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== 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. The equal temperament [[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. It [[support]]s [[qak]] and [[octowerck]].  


=== Odd harmonics ===
=== Odd harmonics ===
Line 13: Line 13:
== 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|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|-621 392}}
| {{monzo| -621 392 }}
|{{mapping|392 621}}
| {{mapping| 392 621 }}
| 0.2948
| 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
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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
|-
|-
|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>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

Revision as of 09:28, 1 January 2024

← 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

Template:EDO intro

Theory

392et is consistent to the 7-odd-limit with a flat tendency in the prime harmonics. The equal temperament 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. 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 it is distinct

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