205edo: Difference between revisions
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== Theory == | == Theory == | ||
Since {{nowrap|205 {{=}} 5 × 41}}, | Since {{nowrap|205 {{=}} 5 × 41}}, 205edo shares its [[3/2|fifth]] with [[41edo]]. It can serve as a tuning for various temperaments, such as [[amity]] or [[laka]], and supplies the [[optimal patent val]] for [[quanic]] in the 7-, 11-, 13-, 17- and 19-limit, and for 13-limit amity, as well as other temperaments [[tempering out]] the huntma, [[640/637]], the rank-5 temperament for which it also supplies the optimal patent val. | ||
In the 5-limit it tempers out 1600000/1594323, the [[amity comma]], and {{monzo| 38 -2 -15 }}, the [[hemithirds comma]], and is an excellent tuning for 5-limit amity. The [[patent val]] {{val| 205 325 476 576 709 759 }} tempers out [[4375/4374]], [[5120/5103]], [[6144/6125]] in the 7-limit; [[540/539]], 1331/1323, and 2420/2401 in the 11-limit; [[352/351]], [[640/637]], [[729/728]], [[847/845]], and [[1188/1183]] in the 13-limit. | In the 5-limit it tempers out 1600000/1594323, the [[amity comma]], and {{monzo| 38 -2 -15 }}, the [[hemithirds comma]], and is an excellent tuning for 5-limit amity. The [[patent val]] {{val| 205 325 476 576 709 759 }} tempers out [[4375/4374]], [[5120/5103]], [[6144/6125]] in the 7-limit; [[540/539]], 1331/1323, and 2420/2401 in the 11-limit; [[352/351]], [[640/637]], [[729/728]], [[847/845]], and [[1188/1183]] in the 13-limit. | ||
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== Notation == | == Notation == | ||
=== Ups and downs === | === Ups and downs === | ||
205edo can be notated with [[Ups and downs notation|ups and downs]] representing 5\205 = 1\41, and lifts and drops (written as / and \) representing 1\205. Alternatively, ups and downs represent 1\205 and the quintuple-arrow symbols quip and quid (> and <) represent 5\205 = 1\41. Both notations have the advantage of building on a familiarity with 41edo The first notation is especially useful for [[Kite guitar | 205edo can be notated with [[Ups and downs notation|ups and downs]] representing {{nowrap|5\205 {{=}} 1\41}}, and lifts and drops (written as / and \) representing 1\205. Alternatively, ups and downs represent 1\205 and the quintuple-arrow symbols quip and quid (> and <) represent {{nowrap|5\205 {{=}} 1\41}}. Both notations have the advantage of building on a familiarity with 41edo The first notation is especially useful for [[Kite guitar]]ists who want to notate microbends more precisely. | ||
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
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| P1 || /1 || //1 || ^\\1 || ^\1 || ^1 || ^/1 || ^//1 || v\\m2 || v\m2 || vm2 || v/m2 || v//m2 || \\m2 || \m2 || m2 || /m2 || … | | P1 || /1 || //1 || ^\\1 || ^\1 || ^1 || ^/1 || ^//1 || v\\m2 || v\m2 || vm2 || v/m2 || v//m2 || \\m2 || \m2 || m2 || /m2 || … | ||
|- | |- | ||
|P1 | | P1 | ||
|^1 | | ^1 | ||
|^^1 | | ^^1 | ||
|^^^1 | | ^^^1 | ||
|v>1 | | v>1 | ||
|>1 | | >1 | ||
|^>1 | | ^>1 | ||
|^^>1 | | ^^>1 | ||
|vv<m2 | | vv<m2 | ||
|v<m2 | | v<m2 | ||
|<m2 | | <m2 | ||
|^<m2 | | ^<m2 | ||
|vvvm2 | | vvvm2 | ||
|vvm2 | | vvm2 | ||
|vm2 | | vm2 | ||
|m2 | | m2 | ||
|^m2 | | ^m2 | ||
| | | | ||
|} | |} | ||
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| 1600000/1594323, {{monzo| 38 -2 -15 }} | | 1600000/1594323, {{monzo| 38 -2 -15 }} | ||
| {{mapping| 205 325 476 }} | | {{mapping| 205 325 476 }} | ||
| | | −0.106 | ||
| 0.141 | | 0.141 | ||
| 2.41 | | 2.41 | ||
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| 5632/5625, 14641/14580, 1600000/1594323 | | 5632/5625, 14641/14580, 1600000/1594323 | ||
| {{mapping| 181 287 420 508 }} | | {{mapping| 181 287 420 508 }} | ||
| | | −0.002 | ||
| 0.218 | | 0.218 | ||
| 3.72 | | 3.72 | ||
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| [[Countercomp]] | | [[Countercomp]] | ||
|} | |} | ||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if | <nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | ||
== Scales == | == Scales == | ||