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 {{nowrap|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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| 83.05

| 21/20

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

| [[Sextilifourths]] (419f)

|-

| 1

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| 498.33

| 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 ==

* [[~~Sextilififths13~~]]

* [[Sextilifourths13]]

== Music ==

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[[Category:Listen]]

[[Category:~~Sextilififths~~]]

[[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 {{nowrap|130 &amp; 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 70: / Line 70: @@
 | 83.05
 | 21/20
-| [[Sextilififths]] (419f)
+| [[Sextilifourths]] (419f)
 |-
 | 1
@@ Line 82: / Line 82: @@
 | 498.33
 | 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 ==
-* [[Sextilififths13]]
+* [[Sextilifourths13]]
 == Music ==
@@ Line 94: / Line 94: @@
 [[Category:Listen]]
-[[Category:Sextilififths]]
+[[Category:Sextilifourths]]