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Augmented (temperament): Difference between revisions

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{{Infobox Regtemp
{{Infobox regtemp
| Title = Augmented
| Title = Augmented
| Subgroups = 2.3.5, 2.3.5.7, 2.3.5.7.11
| Subgroups = 2.3.5, 2.3.5.7, 2.3.5.7.11
| Comma basis = [[128/125]] (5-limit);<br>[[64/63]], [[126/125]] (7-limit);<br>[[56/55]], [[64/63]], [[100/99]] (11-limit)
| Comma basis = [[128/125]] (5-limit); <br>[[64/63]], [[126/125]] (7-limit); <br>[[56/55]], [[64/63]], [[100/99]] (11-limit)
| Edo join 1 = 12 | Edo join 2 = 15
| Edo join 1 = 12 | Edo join 2 = 15
| Mapping = 3; 1 0 -2 -2
| Mapping = 3; 1 0 -2 -2
| Generator = 3/2
| Generators = 3/2
| Generator tuning = 711.6
| Generators tuning = 711.6
| Optimization method = CWE
| Optimization method = CWE
| MOS scales = [[3L 3s]], [[3L 6s]], [[3L 9s]], [[12L 3s]]
| MOS scales = [[3L 3s]], [[3L 6s]], [[3L 9s]], [[12L 3s]]
| Pergen = (P8/3, P5)
| Pergen = (P8/3, P5)
| Odd limit 1 = 7 | Mistuning 1 = 13.7 | Complexity 1 = 12
| Odd limit 1 = 7 | Mistuning 1 = 13.7 | Complexity 1 = 12
| Odd limit 2 = (7-limit) 21 | Mistuning 2 = 20.5 | Complexity 2 = 15
| Odd limit 2 = 11 | Mistuning 2 = 22.4 | Complexity 2 = 15
}}
}}
'''Augmented''' is a [[regular temperament|temperament]] that sets [[5/4]] to one third of an [[2/1|octave]], [[tempering out]] the diesis, [[128/125]], and has a [[generator]] of a [[3/2|perfect fifth]]. The fifth can be tuned to [[12edo]], but a sharper tuning is often perferred for it to blend with the sharp 5/4. This gives rise to the natural [[7-limit]] [[extension]] (known as '''augene''') that tempers out [[64/63]] and [[126/125]], where the whole tone stands in for [[8/7]][[~]][[9/8]].  
'''Augmented''' is a [[regular temperament|temperament]] that sets [[5/4]] to one third of an [[2/1|octave]], [[tempering out]] the diesis, [[128/125]], and has a [[generator]] of a [[3/2|perfect fifth]]. The fifth can be tuned to [[12edo]], but a sharper tuning is often perferred for it to blend with the sharp 5/4. This gives rise to the natural [[7-limit]] [[extension]] (known as '''augene''') that tempers out [[64/63]] and [[126/125]], where the whole tone stands in for [[8/7]][[~]][[9/8]].  
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| 0
| 0
| 0.0
| 0.0
| 1/1
| '''1/1'''
| 400.0
| 400.0
| '''5/4''', 14/11
| '''5/4''', 14/11
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|-
|-
| 1
| 1
| 1113.1
| 711.6
| '''3/2'''
| 1111.6
| 15/8, 40/21
| 15/8, 40/21
| 313.1
| 311.6
| 6/5
| 6/5
| 713.1
| '''3/2'''
|-  
|-  
| 2
| 2
| 1026.3
| 223.2
| 9/5, 20/11
| 226.3
| '''8/7''', '''9/8'''
| '''8/7''', '''9/8'''
| 626.3
| 623.2
| 10/7, '''16/11'''
| 10/7, '''16/11'''
| 1023.2
| 9/5, 20/11
|-
|-
| 3
| 3
| 939.4
| 934.8
| 12/7
| 12/7
| 139.4
| 134.8
| 12/11, 15/14
| 12/11, 15/14
| 539.4
| 534.8
| 15/11
| 15/11
|-
|-
| 4
| 4
| 852.6
| 446.4
| 9/7
| 846.4
| 18/11
| 18/11
| 52.6
| 46.4
| 45/44
| 45/44
| 452.6
| 9/7
|-
|-
| 5
| 5
| 765.7
| 1158.0
| 54/35
| 1165.7
| 27/14, 96/49
| 27/14, 96/49
| 365.7
| 358.0
| 27/22
| 27/22
| 758.0
| 54/35
|}
|}
<nowiki/>* In 11-odd-limit minimax tuning, octave reduced
<nowiki/>* In 11-limit CWE tuning, octave reduced


== Scales ==
== Scales ==
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== Tunings ==
== Tunings ==
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 5-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~3/2 = 701.9550{{c}}
| CWE: ~3/2 = 705.0691{{c}}
| POTE: ~3/2 = 706.6376{{c}}
|}
{| class="wikitable mw-collapsible mw-collapsed"
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
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|  
|  
| 715.789
| 715.789
| 53bce val
| 57bce val
|-
|-
|  
|  
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