444edo: Difference between revisions
ArrowHead294 (talk | contribs) mNo edit summary |
ArrowHead294 (talk | contribs) m Partial undo |
||
| Line 12: | Line 12: | ||
== Regular temperament properties == | == Regular temperament properties == | ||
{ | {| class="wikitable center-4 center-5 center-6" | ||
|- | |||
! rowspan="2" | [[Subgroup]] | |||
! rowspan="2" | [[Comma list]] | |||
! rowspan="2" | [[Mapping]] | |||
! rowspan="2" | Optimal<br />8ve stretch (¢) | |||
! colspan="2" | Tuning error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |- | ||
| 2.3.5 | | 2.3.5 | ||
| {{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 | ||
|} | |||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{ | {| class="wikitable center-all left-5" | ||
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |||
|- | |||
! Periods<br />per 8ve | |||
! Generator* | |||
! Cents* | |||
! Associated<br />ratio* | |||
! Temperaments | |||
|- | |- | ||
| 1 | | 1 | ||
| Line 36: | Line 52: | ||
| 4/3<br>(35/33) | | 4/3<br>(35/33) | ||
| [[Undim]] (444d) | | [[Undim]] (444d) | ||
|} | |||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | |||
Revision as of 13:20, 16 November 2024
| ← 443edo | 444edo | 445edo → |
Theory
444edo is only consistent to the 5-odd-limit since harmonic 7 is about halfway between its steps. 444 = 4 × 111, and 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 it is distinct