385edo: Difference between revisions

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== 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 ==
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! rowspan="2" | [[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
|-
|-
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|-
|-
| 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
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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
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|+ style="font-size: 105%;" | 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 (temperament)|Pental]] (5-limit)
| [[Quintile]] (5-limit)
|}
|}
<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
<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]]


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

Revision as of 05:51, 4 September 2026

← 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

385 equal divisions of the octave (abbreviated 385edo or 385ed2), also called 385-tone equal temperament (385tet) or 385 equal temperament (385et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 385 equal parts of about 3.12 ¢ each. Each step represents a frequency ratio of 21/385, or the 385th root of 2.

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, 6656/6655, 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\385 193.25 262144/234375 Luna
1 162/385 504.94 4/3 Countermeantone
5 3\385 9.35 176/175 Hemiquintile
5 6\385 18.70 81/80 Quintile (5-limit)

* In minimal-generator form

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