443edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
443edo is in[[consistent]] to the [[5-odd-limit]] and the error of [[harmonic]] [[5/1|5]] is quite large. To start with, the [[patent val]] {{val| 443 702 '''1029''' '''1244''' '''1533''' }} as well as the 443cde [[val]] {{val| 443 702 '''1028''' '''1243''' '''1532''' }} are worth considering. | 443edo is in[[consistent]] to the [[5-odd-limit]] and the error of [[harmonic]] [[5/1|5]] is quite large. To start with, the [[patent val]] {{val| 443 702 '''1029''' '''1244''' '''1533''' }} as well as the 443cde [[val]] {{val| 443 702 '''1028''' '''1243''' '''1532''' }} are worth considering. | ||
Using the patent val, the equal temperament [[tempering out|tempers out]] [[6144/6125]], [[32805/32768]], and 67108864/66976875 in the 7-limit; [[540/539]], [[5632/5625]], [[8019/8000]], and [[131072/130977]] in the 11-limit. It [[support]]s [[hemischis]], the 130 & 313 temperament. | Using the patent val, the equal temperament [[tempering out|tempers out]] [[6144/6125]], [[32805/32768]], and 67108864/66976875 in the 7-limit; [[540/539]], [[5632/5625]], [[8019/8000]], and [[131072/130977]] in the 11-limit. It [[support]]s [[hemischis]], the {{nowrap|130 & 313}} temperament. | ||
=== Prime harmonics === | === Prime harmonics === | ||
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== 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" | [[Subgroup]] | ||
! rowspan="2" | [[Comma list | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br>8ve | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
! colspan="2" | Tuning | ! colspan="2" | Tuning error | ||
|- | |- | ||
! [[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
! [[TE simple badness|Relative]] (%) | ! [[TE simple badness|Relative]] (%) | ||
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| {{monzo|-702 443}} | | {{monzo|-702 443}} | ||
| {{mapping| 443 702 }} | | {{mapping| 443 702 }} | ||
| 0.1183 | | +0.1183 | ||
| 0.1183 | | 0.1183 | ||
| 4.37 | | 4.37 | ||
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=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
|+Table of rank-2 temperaments by generator | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | |- | ||
! Periods<br />per 8ve | |||
! Generator* | ! Generator* | ||
! Cents* | ! Cents* | ||
! Associated<br> | ! Associated<br />ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
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| [[Hemischis]] (443) | | [[Hemischis]] (443) | ||
|} | |} | ||
<nowiki>* | <nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | ||
== Scales == | == Scales == | ||
| Line 53: | Line 55: | ||
== Music == | == Music == | ||
; [[ | ; [[Francium]] | ||
* "Confusion" from ''HemischisMatic EP'' (2023) | * "Confusion" from ''HemischisMatic EP'' (2023) – [https://open.spotify.com/track/72Pc4tty5rlLazQXW3tcCe Spotify] | [https://francium223.bandcamp.com/track/confusion Bandcamp] | [https://youtu.be/et0Qd4YAEP4?si=yRYbMo8sVz1qRvEE YouTube] – [[hemischis]] in 443edo tuning | ||
[[Category:Listen]] | |||
Latest revision as of 12:51, 21 February 2025
| ← 442edo | 443edo | 444edo → |
443 equal divisions of the octave (abbreviated 443edo or 443ed2), also called 443-tone equal temperament (443tet) or 443 equal temperament (443et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 443 equal parts of about 2.71 ¢ each. Each step represents a frequency ratio of 21/443, or the 443rd root of 2.
Theory
443edo is inconsistent to the 5-odd-limit and the error of harmonic 5 is quite large. To start with, the patent val ⟨443 702 1029 1244 1533] as well as the 443cde val ⟨443 702 1028 1243 1532] are worth considering.
Using the patent val, the equal temperament tempers out 6144/6125, 32805/32768, and 67108864/66976875 in the 7-limit; 540/539, 5632/5625, 8019/8000, and 131072/130977 in the 11-limit. It supports hemischis, the 130 & 313 temperament.
Prime harmonics
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.00 | -0.37 | +1.05 | +0.93 | +1.28 | -0.80 | +0.69 | +0.46 | +0.17 | -0.23 | +0.79 |
| Relative (%) | +0.0 | -13.8 | +38.6 | +34.2 | +47.2 | -29.5 | +25.4 | +16.8 | +6.2 | -8.6 | +29.1 | |
| Steps (reduced) |
443 (0) |
702 (259) |
1029 (143) |
1244 (358) |
1533 (204) |
1639 (310) |
1811 (39) |
1882 (110) |
2004 (232) |
2152 (380) |
2195 (423) | |
Subsets and supersets
443edo is the 86th prime edo. 886edo, which doubles it, gives a good correction until the 11-limit.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3 | [-702 443⟩ | [⟨443 702]] | +0.1183 | 0.1183 | 4.37 |
Rank-2 temperaments
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperaments |
|---|---|---|---|---|
| 1 | 92\443 | 249.21 | 15/13 | Hemischis (443) |
* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct