420edo: Difference between revisions

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== Theory ==

~~420~~ is ~~a largely composite number~~, ~~being divisible by all numbers inclusively from 2 to 7. It'~~s ~~other divisors are {{EDOs~~| 10, 12, 14, 15, 20, 21, 28, ~~30, 35, 42, 60, 70, 84, 105, 140~~, and ~~210 }}~~.

420edo is [[enfactoring|enfactored]] in the 7-limit, with the same tuning of [[harmonic]]s [[3/1|3]], [[5/1|5]], and [[7/1|7]] as [[140edo]]. The [[13/1|13th harmonic]] is also present in 140edo, and ultimately derives from [[10edo]]. The [[29/1|29th harmonic]], while having significantly drifted in terms of [[relative interval error]], has retained its step position from [[7edo]]. In addition, in the 29-limit, only the harmonics 11 and 17 have step numbers coprime with 420. This means that all other approximations are preserved from smaller edos, thus enabling edo mergers and mashups.

~~Remarkably~~, ~~approximation to~~ the ~~third harmonic~~, ~~which~~ it ~~derives from 70edo~~, ~~constitutes 666 steps of 420edo. Nice~~.

420edo is good at the 2.5.7.11.13.19.23 [[subgroup]], and has a great potential as a near-just xenharmonic system. It also works satisfactorily with the 29-limit as a whole, though in[[consistent]]. In the 11-limit, it notably tempers out [[4000/3993]], and in the 13-limit, [[10648/10647]].

=== ~~Largely composite number theory~~ ===

=== Odd harmonics ===

~~Being a largely composite number of steps, 420edo is rich~~ in ~~modulation circles. 420edo is [[enfactoring~~|enfactored]] in the 7-limit, with the same tuning of 3, 5, and 7 as [[140edo]]. The 13th harmonic is also present in 140edo, and ultimately derives from [[10edo]]. The 29th harmonic, while having significantly drifted, has retained its step position from [[7edo]].

~~In addition, in the 29-limit, only 11~~ and ~~17 have step correspondences coprime with~~ 420~~. This means that all other approximations are preserved from smaller~~ edos, ~~thus enabling EDO mergers~~ and ~~mashups~~.

=== Subsets and supersets ===

420 is a largely composite number, its nontrivial subset edos being {{EDOs| 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, and 210 }}. For this reason 420edo is rich in modulation circles.

==~~= Regular temperament theory =~~==

== Trivia ==

~~420edo can be adapted for use with 2.5.7.11.13.19.23 subgroup, and it works satisfactorily with the 29-limit as a whole, although due~~ to ~~over 25% error on some harmonics, it's inconsistent. In~~ the ~~11-limit~~, ~~it notably tempers out [[4000/3993]]~~, ~~and in the 13-limit, [[10648/10647]]~~.

The approximation to the third harmonic, which derives from 70edo, constitutes 666 steps of 420edo.

=== ~~Harmonics~~ ===

== Regular temperament properties ==

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

|-

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

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

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

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

! colspan="2" | Tuning Error

|-

! [[TE error|Absolute]] (¢)

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

|-

| 2.3.5.7.11

| 2401/2400, 4000/3993, 5120/5103, 15625/15552

| {{mapping| 420 666 975 1179 1453 }}

| −0.051

| 0.278

| 9.74

|- style="border-top: double;"

| 2.5.7.11.13.19.23

| 875/874, 5635/5632, 10241/10240, 12103/12100, 11875/11858, 10985/10976

| {{mapping| 420 975 1179 1453 1554 1784 1900 }}

| +0.069

| 0.104

| 3.62

|}

== Music ==

* [https://~~www.youtube~~.~~com~~/~~watch~~?v=~~X1deLLCJD64~~ Follow In Is] ~~by Mandrake~~

; [[Mandrake]]

[[Category:~~Equal divisions of the octave|###~~]] ~~~~

* [https://youtu.be/X1deLLCJD64?si=baHHYZQV9VFMaJZs ''Follow In Is''] (2022) – a superset of [[12edo]], [[5edo]], and [[7edo]], least common multiple of which is 420edo.

