420edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
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 | 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. | ||
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]]. | 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]]. | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
420 is a largely composite number, being | 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. | ||
== Trivia == | == Trivia == | ||
The approximation to the third harmonic, which derives from 70edo, constitutes 666 steps of 420edo | The approximation to the third harmonic, which derives from 70edo, constitutes 666 steps of 420edo. | ||
== 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" | [[Subgroup]] | ||
! rowspan="2" | [[Comma list|Comma List]] | ! rowspan="2" | [[Comma list|Comma List]] | ||
| Line 29: | Line 30: | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 2401/2400, 4000/3993, 5120/5103, 15625/15552 | | 2401/2400, 4000/3993, 5120/5103, 15625/15552 | ||
| | | {{mapping| 420 666 975 1179 1453 }} | ||
| | | −0.051 | ||
| 0.278 | | 0.278 | ||
| 9.74 | | 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 == | == Music == | ||
; [[Mandrake]] | ; [[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:Listen]] | |||