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 [[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 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.

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

420edo is better at the 2.5.7.11.13.19.23 [[subgroup]], and 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, being divisible by {{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. Nice.

~~=== Harmonics ===~~

~~{{Harmonics in equal|420}}~~

== Music ==

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 == 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 [[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 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.
-Remarkably, approximation to the third harmonic, which it derives from 70edo, constitutes 666 steps of 420edo. Nice.
+edo is better at the 2.5.7.11.13.19.23 [[subgroup]], and 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, being divisible by {{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. Nice.
-=== Harmonics ===
-{{Harmonics in equal|420}}
 == Music ==