[[Category:Listen]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|420}}
+{{ED intro}}
 == Theory ==
-is a largely composite number, being divisible by all numbers inclusively from 2 to 7. It's other divisors are {{EDOs| 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, and 210 }}.
+edo is [[enfactoring|enfactored]] in the 7-limit, with the same tuning of [[harmonic]]s [[3/1|3]], [[5/1|5]], and [[7/1|7]] as [[140edo]]. The [[13/1|13th harmonic]] is also present in 140edo, and ultimately derives from [[10edo]]. The [[29/1|29th harmonic]], while having significantly drifted in terms of [[relative interval error]], has retained its step position from [[7edo]]. In addition, in the 29-limit, only the harmonics 11 and 17 have step numbers coprime with 420. This means that all other approximations are preserved from smaller edos, thus enabling edo mergers and mashups.
-Remarkably, approximation to the third harmonic, which it derives from 70edo, constitutes 666 steps of 420edo. Nice.
+edo is good at the 2.5.7.11.13.19.23 [[subgroup]], and has a great potential as a near-just xenharmonic system. It also works satisfactorily with the 29-limit as a whole, though in[[consistent]]. In the 11-limit, it notably tempers out [[4000/3993]], and in the 13-limit, [[10648/10647]].
-=== Largely composite number theory ===
+=== Odd harmonics ===
-Being a largely composite number of steps, 420edo is rich in modulation circles. 420edo is [[enfactoring|enfactored]] in the 7-limit, with the same tuning of 3, 5, and 7 as [[140edo]]. The 13th harmonic is also present in 140edo, and ultimately derives from [[10edo]]. The 29th harmonic, while having significantly drifted, has retained its step position from [[7edo]].
+{{Harmonics in equal|420}}
-In addition, in the 29-limit, only 11 and 17 have step correspondences coprime with 420. This means that all other approximations are preserved from smaller edos, thus enabling EDO mergers and mashups.
+=== Subsets and supersets ===
+is a largely composite number, its nontrivial subset edos being {{EDOs| 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, and 210 }}. For this reason 420edo is rich in modulation circles.
-=== Regular temperament theory ===
+== Trivia ==
-edo can be adapted for use with 2.5.7.11.13.19.23 subgroup, and it works satisfactorily with the 29-limit as a whole, although due to over 25% error on some harmonics, it's inconsistent. In the 11-limit, it notably tempers out [[4000/3993]], and in the 13-limit, [[10648/10647]].
+The approximation to the third harmonic, which derives from 70edo, constitutes 666 steps of 420edo.
-=== Harmonics ===
+== Regular temperament properties ==
-{{Harmonics in equal|420}}
+{| class="wikitable center-4 center-5 center-6"
+|-
+! rowspan="2" | [[Subgroup]]
+! rowspan="2" | [[Comma list|Comma List]]
+! rowspan="2" | [[Mapping]]
+! rowspan="2" | Optimal 8ve <br>Stretch (¢)
+! colspan="2" | Tuning Error
+|-
+! [[TE error|Absolute]] (¢)
+! [[TE simple badness|Relative]] (%)
+|-
+| 2.3.5.7.11
+| 2401/2400, 4000/3993, 5120/5103, 15625/15552
+| {{mapping| 420 666 975 1179 1453 }}
+| −0.051
+| 0.278
+| 9.74
+|- style="border-top: double;"
+| 2.5.7.11.13.19.23
+| 875/874, 5635/5632, 10241/10240, 12103/12100, 11875/11858, 10985/10976
+| {{mapping| 420 975 1179 1453 1554 1784 1900 }}
+| +0.069
+| 0.104
+| 3.62
+|}
 == Music ==
-* [https://www.youtube.com/watch?v=X1deLLCJD64 Follow In Is] by Mandrake
+; [[Mandrake]]
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
+* [https://youtu.be/X1deLLCJD64?si=baHHYZQV9VFMaJZs ''Follow In Is''] (2022) – a superset of [[12edo]], [[5edo]], and [[7edo]], least common multiple of which is 420edo.
+[[Category:Listen]]