392edo: 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}}
{{EDO intro|392}}
{{ED intro}}


== Theory ==
== Theory ==
392et is consistent to the [[7-odd-limit]], although it has a flat tendency in the prime harmonics. It tempers out the parakleisma in the 5-limit; 184528125/184473632, [[703125/702464]], 95703125/95551488 and 420175/419904 in the 7-limit. 392edo supports [[misneb]].  
392et is [[consistent]] to the [[7-odd-limit]] with a flat tendency in the [[prime harmonic]]s. As an equal temperament, it [[tempering out|tempers out]] the [[parakleisma]] in the 5-limit; 321489/320000 ([[varunisma]]), 420175/419904 ([[wizma]]), 703125/702464 ([[meter]]), and 823543/819200 ([[quince comma]]) in the 7-limit; and [[441/440]], [[8019/8000]], [[9801/9800]], and [[41503/41472]] in the 11-limit. It [[support]]s [[qak]] and [[octowerck]].  


=== Odd harmonics ===
=== Odd harmonics ===
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=== Subsets and supersets ===
=== Subsets and supersets ===
392 factors into 2<sup>3</sup> × 7<sup>2</sup>, with subset edos {{EDOs|2, 4, 7, 8, 14, 28, 49, 56, 98, and 196}}.
392 factors into 2<sup>3</sup> × 7<sup>2</sup>, with subset edos {{EDOs| 2, 4, 7, 8, 14, 28, 49, 56, 98, and 196 }}.


== 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|-621 392}}
! rowspan="2" | [[Comma list]]
|{{mapping|392 621}}
! rowspan="2" | [[Mapping]]
| 0.2948
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -621 392 }}
| {{mapping| 392 621 }}
| +0.2948
| 0.2949
| 0.2949
| 9.63
| 9.63
|-
|-
|2.3.5
| 2.3.5
|{{monzo|8 14 -13}}, {{monzo|-49 28 2}}
| {{monzo| 8 14 -13 }}, {{monzo| -49 28 2 }}
|{{mapping|392 621 910}}
| {{mapping| 392 621 910 }}
| 0.2826
| +0.2826
| 0.2414
| 0.2414
| 7.89
| 7.89
|-
|-
|2.3.5.7
| 2.3.5.7
|321489/320000, 420175/419904, 703125/702464
| 321489/320000, 420175/419904, 703125/702464
|{{mapping|392 621 910 1100}}
| {{mapping| 392 621 910 1100 }}
| 0.3437
| +0.3437
| 0.2343
| 0.2343
| 7.65
| 7.65
|-
|-
|2.3.5.7.11
| 2.3.5.7.11
|441/440, 8019/8000, 41503/41472, 1265625/1261568
| 441/440, 8019/8000, 41503/41472, 703125/702464
|{{mapping|392 621 910 1100 1356}}
| {{mapping| 392 621 910 1100 1356 }}
| 0.2922
| +0.2922
| 0.2335
| 0.2335
| 7.63
| 7.63
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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
|-
! Generator<br>(reduced)*
! Periods<br />per 8ve
! Cents<br>(reduced)*
! Generator*
! Associated<br>Ratio*
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
|1
| 1
|37\392
| 37\392
|113.27
| 113.27
|16/15
| 16/15
|[[Misneb]]
| [[Misneb]] (5-limit)
|-
|-
|1
| 1
|103\392
| 103\392
|315.31
| 315.31
|6/5
| 6/5
|[[Parakleismic]]
| [[Parakleismic]] (5-limit)
|-
|-
|1
| 1
|149\392
| 149\392
|456.12
| 456.12
|125/96
| 125/96
|[[Qak]]
| [[Qak]]
|-
|-
|8
| 8
|185\392<br>(11\392)
| 185\392<br />(11\392)
|566.33<br>(33.67)
| 566.33<br />(33.67)
|104/75<br>(55/54)
| 104/75<br />(55/54)
|[[Octowerck]]
| [[Octowerck]] (392f)
|-
|-
|28
| 28
|163\392<br>(5\392)
| 163\392<br />(5\392)
|498.98<br>(15.31)
| 498.98<br />(15.31)
|4/3<br>(105/104)
| 4/3<br />(105/104)
|[[Oquatonic]]
| [[Oquatonic]] (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

Latest revision as of 13:32, 13 March 2026

← 391edo 392edo 393edo →
Prime factorization 23 × 72
Step size 3.06122 ¢ 
Fifth 229\392 (701.02 ¢)
Semitones (A1:m2) 35:31 (107.1 ¢ : 94.9 ¢)
Consistency limit 7
Distinct consistency limit 7

392 equal divisions of the octave (abbreviated 392edo or 392ed2), also called 392-tone equal temperament (392tet) or 392 equal temperament (392et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 392 equal parts of about 3.06 ¢ each. Each step represents a frequency ratio of 21/392, or the 392nd root of 2.

Theory

392et is consistent to the 7-odd-limit with a flat tendency in the prime harmonics. As an equal temperament, it tempers out the parakleisma in the 5-limit; 321489/320000 (varunisma), 420175/419904 (wizma), 703125/702464 (meter), and 823543/819200 (quince comma) in the 7-limit; and 441/440, 8019/8000, 9801/9800, and 41503/41472 in the 11-limit. It supports qak and octowerck.

Odd harmonics

Approximation of odd harmonics in 392edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -0.93 -0.60 -1.48 +1.19 -0.30 +1.31 +1.53 -0.87 -0.57 +0.65 -0.72
Relative (%) -30.5 -19.6 -48.3 +38.9 -9.7 +42.8 +49.9 -28.5 -18.8 +21.2 -23.6
Steps
(reduced) 		621
(229) 		910
(126) 		1100
(316) 		1243
(67) 		1356
(180) 		1451
(275) 		1532
(356) 		1602
(34) 		1665
(97) 		1722
(154) 		1773
(205)

Subsets and supersets

392 factors into 23 × 72, with subset edos 2, 4, 7, 8, 14, 28, 49, 56, 98, and 196.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-621 392⟩ [⟨392 621]] +0.2948 0.2949 9.63
2.3.5 [8 14 -13⟩, [-49 28 2⟩ [⟨392 621 910]] +0.2826 0.2414 7.89
2.3.5.7 321489/320000, 420175/419904, 703125/702464 [⟨392 621 910 1100]] +0.3437 0.2343 7.65
2.3.5.7.11 441/440, 8019/8000, 41503/41472, 703125/702464 [⟨392 621 910 1100 1356]] +0.2922 0.2335 7.63

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 37\392 113.27 16/15 Misneb (5-limit)
1 103\392 315.31 6/5 Parakleismic (5-limit)
1 149\392 456.12 125/96 Qak
8 185\392
(11\392) 		566.33
(33.67) 		104/75
(55/54) 		Octowerck (392f)
28 163\392
(5\392) 		498.98
(15.31) 		4/3
(105/104) 		Oquatonic (5-limit)

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

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