421edo: Difference between revisions
Created page with "{{Infobox ET}} {{EDO intro|421}} == Theory == 421et is only consistent to the 3-odd-limit, with its harmonic 5 being way too sharp. It is suitable for the 2.3.7.11.13.29...." |
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== Theory == | == Theory == | ||
421edo is in[[consistent]] to the [[5-odd-limit]], with its [[harmonic]] [[5/1|5]] being way too sharp. To start with, consider the following [[breed]]s: | |||
* {{val| 421 667 '''977''' 1182 }} (421c) | |||
* {{val| 421 667 '''977''' '''1181''' }} (421cd) | |||
* {{val| 421 667 978 1182 }} ([[patent val]]) | |||
The 421c val [[tempering out|tempers out]] [[4375/4374]] and [[2100875/2097152]], [[support]]ing [[mitonic]]. | |||
The 421cd val tempers out [[1029/1024]] and 823543/820125. | |||
The 421 val tempers out [[2401/2400]] and [[3136/3125]], supporting [[hemiwürschmidt]]. | |||
Omitting harmonic 5, it is suitable for the 2.3.7.11.13.29.37 [[subgroup]], where it tempers out 638/637, [[5292/5291]], 24192/24167, 53361/53248, 88209/87808 and 85293/85184. | |||
=== Odd harmonics === | === Odd 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|Comma List]] | ! rowspan="2" | [[Comma list|Comma List]] | ||
! rowspan="2" |[[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" |Optimal<br>8ve Stretch (¢) | ! rowspan="2" | Optimal<br>8ve Stretch (¢) | ||
! colspan="2" |Tuning Error | ! colspan="2" | Tuning Error | ||
|- | |- | ||
![[TE error|Absolute]] (¢) | ! [[TE error|Absolute]] (¢) | ||
![[TE simple badness|Relative]] (%) | ! [[TE simple badness|Relative]] (%) | ||
|- | |- | ||
|2.3 | | 2.3 | ||
|{{monzo|-667 421}} | | {{monzo| -667 421 }} | ||
|{{mapping|421 667}} | | {{mapping| 421 667 }} | ||
| 0.2421 | | 0.2421 | ||
| 0.2421 | | 0.2421 | ||
| 8.49 | | 8.49 | ||
|- | |- | ||
|2.3.7 | | 2.3.7 | ||
|{{monzo|-44 26 1}}, {{monzo|37 5 -16}} | | {{monzo| -44 26 1 }}, {{monzo| 37 5 -16 }} | ||
|{{mapping|421 667 1182}} | | {{mapping| 421 667 1182 }} | ||
| 0.1263 | | 0.1263 | ||
| 0.2567 | | 0.2567 | ||
| 9.01 | | 9.01 | ||
|- | |- | ||
|2.3.7.11 | | 2.3.7.11 | ||
|88209/87808, 2893401/2883584, 208971104256/208422380089 | | 88209/87808, 2893401/2883584, 208971104256/208422380089 | ||
|{{mapping|421 667 1182 1456}} | | {{mapping| 421 667 1182 1456 }} | ||
| 0.1814 | | 0.1814 | ||
| 0.2419 | | 0.2419 | ||
| 8.49 | | 8.49 | ||
|- | |- | ||
|2.3.7.11.13 | | 2.3.7.11.13 | ||
|53361/53248, 88209/87808 | | 24192/24167, 53361/53248, 85293/85184, 88209/87808 | ||
|{{mapping|421 667 1182 1456 1558}} | | {{mapping| 421 667 1182 1456 1558 }} | ||
| 0.1274 | | 0.1274 | ||
| 0.2418 | | 0.2418 | ||
|8.48 | | 8.48 | ||
|} | |} | ||