416edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
416et is consistent to the [[7-odd-limit]] | 416et is [[consistent]] to the [[7-odd-limit]], but the error of [[harmonic]] [[3/1|3]] is quite large. Nonetheless, it gives reasonable approximations to harmonics [[5/1|5]], [[7/1|7]], [[9/1|9]], [[11/1|11]], [[19/1|19]], and [[23/1|23]], making it suitable for a 2.9.5.7.11.19.23 [[subgroup]] interpretation, where it notably [[tempering out|tempers out]] [[1331/1330]], [[1540/1539]], [[5632/5625]], [[9801/9800]], and 10241/10240. | ||
Using the patent val regardless, it tempers out 321489/320000, 589824/588245, and 703125/702464, [[support]]ing [[tridecatonic]] and [[fermionic]]. | |||
=== Odd harmonics === | === Odd harmonics === | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
416 factors into 2<sup>5</sup> × 13, | Since 416 factors into 2<sup>5</sup> × 13, 416edo subset edos {{EDOs| 2, 4, 8, 13, 16, 26, 32, 52, 104, and 208 }}. [[832edo]], which doubles it, gives a good correction to the harmonic 3. | ||
== 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" |[[Comma list|Comma List]] | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" |[[Mapping]] | ! rowspan="2" | [[Comma list|Comma List]] | ||
! rowspan="2" |Optimal<br>8ve Stretch (¢) | ! rowspan="2" | [[Mapping]] | ||
! colspan="2" |Tuning Error | ! rowspan="2" | Optimal<br>8ve Stretch (¢) | ||
! colspan="2" | Tuning Error | |||
|- | |- | ||
![[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
![[TE simple badness|Relative]] (%) | ! [[TE simple badness|Relative]] (%) | ||
|- | |- | ||
|2.9 | | 2.9 | ||
|{{monzo|1319 -416}} | | {{monzo| 1319 -416 }} | ||
|{{mapping|416 1319}} | | {{mapping| 416 1319 }} | ||
| | | −0.1416 | ||
| 0.1416 | | 0.1416 | ||
| 4.91 | | 4.91 | ||
|- | |- | ||
|2.9.5 | | 2.9.5 | ||
{{monzo|56 -14 -5}}, {{monzo|-5 -16 24}} | | {{monzo| 56 -14 -5 }}, {{monzo| -5 -16 24 }} | ||
|{{mapping|416 1319 966}} | | {{mapping| 416 1319 966 }} | ||
| | | −0.1267 | ||
| 0.1175 | | 0.1175 | ||
| 4.07 | | 4.07 | ||
|- | |- | ||
|2.9.5.7 | | 2.9.5.7 | ||
|420175/419904, 102760448/102515625, | | 420175/419904, 102760448/102515625, {{monzo| 14 -6 7 -4 }} | ||
|{{mapping|416 1319 966 1168}} | | {{mapping| 416 1319 966 1168 }} | ||
| | | −0.1310 | ||
| 0.1021 | | 0.1021 | ||
| 3.54 | | 3.54 | ||
|- | |- | ||
|2.9.5.7.11 | | 2.9.5.7.11 | ||
|5632/5625, 9801/9800, 41503/41472, 774400000/771895089 | | 5632/5625, 9801/9800, 41503/41472, 774400000/771895089 | ||
|{{mapping|416 1319 966 1168 1439}} | | {{mapping| 416 1319 966 1168 1439 }} | ||
| | | −0.0842 | ||
| 0.1308 | | 0.1308 | ||
| 4.53 | | 4.53 | ||
|} | |} | ||
== Music == | |||
; [[birdshite stalactite]] | |||
* [https://www.youtube.com/watch?v=nmUCjbfGpKU ''palate cleanser''] (2024) – tetracot[13] in 416edo tuning | |||