419edo: Difference between revisions

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== 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 [[~~sextilififths~~]], the 130 & 289 temperament, in the 7-limit.

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 ~~sextilififths~~, though [[289edo]] is better suited for that purpose.

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.

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== Regular temperament properties ==

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

|-

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

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

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

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

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

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

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

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

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| 441/440, 4000/3993, 32805/32768, 420175/419904

| {{mapping| 419 664 973 1176 1450 }} (419)

| -0.0168

| −0.0168

| 0.2460

| 8.59

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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

|-

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| 83.05

| 21/20

| [[~~Sextilififths~~]] (419f)

| [[Sextilifourths]] (419f)

|-

| 1

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| 498.33

| 162/125

| [[Helmholtz]]

| [[Helmholtz (temperament)|Helmholtz]]

|}

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

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

== Scales ==

* [[~~Sextilififths13~~]]

* [[Sextilifourths13]]

== Music ==

; [[~~User:Francium|~~Francium]]

; [[Francium]]

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

[[Category:~~Sextilififths~~]]

[[Category:Listen]]

[[Category:Sextilifourths]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|419}}
+{{ED intro}}
 == Theory ==
-edo 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 [[sextilififths]], the 130 & 289 temperament, in the 7-limit.
+edo 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 &amp; 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 sextilififths, though [[289edo]] is better suited for that purpose.
+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.
@@ Line 17: / Line 17: @@
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
-! rowspan="2" | [[Comma list|Comma List]]
+! rowspan="2" | [[Comma list]]
 ! 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]] (¢)
@@ Line 50: / Line 51: @@
 | 441/440, 4000/3993, 32805/32768, 420175/419904
 | {{mapping| 419 664 973 1176 1450 }} (419)
-| -0.0168
+| −0.0168
 | 0.2460
 | 8.59
@@ Line 57: / Line 58: @@
 === 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
 |-
@@ Line 68: / Line 70: @@
 | 83.05
 | 21/20
-| [[Sextilififths]] (419f)
+| [[Sextilifourths]] (419f)
 |-
 | 1
@@ Line 80: / Line 82: @@
 | 498.33
 | 162/125
-| [[Helmholtz]]
+| [[Helmholtz (temperament)|Helmholtz]]
 |}
-<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 == Scales ==
-* [[Sextilififths13]]
+* [[Sextilifourths13]]
 == Music ==
-; [[User:Francium|Francium]]
+; [[Francium]]
 * [https://www.youtube.com/watch?v=cXvlQxvwUIM ''Cultural Appropriation?''] (2023)
-[[Category:Sextilififths]]
+[[Category:Listen]]
+[[Category:Sextilifourths]]