385edo: Difference between revisions

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Doesn't support septimal pental so the limit must be specified
Rework; cleanup; clarify the title row of the rank-2 temp table
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== 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]], [[6656/665]] 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
 
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 ===
=== Prime harmonics ===
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=== 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 ==
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| 2.3
| 2.3
| {{monzo| -122 77 }}
| {{monzo| -122 77 }}
| {{val| 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 }}
| {{val| 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
| {{val| 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, 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
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|-
|-
| 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
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|-
|-
| 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
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|+Table of rank-2 temperaments by generator
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator<br>(Reduced)
! Generator*
! Cents<br>(Reduced)
! Cents*
! Associated<br>Ratio
! Associated<br>Ratio*
! Temperaments
! Temperaments
|-
|-
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| [[Pental]] (5-limit)
| [[Pental]] (5-limit)
|}
|}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct


{{Todo| review }}
[[Category:Hemipental]]

Revision as of 17:17, 6 November 2023

← 384edo 385edo 386edo →
Prime factorization 5 × 7 × 11
Step size 3.11688 ¢ 
Fifth 225\385 (701.299 ¢) (→ 45\77)
Semitones (A1:m2) 35:30 (109.1 ¢ : 93.51 ¢)
Consistency limit 7
Distinct consistency limit 7

Template:EDO intro

Theory

385edo has a reasonable approximation to the 11-limit, and perhaps beyond. The equal temperament 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 supports 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, 6656/665 and 10648/10647 in the 13-limit; and 936/935, 1275/1274, 1377/1375, and 2601/2600 in the 17-limit.

Prime harmonics

Approximation of prime harmonics in 385edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -0.66 +0.18 +0.52 +0.37 +1.03 +1.02 -1.41 +1.34 -1.01 -1.14
Relative (%) +0.0 -21.1 +5.8 +16.8 +11.9 +33.1 +32.7 -45.2 +42.9 -32.3 -36.6
Steps
(reduced) 		385
(0) 		610
(225) 		894
(124) 		1081
(311) 		1332
(177) 		1425
(270) 		1574
(34) 		1635
(95) 		1742
(202) 		1870
(330) 		1907
(367)

Subsets and supersets

Since 385 factors into 5 × 7 × 11, 385edo has subset edos 5, 7, 11, 35, 55, and 77.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3 [-122 77⟩ [⟨385 610]] +0.2070 0.2071 6.64
2.3.5 [-28 25 -5⟩, [38 -2 -15⟩ [⟨385 610 894]] +0.1122 0.2158 6.92
2.3.5.7 19683/19600, 589824/588245, 703125/702464 [⟨385 610 894 1081]] +0.0374 0.2274 7.30
2.3.5.7.11 540/539, 8019/8000, 151263/151250, 172032/171875 [⟨385 610 894 1081 1332]] +0.0085 0.2114 6.78
2.3.5.7.11.13 540/539, 1575/1573, 2200/2197, 4096/4095, 8019/8000 [⟨385 610 894 1081 1332 1425]] -0.0394 0.2207 7.08
2.3.5.7.11.13.17 540/539, 936/935, 1377/1375, 1575/1573, 2200/2197, 4096/4095 [⟨385 610 894 1081 1332 1425 1574]] -0.0693 0.2171 6.97

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
Ratio* 		Temperaments
1 62\385 193.247 4096/3645 Luna
1 162/385 504.935 4/3 Countermeantone
5 160\385
(6\385) 		498.701
(18.701) 		4/3
(81/80) 		Pental (5-limit)

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct

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