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385edo: Difference between revisions

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-typo; review the temp table; +hemipental
m →Regular temperament properties: quintile isn't defined beyond the 5-limit
 
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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|385}}
{{ED intro}}


== Theory ==
== Theory ==
385edo has a reasonable approximation to the 11-limit, and perhaps beyond. The equal temperament [[tempering out|tempers out]] [[19683/19600]], [[589824/588245]], and [[703125/702464]] in the 7-limit; [[540/539]], [[8019/8000]], 43923/43904, 151263/151250, 160083/160000, 166698/166375, and 172032/171875 in the 11-limit. It [[support]]s [[hemipental]] and provides the [[optimal patent val]] for the 7-limit version thereof. Using the [[patent val]], it tempers out [[1575/1573]], [[1716/1715]], [[2200/2197]], [[4096/4095]], [[6656/6655]] and [[10648/10647]] in the 13-limit; and [[936/935]], [[1275/1274]], 1377/1375, and [[2601/2600]] in the 17-limit.  
385edo has a reasonable approximation to the [[11-limit]], and perhaps beyond. The equal temperament [[tempering out|tempers out]] [[19683/19600]], [[589824/588245]], and [[703125/702464]] in the [[7-limit]]; [[540/539]], [[8019/8000]], [[43923/43904]], [[151263/151250]], [[160083/160000]], 166698/166375, and 172032/171875 in the 11-limit. It [[support]]s [[hemipental]] and provides the [[optimal patent val]] for the 7-limit version thereof.  
 
Using the [[patent val]], it tempers out [[1575/1573]], [[1716/1715]], [[2200/2197]], [[4096/4095]], [[6656/6655]], and [[10648/10647]] in the [[13-limit]]; and [[936/935]], [[1275/1274]], 1377/1375, and [[2601/2600]] in the [[17-limit]].  


=== Prime harmonics ===
=== Prime harmonics ===
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=== Subsets and supersets ===
=== Subsets and supersets ===
Since 385 factors into {{factorization|385}}, 385edo has subset edos {{EDOs| 5, 7, 11, 35, 55, and 77 }}.
Since 385 factors into {{nowrap| 5 × 7 × 11 }}, 385edo has subset edos {{EDOs| 5, 7, 11, 35, 55, and 77 }}.


== 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]]
! 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| -122 77 }}
| {{Monzo| -122 77 }}
| {{mapping| 385 610 }}
| {{Mapping| 385 610 }}
| +0.2070
| +0.2070
| 0.2071
| 0.2071
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|-
|-
| 2.3.5
| 2.3.5
| {{monzo| -28 25 -5 }}, {{monzo| 38 -2 -15 }}
| {{Monzo| -28 25 -5 }}, {{monzo| 38 -2 -15 }}
| {{mapping| 385 610 894 }}
| {{Mapping| 385 610 894 }}
| +0.1122
| +0.1122
| 0.2158
| 0.2158
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| 2.3.5.7
| 2.3.5.7
| 19683/19600, 589824/588245, 703125/702464
| 19683/19600, 589824/588245, 703125/702464
| {{mapping| 385 610 894 1081 }}
| {{Mapping| 385 610 894 1081 }}
| +0.0374
| +0.0374
| 0.2274
| 0.2274
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| 2.3.5.7.11
| 2.3.5.7.11
| 540/539, 8019/8000, 151263/151250, 172032/171875
| 540/539, 8019/8000, 151263/151250, 172032/171875
| {{mapping| 385 610 894 1081 1332 }}
| {{Mapping| 385 610 894 1081 1332 }}
| +0.0085
| +0.0085
| 0.2114
| 0.2114
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| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 540/539, 1575/1573, 2200/2197, 4096/4095, 8019/8000
| 540/539, 1575/1573, 2200/2197, 4096/4095, 8019/8000
| {{mapping| 385 610 894 1081 1332 1425 }}
| {{Mapping| 385 610 894 1081 1332 1425 }}
| -0.0394
| −0.0394
| 0.2207
| 0.2207
| 7.08
| 7.08
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| 2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
| 540/539, 936/935, 1377/1375, 1575/1573, 2200/2197, 4096/4095
| 540/539, 936/935, 1377/1375, 1575/1573, 2200/2197, 4096/4095
| {{mapping| 385 610 894 1081 1332 1425 1574 }}
| {{Mapping| 385 610 894 1081 1332 1425 1574 }}
| -0.0693
| −0.0693
| 0.2171
| 0.2171
| 6.97
| 6.97
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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>Ratio*
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
|-
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|-
|-
| 5
| 5
| 80\385<br>(3\385)
| 3\385
| 249.35<br>(9.35)
| 9.35
| 81/70<br>(176/175)
| 176/175
| [[Hemipental]]
| [[Hemiquintile]]
|-
|-
| 5
| 5
| 160\385<br>(6\385)
| 6\385
| 498.70<br>(18.70)
| 18.70
| 4/3<br>(81/80)
| 81/80
| [[Pental]] (5-limit)
| [[Quintile]]
|}
|}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]]


[[Category:Hemipental]]
[[Category:Hemiquintile]]
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