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385edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|385}}
{{ED intro}}


== Theory ==
== Theory ==
385et tempers out following commas:
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.


7-limit commas: 589824/588245, 134217728/133984375, [[703125/702464]], 1959552/1953125, 5250987/5242880, 200120949/200000000
=== Prime harmonics ===
 
11-limit commas: 1073741824/1071794405, 161280/161051, 25165824/25109315, 234375/234256, 2097152/2096325, 1366875/1362944, 166698/166375, 496125/495616, 151263/151250, 104857600/104825259, [[540/539]], 172032/171875, 369140625/369098752, 825000/823543, 180224/180075, [[8019/8000]], 160083/160000, 539055/537824, 766656/765625, 202397184/201768035, 43923/43904, 20614528/20588575, 39135393/39062500, 781258401/781250000
 
===Prime harmonics===
{{Harmonics in equal|385}}
{{Harmonics in equal|385}}


===Subsets and supersets===
=== Subsets and supersets ===
385 factors into 5 x 7 x 11, with subset edos {{EDOs| 5, 7, 11, 35, 55, and 77}}.
Since 385 factors into {{factorization|385}}, 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" |[[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! colspan="2" |Tuning Error
|-
![[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
|-
|-
|2.3
! rowspan="2" | [[Subgroup]]
|{{monzo|-122 77}}
! rowspan="2" | [[Comma list]]
|{{val|385 610}}
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -122 77 }}
| {{mapping| 385 610 }}
| +0.2070
| +0.2070
| 0.2071
| 0.2071
| 6.64
| 6.64
|-
|-
|2.3.5
| 2.3.5
|{{monzo|-28 25 -5}}, {{monzo|38 -2 -15}}
| {{monzo| -28 25 -5 }}, {{monzo| 38 -2 -15 }}
|{{val|385 610 894}}
| {{mapping| 385 610 894 }}
| +0.1122
| +0.1122
| 0.2158
| 0.2158
| 6.92
| 6.92
|-
|-
|2.3.5.7
| 2.3.5.7
|19683/19600, 703125/702464, 589824/588245
| 19683/19600, 589824/588245, 703125/702464
|{{val|385 610 894 1081}}
| {{mapping| 385 610 894 1081 }}
| +0.0374
| +0.0374
| 0.2274
| 0.2274
| 7.30
| 7.30
|-
|-
|2.3.5.7.11
| 2.3.5.7.11
|540/539, 8019/8000, 496125/495616, 172032/171875
| 540/539, 8019/8000, 151263/151250, 172032/171875
|{{val|385 610 894 1081 1332}}
| {{mapping| 385 610 894 1081 1332 }}
| +0.0085
| +0.0085
| 0.2114
| 0.2114
| 6.78
| 6.78
|-
|-
|2.3.5.7.11.13
| 2.3.5.7.11.13
|540/539, 1716/1715, 8019/8000, 4096/4095, 81675/81536
| 540/539, 1575/1573, 2200/2197, 4096/4095, 8019/8000
|{{val|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
|-
|-
|2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
|540/539, 936/935, 1377/1375, 1716/1715, 4096/4095, 13365/13328
| 540/539, 936/935, 1377/1375, 1575/1573, 2200/2197, 4096/4095
|{{val|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
Line 71: Line 68:
=== 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
|-
! Generator<br>(reduced)
! Periods<br />per 8ve
! Cents<br>(reduced)
! Generator*
! Associated<br>ratio
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
|1
| 1
|62\385
| 62\385
|193.247
| 193.25
|4096/3645
| 262144/234375
|[[Luna]]
| [[Luna]]
|-
|-
|1
| 1
|162/385
| 162/385
|504.935
| 504.94
|4/3
| 4/3
|[[Countermeantone]]
| [[Countermeantone]]
|-
|-
|5
| 5
|160\385<br>(6\385)
| 80\385<br />(3\385)
|498.701<br>(18.701)
| 249.35<br />(9.35)
|4/3<br>(81/80)
| 81/70<br />(176/175)
|[[Pental]]
| [[Hemipental]]
|-
| 5
| 160\385<br />(6\385)
| 498.70<br />(18.70)
| 4/3<br />(81/80)
| [[Pental (temperament)|Pental]] (5-limit)
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
[[Category:Hemipental]]
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