210edo: Difference between revisions
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m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" |
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{{Infobox ET}} | {{Infobox ET}} | ||
{{ | {{ED intro}} | ||
== Theory == | == Theory == | ||
Since {{nowrap|210 {{=}} 3 × 70}}, 210edo shares its [[3/2|fifth]] with [[70edo]]. It is [[consistent]] to the [[9-odd-limit]], but there is a sharp tendency in the lower [[harmonic]]s. The equal temperament [[tempering out|tempers out]] 67108864/66430125 ([[misty comma]]) and 30958682112/30517578125 (trisedodge comma) in the 5-limit; [[3136/3125]], [[5120/5103]], and 118098/117649 in the 7-limit. | |||
Using the | Using the 210e val, which does the best, it tempers out [[540/539]], [[4000/3993]], 6912/6875, and 15488/15435 in the 11-limit; [[351/350]], [[364/363]], [[1001/1000]], [[2197/2187]], and 3584/3575 in the 13-limit. Using the patent val, it tempers out [[176/175]], 1375/1372, [[8019/8000]], and 41503/41472 in the 11-limit; [[351/350]], [[352/351]], [[847/845]], 2197/2187, and 16900/16807 in the 13-limit. | ||
=== Odd harmonics === | === Odd harmonics === | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
Since 210 factors into | Since 210 factors into {{factorisation|210}}, 210edo has subset edos {{EDOs| 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, and 105 }}. | ||
== 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| 26 -12 -3 }}, {{monzo| 19 10 -15 }} | | {{monzo| 26 -12 -3 }}, {{monzo| 19 10 -15 }} | ||
| {{mapping| 210 333 488 }} | | {{mapping| 210 333 488 }} | ||
| | | −0.5138 | ||
| 0.3987 | | 0.3987 | ||
| 6.98 | | 6.98 | ||
| Line 34: | Line 35: | ||
| 3136/3125, 5120/5103, 118098/117649 | | 3136/3125, 5120/5103, 118098/117649 | ||
| {{mapping| 210 333 488 590 }} | | {{mapping| 210 333 488 590 }} | ||
| | | −0.6170 | ||
| 0.3888 | | 0.3888 | ||
| 6.80 | | 6.80 | ||
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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 | ||
|- | |- | ||
| 3 | | 3 | ||
| 123\210<br>(17\210) | | 123\210<br />(17\210) | ||
| 702.86<br>(97.14) | | 702.86<br />(97.14) | ||
| 3/2<br>( | | 3/2<br />(18/17) | ||
| [[Misty]] | | [[Misty]] (210gh) | ||
|- | |- | ||
| 5 | | 5 | ||
| | | 13\210 | ||
| | | 74.29 | ||
| | | 25/24 | ||
| [[ | | [[Countdown]] (210e) | ||
|} | |} | ||
<nowiki>* | <nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct | ||