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Augmented (temperament): Difference between revisions

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Rework for augmented as a single 5- to 11-limit temp
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{{Infobox Regtemp
{{Infobox Regtemp
| Title = Augene
| Title = Augmented
| Subgroups = 2.3.5.7
| Subgroups = 2.3.5, 2.3.5.7, 2.3.5.7.11
| Comma basis = [[64/63]], [[126/125]]
| 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
| Mapping = 3; 1 0 -2 -2
| Generator = 3/2
| Generator = 3/2
| Generator tuning = 709.3
| Generator 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]]
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| Odd limit 2 = (7-limit) 21 | Mistuning 2 = 20.5 | Complexity 2 = 15
| Odd limit 2 = (7-limit) 21 | Mistuning 2 = 20.5 | Complexity 2 = 15
}}
}}
'''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, ….
'''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-odd-limit|5-]] ([[128/125]]) and [[7-odd-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.  


See [[Augmented family #Augene]] for technical data. See [[Augene extensions]] for a discussion on 13-limit extensions.
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{{Clear}}
== Interval chain ==
== Interval chain ==
In the following table, odd harmonics 1–11 are in '''bold'''.  
In the following table, odd harmonics 1–11 are in '''bold'''.  
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