420edo: Difference between revisions

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~~The '''420 equal divisions of the octave''' divides the [[octave]] into parts of 2.857 [[cent]]s each.~~

== Theory ==

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.

~~420~~ is ~~a highly 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 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]].

~~Remarkably, approximation to the third harmonic (perfect fifth plus an octave, or tritave) constitutes 666 steps of 420edo. Nice.~~

=== Odd harmonics ===

~~Being~~ a ~~highly~~ composite number ~~of steps~~, 420edo is rich in modulation circles. In addition, of the first 10 prime harmonics, 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, and showing the vibrant and highly composite nature of 420.

=== 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.

420edo ~~can be adapted for use with 2.7.11.13.19.23 subgroup~~.

== Trivia ==

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

~~420edo is~~ [[~~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 the~~ 11~~-limit~~, ~~it notably tempers out [[~~4000/3993]], ~~and in the~~ 13~~-limit~~, ~~[[10648~~/~~10647]]~~.

== 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

|}

==~~= Prime harmonics =~~==

== Music ==

~~{{Primes in edo|420|columns~~=~~10}}~~

; [[Mandrake]]

* [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:~~Equal divisions of the octave~~]]

[[Category:Listen]]

@@ Line 1: / Line 1: @@
-The '''420 equal divisions of the octave''' divides the [[octave]] into parts of 2.857 [[cent]]s each.
+{{Infobox ET}}
+{{ED intro}}
 == Theory ==
+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.
-is a highly 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 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]].
-Remarkably, approximation to the third harmonic (perfect fifth plus an octave, or tritave) constitutes 666 steps of 420edo. Nice.
+=== Odd harmonics ===
+{{Harmonics in equal|420}}
-Being a highly composite number of steps, 420edo is rich in modulation circles. In addition, of the first 10 prime harmonics, 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, and showing the vibrant and highly composite nature of 420.
+=== 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.
-edo can be adapted for use with 2.7.11.13.19.23 subgroup.
+== Trivia ==
+The approximation to the third harmonic, which derives from 70edo, constitutes 666 steps of 420edo.
-edo is [[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 the 11-limit, it notably tempers out [[4000/3993]], and in the 13-limit, [[10648/10647]].
+== 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
+|}
-=== Prime harmonics ===
+== Music ==
-{{Primes in edo|420|columns=10}}
+; [[Mandrake]]
+* [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:Equal divisions of the octave]]
+[[Category:Listen]]