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Augmented (temperament): Difference between revisions

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'''Augene''' is a [[regular temperament]] of the [[augmented family]], which means that [[128/125]] is [[tempering out|tempered out]] and three [[5/4]]'s make a [[2/1]]. Augene is distinguished from its relative [[august]] by tempering out [[64/63]], which means that tunings in which the fifth is larger than 7\12 (700 cents) are optimal. Therefore the [[optimal ET sequence]] of augene goes 12, 15, 27, … in constrast to august's which goes 9, 12, 21, ….
{{Infobox regtemp
| Title = Augmented
| 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)
| Edo join 1 = 12 | Edo join 2 = 15
| Mapping = 3; 1 0 -2 -2
| Generators = 3/2
| Generators tuning = 711.6
| Optimization method = CWE
| MOS scales = [[3L 3s]], [[3L 6s]], [[3L 9s]], [[12L 3s]]
| Pergen = (P8/3, P5)
| Odd limit 1 = 7 | Mistuning 1 = 13.7 | Complexity 1 = 12
| 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]].  


It is also the unique [[7-limit]] [[regular temperament]] that tempers out the [[essential tempering commas]] of the 5 ([[128/125]])- and 7-limit ([[64/63]], [[126/125]]).
Further extension to the [[11-limit]] is available by noticing 128/125 = ([[56/55]])⋅([[176/175]]), and tempering out both commas means [[14/11]] is equated to the 1/3 octave as well. This does imply more [[damage]] in the 11-limit however, as the [[11/8]] is now conflated with [[7/5]].  


The first few augene [[mos scale]]s are [[3L 3s]], [[3L 6s]], [[3L 9s]], [[12L 3s]], … and the first edos that reasonably support augene are [[12edo]], [[15edo]], and especially [[27edo]].
The first few augmented [[mos scale]]s are [[3L 3s]], [[3L 6s]], [[3L 9s]], [[12L 3s]], … and the first edos that reasonably [[support]] it are [[12edo]], [[15edo]], and [[27edo]].
 
Alternative 7-limit extensions of augmented include [[august]] (9 & 12), which tempers out [[36/35]] and favors a flat-of-just fifth.
Therefore the [[optimal ET sequence]] of august goes 9, 12, 21, … in constrast to augmented's which goes 12, 15, 27, ….
 
See [[Augmented family #Augmented]] and [[Augmented family #Septimal augmented (augene)|#Septimal augmented (augene)]] for technical data. See [[Augmented extensions]] for a discussion on 13-limit extensions.  


== Interval chain ==
== Interval chain ==
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{| class="wikitable center-1 right-2 right-4 right-6"
{| class="wikitable center-1 right-2 right-4 right-6"
|-
|-
! rowspan="2" | &#35;
! rowspan="2" | #
! colspan="2" | Period 0
! colspan="2" | Period 0
! colspan="2" | Period 1
! colspan="2" | Period 1
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|-
|-
! Cents*
! Cents*
! Approx. Ratios
! Approx. ratios
! Cents*
! Cents*
! Approx. Ratios
! Approx. ratios
! Cents*
! Cents*
! Approx. Ratios
! Approx. ratios
|-
|-
| 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
<nowiki/>* In 11-limit CWE tuning, octave reduced


== Tuning spectrum ==
== Scales ==
=== Scala files ===
* [[12-27]] – 27edo tuning


== 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"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~3/2 = 709.5949{{c}}
| CWE: ~3/2 = 709.3249{{c}}
| POTE: ~3/2 = 709.2568{{c}}
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~3/2 = 713.5701{{c}}
| CWE: ~3/2 = 711.6031{{c}}
| POTE: ~3/2 = 711.1766{{c}}
|}
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
|-
|-
! Edo<br />Generator
! Edo<br>generator
! [[Eigenmonzo|Eigenmonzo<br />(Unchanged-interval)]]*
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]*
! Fifth (¢)
! Generator (¢)
! Comments
! Comments
|-
|
| 13/7
| 671.702
|
|-
|-
|  
|  
| 15/8
| 15/8
| 688.269
| 688.269
|
| -1/3 comma
|-
|
| 13/11
| 689.210
|
|-
|-
| 7\12
| '''7\12'''
|
| 700.000
|  
|  
| '''700.000'''
| '''Lower bound of 7- to 11-odd-limit diamond monotone'''
|-
|-
|  
|  
| 3/2
| 3/2
| 701.955
| 701.955
|  
| Untempered
|-
|-
| 30\51
| 30\51
|  
|  
| 705.882
| 705.882
|  
| 51cdeee val
|-
|-
|  
|  
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|  
|  
| 707.692
| 707.692
|  
| 39dee val
|-
|-
|  
|  
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| 9/5
| 9/5
| 708.798
| 708.798
| 9-odd-limit minimax
| 1/6 comma, 9-odd-limit minimax
|-
|-
| 39\66
| 39\66
|  
|  
| 709.091
| 709.091
|  
| 66cdeee val
|-
|-
|  
|  
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|  
|  
| 711.111
| 711.111
|  
| 27e val
|-
|-
|  
|  
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|  
|  
|-
|-
| 57\96
| 41\69
|
| 712.500
|  
|  
| 713.043
| 69bcee val
|-
|-
|  
|  
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|  
|  
| 714.286
| 714.286
|  
| 42e val
|-
|-
|  
|  
| 7/4
| 7/4
| 715.587
| 715.587
| 7-, 13- and 15-odd-limit minimax
| 7-odd-limit minimax
|-
|-
|  
|  
| 5/3
| 5/3
| 715.641
| 715.641
| 5-odd-limit minimax
| 1/3 comma, 5-odd-limit minimax
|-
|-
| 34\57
| 34\57
|  
|  
| 715.789
| 715.789
|  
| 57bce val
|-
| 25\72
|
| 716.667
|
|-
|-
|  
|  
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|  
|  
|-
|-
| 9\15
| '''9\15'''
|
| 720.000
|
|-
|
| 13/9
| 721.127
|
|-
|
| 15/13
| 723.871
|  
|  
| '''720.000'''
| '''Upper bound of 7- to 11-odd-limit diamond monotone'''
|-
|-
|  
|  
| 11/8
| 11/8
| 724.341
| 724.341
|
|-
|
| 13/12
| 730.714
|
|-
|
| 13/10
| 745.786
|
|-
|
| 13/8
| 759.472
|  
|  
|}
|}
<nowiki />* Besides the octave
<nowiki/>* Besides the octave


== Music ==
== Music ==
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* [https://web.archive.org/web/20201127012922/http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of27sonatina.mp3 ''Galticeran Sonatina''] &ndash; in Augene[12] tuned to 27edo
* [https://web.archive.org/web/20201127012922/http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of27sonatina.mp3 ''Galticeran Sonatina''] &ndash; in Augene[12] tuned to 27edo


[[Category:Augene| ]] <!-- Main article -->
[[Category:Augmented| ]] <!-- Main article -->
[[Category:Rank-2 temperaments]]
[[Category:Rank-2 temperaments]]
[[Category:Augmented family]]
[[Category:Augmented family]]
[[Category:Archytas clan]]
[[Category:Archytas clan]]
[[Category:Starling temperaments]]
[[Category:Starling temperaments]]
Retrieved from "https://en.xen.wiki/w/Augmented_(temperament)"