420edo: Difference between revisions
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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. | 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. | ||
420edo is | 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]]. | ||
=== Odd harmonics === | === Odd harmonics === | ||
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=== Subsets and supersets === | === 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. | 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. | ||
== Trivia == | == Trivia == | ||
Revision as of 14:19, 25 May 2023
| ← 419edo | 420edo | 421edo → |
Theory
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 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.
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 inconsistent. In the 11-limit, it notably tempers out 4000/3993, and in the 13-limit, 10648/10647.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.90 | -0.60 | -0.25 | -1.05 | +0.11 | -0.53 | +0.30 | +0.76 | -0.37 | +0.65 | +0.30 |
| Relative (%) | +31.6 | -21.0 | -8.9 | -36.9 | +3.9 | -18.5 | +10.6 | +26.6 | -13.0 | +22.7 | +10.4 | |
| Steps (reduced) |
666 (246) |
975 (135) |
1179 (339) |
1331 (71) |
1453 (193) |
1554 (294) |
1641 (381) |
1717 (37) |
1784 (104) |
1845 (165) |
1900 (220) | |
Subsets and supersets
420 is a largely composite number, being divisible by 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
The approximation to the third harmonic, which derives from 70edo, constitutes 666 steps of 420edo. Nice.
Regular temperament properties
| Subgroup | Comma List | Mapping | Optimal 8ve Stretch (¢) |
Tuning Error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5.7.11 | 2401/2400, 4000/3993, 5120/5103, 15625/15552 | [⟨420 666 975 1179 1453]] | -0.051 | 0.278 | 9.74 |
Music
- Follow In Is – a superset of 12edo, 5edo, and 7edo, least common multiple of which is 420edo.