419edo: Difference between revisions

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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
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
|-
|2.3
| 2.3
|{{monzo| -664 419}}
| {{monzo| -664 419 }}
|{{val| 419​ 664​}}
| {{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
|Subneutral
| [[Sextilifourths]] (419f)
|-
| 1
| 49\419
| 140.33
| 243/224
| [[Tsaharuk]] (7-limit)
|-
| 1
| 174\419
| 498.33
| 162/125
| [[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]]
== Music ==
; [[Francium]]
* [https://www.youtube.com/watch?v=cXvlQxvwUIM ''Cultural Appropriation?''] (2023)


==Scales==
[[Category:Listen]]
*[[Sextilififths13]]
[[Category:Sextilifourths]]

Latest revision as of 02:29, 17 April 2025

← 418edo 419edo 420edo →
Prime factorization 419 (prime)
Step size 2.86396 ¢ 
Fifth 245\419 (701.671 ¢)
Semitones (A1:m2) 39:32 (111.7 ¢ : 91.65 ¢)
Consistency limit 9
Distinct consistency limit 9

419 equal divisions of the octave (abbreviated 419edo or 419ed2), also called 419-tone equal temperament (419tet) or 419 equal temperament (419et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 419 equal parts of about 2.86 ¢ each. Each step represents a frequency ratio of 21/419, or the 419th root of 2.

Theory

419edo is a decent 7-limit system, and is consistent to the 9-odd-limit. The equal temperament tempers out 32805/32768 (schisma) in the 5-limit; 235298/234375 (triwellisma), 420175/419904 (wizma) in the 7-limit. It supports and provides the optimal patent val for sextilifourths, the 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

Approximation of prime harmonics in 419edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -0.28 +0.32 -0.81 +1.43 -1.39 +1.01 +0.34 -1.07 -1.41 +0.55
Relative (%) +0.0 -9.9 +11.2 -28.2 +49.8 -48.4 +35.3 +11.8 -37.2 -49.4 +19.2
Steps
(reduced) 		419
(0) 		664
(245) 		973
(135) 		1176
(338) 		1450
(193) 		1550
(293) 		1713
(37) 		1780
(104) 		1895
(219) 		2035
(359) 		2076
(400)

Subsets and supersets

419edo is the 81th prime edo.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-664 419⟩ [⟨419​ 664​]] +0.0897 0.0897 3.13
2.3.5 32805/32768, [41 43 -47⟩ [⟨419 ​664​ 973]] +0.0137 0.1301 4.54
2.3.5.7 32805/32768, 235298/234375, 420175/419904 [⟨419 ​664​ 973 ​1176 ​]] +0.0821 0.1635 5.71
2.3.5.7.11 441/440, 4000/3993, 32805/32768, 420175/419904 [⟨419​ 664 ​973 ​1176​ 1450]] (419) −0.0168 0.2460 8.59

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 29\419 83.05 21/20 Sextilifourths (419f)
1 49\419 140.33 243/224 Tsaharuk (7-limit)
1 174\419 498.33 162/125 Helmholtz

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Scales

Music

Francium
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