| 56 -14 -5 }}, {{monzo| -5 -16 24 }}

|{{mapping|416 1319 966}}

| {{mapping| 416 1319 966 }}

| -0.1267

| −0.1267

| 0.1175

| 4.07

|-

|2.9.5.7

| 2.9.5.7

|420175/419904, 102760448/102515625, ~~1280000000/1275989841~~

| 420175/419904, 102760448/102515625, {{monzo| 14 -6 7 -4 }}

|{{mapping|416 1319 966 1168}}

| {{mapping| 416 1319 966 1168 }}

| -0.1310

| −0.1310

| 0.1021

| 3.54

|-

|2.9.5.7.11

| 2.9.5.7.11

|5632/5625, 9801/9800, 41503/41472, 774400000/771895089

| 5632/5625, 9801/9800, 41503/41472, 774400000/771895089

|{{mapping|416 1319 966 1168 1439}}

| {{mapping| 416 1319 966 1168 1439 }}

| -0.0842

| −0.0842

| 0.1308

| 4.53

|}

== Music ==

; [[birdshite stalactite]]

* [https://www.youtube.com/watch?v=nmUCjbfGpKU ''palate cleanser''] (2024) – tetracot[13] in 416edo tuning

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|416}}
+{{ED intro}}
 == Theory ==
-et is consistent to the [[7-odd-limit]] and the [[harmonic]] [[3/1|3]] is about halfway its steps. It is suitable for the 2.9.5.7.11.19.23.29.31.37 [[subgroup]], tempering out [[1540/1539]], [[5632/5625]], [[9801/9800]], 10241/10240, 1045/1044, 26125/26082, 46000/45927, 17600/17577 and 1036/1035. It [[support]]s [[tridecatonic]], [[fermionic]], [[embankment]] and [[polder]].
+et is [[consistent]] to the [[7-odd-limit]], but the error of [[harmonic]] [[3/1|3]] is quite large. Nonetheless, it gives reasonable approximations to harmonics [[5/1|5]], [[7/1|7]], [[9/1|9]], [[11/1|11]], [[19/1|19]], and [[23/1|23]], making it suitable for a 2.9.5.7.11.19.23 [[subgroup]] interpretation, where it notably [[tempering out|tempers out]] [[1331/1330]], [[1540/1539]], [[5632/5625]], [[9801/9800]], and 10241/10240.
+Using the patent val regardless, it tempers out 321489/320000, 589824/588245, and 703125/702464, [[support]]ing [[tridecatonic]] and [[fermionic]].
 === Odd harmonics ===
@@ Line 9: / Line 11: @@
 === Subsets and supersets ===
-factors into 2<sup>5</sup> × 13, with subset edos {{EDOs|2, 4, 8, 13, 16, 26, 32, 52, 104, and 208}}. [[832edo]], which doubles it, gives a good correction to the harmonic 3.
+Since 416 factors into 2<sup>5</sup> × 13, 416edo subset edos {{EDOs| 2, 4, 8, 13, 16, 26, 32, 52, 104, and 208 }}. [[832edo]], which doubles it, gives a good correction to the harmonic 3.
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
-! rowspan="2" |[[Subgroup]]
+|-
-! rowspan="2" |[[Comma list|Comma List]]
+! rowspan="2" | [[Subgroup]]
-! rowspan="2" |[[Mapping]]
+! rowspan="2" | [[Comma list|Comma List]]
-! rowspan="2" |Optimal<br>8ve Stretch (¢)
+! rowspan="2" | [[Mapping]]
-! colspan="2" |Tuning Error
+! rowspan="2" | Optimal<br>8ve Stretch (¢)
+! colspan="2" | Tuning Error
 |-
-![[TE error|Absolute]] (¢)
+! [[TE error|Absolute]] (¢)
-![[TE simple badness|Relative]] (%)
+! [[TE simple badness|Relative]] (%)
 |-
-|2.9
+| 2.9
-|{{monzo|1319 -416}}
+| {{monzo| 1319 -416 }}
-|{{mapping|416 1319}}
+| {{mapping| 416 1319 }}
-| -0.1416
+| −0.1416
 | 0.1416
 | 4.91
 |-
-|2.9.5
+| 2.9.5
-|{{monzo|56 -14 -5}}, {{monzo|-5 -16 24}}
+| {{monzo| 56 -14 -5 }}, {{monzo| -5 -16 24 }}
-|{{mapping|416 1319 966}}
+| {{mapping| 416 1319 966 }}
-| -0.1267
+| −0.1267
 | 0.1175
 | 4.07
 |-
-|2.9.5.7
+| 2.9.5.7
-|420175/419904, 102760448/102515625, 1280000000/1275989841
+| 420175/419904, 102760448/102515625, {{monzo| 14 -6 7 -4 }}
-|{{mapping|416 1319 966 1168}}
+| {{mapping| 416 1319 966 1168 }}
-| -0.1310
+| −0.1310
 | 0.1021
 | 3.54
 |-
-|2.9.5.7.11
+| 2.9.5.7.11
-|5632/5625, 9801/9800, 41503/41472, 774400000/771895089
+| 5632/5625, 9801/9800, 41503/41472, 774400000/771895089
-|{{mapping|416 1319 966 1168 1439}}
+| {{mapping| 416 1319 966 1168 1439 }}
-| -0.0842
+| −0.0842
 | 0.1308
 | 4.53
 |}
+== Music ==
+; [[birdshite stalactite]]
+* [https://www.youtube.com/watch?v=nmUCjbfGpKU ''palate cleanser''] (2024) – tetracot[13] in 416edo tuning