444edo: Difference between revisions
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== Theory == | == Theory == | ||
444edo is only [[consistent]] to the [[5-odd-limit]] since [[harmonic]] [[7/1|7]] is about halfway between its steps. 444 = 4 × 111, | 444edo is only [[consistent]] to the [[5-odd-limit]] since [[harmonic]] [[7/1|7]] is about halfway between its steps. Since {{nowrap|444 {{=}} 4 × 111}}, its harmonic [[3/1|3]] derives from [[111edo]]. Using the [[patent val]], the equal temperament [[tempering out|tempers out]] [[250047/250000]], 29360128/29296875, 67108864/66976875 and in the 7-limit; [[3025/3024]], [[5632/5625]], 42592/42525, 102487/102400, [[131072/130977]], 160083/160000, 172032/171875, 322102/321489, 391314/390625 and [[1771561/1769472]] in the 11-limit. It [[support]]s the [[magnesium]] temperament. | ||
=== Odd harmonics === | === Odd harmonics === | ||
| Line 26: | Line 26: | ||
| {{monzo| 41 -20 -4 }}, {{monzo| -29 -11 20 }} | | {{monzo| 41 -20 -4 }}, {{monzo| -29 -11 20 }} | ||
| {{mapping| 444 704 1031 }} | | {{mapping| 444 704 1031 }} | ||
| | | −0.1821 | ||
| 0.2071 | | 0.2071 | ||
| 7.66 | | 7.66 | ||
| Line 53: | Line 53: | ||
| [[Undim]] (444d) | | [[Undim]] (444d) | ||
|} | |} | ||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if | <nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | ||
Revision as of 16:57, 15 January 2025
| ← 443edo | 444edo | 445edo → |
Theory
444edo is only consistent to the 5-odd-limit since harmonic 7 is about halfway between its steps. Since 444 = 4 × 111, its harmonic 3 derives from 111edo. Using the patent val, the equal temperament tempers out 250047/250000, 29360128/29296875, 67108864/66976875 and in the 7-limit; 3025/3024, 5632/5625, 42592/42525, 102487/102400, 131072/130977, 160083/160000, 172032/171875, 322102/321489, 391314/390625 and 1771561/1769472 in the 11-limit. It supports the magnesium temperament.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.75 | +0.17 | -1.26 | -1.21 | +0.03 | +0.01 | +0.92 | +0.45 | -0.22 | -0.51 | -1.25 |
| Relative (%) | +27.7 | +6.4 | -46.6 | -44.7 | +1.2 | +0.5 | +34.1 | +16.6 | -8.0 | -18.9 | -46.2 | |
| Steps (reduced) |
704 (260) |
1031 (143) |
1246 (358) |
1407 (75) |
1536 (204) |
1643 (311) |
1735 (403) |
1815 (39) |
1886 (110) |
1950 (174) |
2008 (232) | |
Subsets and supersets
Since 444 factors into 22 × 3 × 37, 444edo has subset edos 2, 3, 4, 6, 12, 37, 74, 111, 148, and 222. 1332edo, which triples it, gives a good correction to the harmonic 7.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5 | [41 -20 -4⟩, [-29 -11 20⟩ | [⟨444 704 1031]] | −0.1821 | 0.2071 | 7.66 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 13\444 | 35.14 | 1990656/1953125 | Gammic (5-limit) |
| 4 | 184\444 (38\444) |
497.30 (102.70) |
4/3 (35/33) |
Undim (444d) |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct