392edo: Difference between revisions

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== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{{comma basis begin}}
! 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
| 2.3
Line 49: Line 41:
| 0.2335
| 0.2335
| 7.63
| 7.63
|}
{{comma basis end}}


=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{{rank-2 begin}}
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>Ratio*
! Temperaments
|-
|-
| 1
| 1
Line 79: Line 65:
|-
|-
| 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]] (392f)
| [[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]] (5-limit)
| [[Oquatonic]] (5-limit)
|}
{{rank-2 end}}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
{{orf}}

Revision as of 02:45, 16 November 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; 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

Template:Comma basis begin |- | 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 Template:Comma basis end

Rank-2 temperaments

Template:Rank-2 begin |- | 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) Template:Rank-2 end Template:Orf

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