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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|419}}.
{{ED intro}}
==Theory==
 
419et tempers out [[32805/32768]], 29360128/29296875, 1959552/1953125, 420175/419904 and [[2100875/2097152]] in the 7-limit and 35156250/35153041, 25165824/25109315, 26214400/26198073, [[4000/3993]], 1366875/1362944, 759375/758912, 21437500/21434787, 2359296/2358125, 472392/471625, 369140625/369098752, [[200704/200475]], [[441/440]], 43046721/43025920, 17537553/17500000, 422576/421875, 160083/160000, 199297406/199290375, 244515348/244140625 and 67110351/67108864 in the 11-limit. It supports the [[sextilififths]] temperament.
== Theory ==
419edo is a decent 7-limit system, and is [[consistent]] to the [[9-odd-limit]]. The equal temperament [[tempering out|tempers out]] 32805/32768 ([[schisma]]) in the 5-limit; 235298/234375 (triwellisma), 420175/419904 (wizma) in the 7-limit. It [[support]]s and provides the [[optimal patent val]] for [[sextilifourths]], the {{nowrap|130 & 289}} temperament, in the 7-limit.
 
Extending it to the 11-limit requires choosing which mapping one wants to use, as both are nearly equally far off the mark. Using the 419e [[val]], it tempers out [[3025/3024]], [[5632/5625]], and [[8019/8000]]. Using the [[patent val]], it tempers out [[441/440]], [[4000/3993]], and 14700/14641 in the 11-limit. The patent val supports 11-limit sextilifourths, though [[289edo]] is better suited for that purpose.
 
The same can be said of the mapping for 13, with the 419e val tempering out [[676/675]], [[1716/1715]], [[4225/4224]], and 4459/4455, and the 419f val tempering out [[729/728]], [[2200/2197]], 2205/2197, 3584/3575, and 4459/4455.
 
=== Odd harmonics ===
{{Harmonics in equal|419}}
 
=== Subsets and supersets ===
419edo is the 81th [[prime edo]].
419edo is the 81th [[prime edo]].
{{Harmonics in equal|419}}
 
==Regular temperament properties==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| 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]] (¢)
! rowspan="2" | [[Subgroup]]
![[TE simple badness|Relative]] (%)
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
|-
|2.3
! [[TE error|Absolute]] (¢)
|{{monzo| -664 419}}
! [[TE simple badness|Relative]] (%)
|{{val| 419​ 664​}}
|-
| 2.3
| {{monzo| -664 419 }}
| {{mapping| 419​ 664​ }}
| +0.0897
| +0.0897
| 0.0897
| 0.0897
| 3.13
| 3.13
|-
|-
|2.3.5
| 2.3.5
|32805/32768, {{monzo| 41 43 -47}}
| 32805/32768, {{monzo| 41 43 -47 }}
|{{val| 419 ​664​ 973}}
| {{mapping| 419 ​664​ 973 }}
| +0.0137
| +0.0137
| 0.1301
| 0.1301
| 4.54
| 4.54
|-
|-
|2.3.5.7
| 2.3.5.7
|32805/32768, 420175/419904, 1959552/1953125
| 32805/32768, 235298/234375, 420175/419904
|{{val| 419 ​664​ 973 ​1176​}}
| {{mapping| 419 ​664​ 973 ​1176 ​}}
| +0.0821
| +0.0821
| 0.1635
| 0.1635
| 5.71
| 5.71
|-
|-
|2.3.5.7.11
| 2.3.5.7.11
|441/440, 200704/200475, 160083/160000, 472392/471625
| 441/440, 4000/3993, 32805/32768, 420175/419904
|{{val| 419​ 664 ​973 ​1176​ 1450}}
| {{mapping| 419​ 664 ​973 ​1176​ 1450 }} (419)
| -0.0168
| −0.0168
| 0.2460
| 0.2460
| 8.59
| 8.59
|-
|2.3.5.7.11.13
|441/440, 2080/2079, 1001/1000, 78975/78848, 472392/471625
|{{val| 419​ 664 ​973 ​1176​ 1450 1550}}
| +0.0484
| 0.2678
| 9,35
|-
|}
|}


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| 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
! Temperaments
|-
|-
|1
| 1
|29\419
| 29\419
|83.05
| 83.05
|21/20
| 21/20
|[[Sextilififths]]
| [[Sextilifourths]] (419f)
|-
|-
|1
| 1
|49\419
| 49\419
|140.33
| 140.33
|243/224
| 243/224
|[[Tsaharuk]]
| [[Tsaharuk]] (7-limit)
|-
|-
|1
| 1
|174\419
| 174\419
|498.33
| 498.33
|162/125
| 162/125
|[[Helmholtz]]
| [[Helmholtz (temperament)|Helmholtz]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
== Scales ==
* [[Sextilifourths13]]


==Scales==
== Music ==
*[[Sextilififths13]]
; [[Francium]]
* [https://www.youtube.com/watch?v=cXvlQxvwUIM ''Cultural Appropriation?''] (2023)


==Music==
[[Category:Listen]]
* [https://www.youtube.com/watch?v=cXvlQxvwUIM Cultural Appropriation?] by Francium
[[Category:Sextilifourths]]
Retrieved from "https://en.xen.wiki/w/419edo"