437edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
==Theory== | |||
== Theory == | |||
===Odd harmonics=== | 437edo is [[consistent]] to the [[7-odd-limit]], but the errors of [[harmonic]]s [[3/1|3]] and [[5/1|5]] are quite large, giving us the option of treating it as either a full 11-limit temperament, or a 2.9.15.21.13.17 [[subgroup]] temperament. | ||
Using the [[patent val]], the equal temperament [[tempering out|tempers out]] [[2401/2400]] and 4096000/4084101 in the 7-limit; [[3025/3024]], [[41503/41472]], [[16384/16335]], and 151263/151250 in the 11-limit. | |||
=== Odd harmonics === | |||
{{Harmonics in equal|437}} | {{Harmonics in equal|437}} | ||
===Subsets and supersets=== | |||
437 factors into | === Subsets and supersets === | ||
==Regular temperament properties== | Since 437 factors into {{factorization|437}}, 437edo contains [[19edo]] and [[23edo]] as subsets. [[874edo]], which doubles it, gives a good correction to the harmonic 3. | ||
== Regular temperament properties == | |||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
|- | |- | ||
|2.9 | ! rowspan="2" | [[Subgroup]] | ||
|{{monzo|-1385 437}} | ! rowspan="2" | [[Comma list]] | ||
|{{ | ! rowspan="2" | [[Mapping]] | ||
| 0.1114 | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
! colspan="2" | Tuning error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |||
| 2.9 | |||
| {{monzo| -1385 437 }} | |||
| {{mapping| 437 1385 }} | |||
| +0.1114 | |||
| 0.1114 | | 0.1114 | ||
| 4.06 | | 4.06 | ||
|} | |} | ||
==Scales== | |||
== Scales == | |||
* [[Namo7]] | * [[Namo7]] | ||
* [[Namo10]] | * [[Namo10]] | ||
* [[Namo17]] | * [[Namo17]] | ||
Latest revision as of 12:40, 21 February 2025
| ← 436edo | 437edo | 438edo → |
437 equal divisions of the octave (abbreviated 437edo or 437ed2), also called 437-tone equal temperament (437tet) or 437 equal temperament (437et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 437 equal parts of about 2.75 ¢ each. Each step represents a frequency ratio of 21/437, or the 437th root of 2.
Theory
437edo is consistent to the 7-odd-limit, but the errors of harmonics 3 and 5 are quite large, giving us the option of treating it as either a full 11-limit temperament, or a 2.9.15.21.13.17 subgroup temperament.
Using the patent val, the equal temperament tempers out 2401/2400 and 4096000/4084101 in the 7-limit; 3025/3024, 41503/41472, 16384/16335, and 151263/151250 in the 11-limit.
Odd harmonics
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.02 | +0.87 | +0.51 | -0.71 | +0.63 | -0.25 | -0.85 | -0.61 | -0.95 | -1.22 | +0.56 |
| Relative (%) | +37.1 | +31.7 | +18.6 | -25.7 | +22.8 | -9.2 | -31.1 | -22.1 | -34.4 | -44.3 | +20.3 | |
| Steps (reduced) |
693 (256) |
1015 (141) |
1227 (353) |
1385 (74) |
1512 (201) |
1617 (306) |
1707 (396) |
1786 (38) |
1856 (108) |
1919 (171) |
1977 (229) | |
Subsets and supersets
Since 437 factors into 19 × 23, 437edo contains 19edo and 23edo as subsets. 874edo, which doubles it, gives a good correction to the harmonic 3.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.9 | [-1385 437⟩ | [⟨437 1385]] | +0.1114 | 0.1114 | 4.06 |