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-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]].
-See [[Augmented family #Augene]] for technical data. See [[Augene extensions]] for a discussion on 13-limit extensions.
+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 ==
@@ Line 26: / Line 43: @@
 | 0
 | 0.0
-| 1/1
+| '''1/1'''
 | 400.0
 | '''5/4''', 14/11
@@ Line 33: / Line 50: @@
 |-
 | 1
-| 1113.1
+| 711.6
+| '''3/2'''
+| 1111.6
 | 15/8, 40/21
-| 313.1
+| 311.6
 | 6/5
-| 713.1
-| '''3/2'''
 |-
 | 2
-| 1026.3
+| 223.2
-| 9/5, 20/11
-| 226.3
 | '''8/7''', '''9/8'''
-| 626.3
+| 623.2
 | 10/7, '''16/11'''
+| 1023.2
+| 9/5, 20/11
 |-
 | 3
-| 939.4
+| 934.8
 | 12/7
-| 139.4
+| 134.8
 | 12/11, 15/14
-| 539.4
+| 534.8
 | 15/11
 |-
 | 4
-| 852.6
+| 446.4
+| 9/7
+| 846.4
 | 18/11
-| 52.6
+| 46.4
 | 45/44
-| 452.6
-| 9/7
 |-
 | 5
-| 765.7
+| 1158.0
-| 54/35
-| 1165.7
 | 27/14, 96/49
-| 365.7
+| 358.0
 | 27/22
+| 758.0
+| 54/35
 |}
-<nowiki/>* In 11-odd-limit minimax tuning, octave reduced
+<nowiki/>* In 11-limit CWE tuning, octave reduced
+== 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"
@@ Line 88: / Line 155: @@
 | -1/3 comma
 |-
-| 7\12
+| '''7\12'''
-|
-| 700.000
 |
+| '''700.000'''
+| '''Lower bound of 7- to 11-odd-limit diamond monotone'''
 |-
 |
@@ Line 101: / Line 168: @@
 |
 | 705.882
-|
+| 51cdeee val
 |-
 |
@@ Line 176: / Line 243: @@
 |
 | 715.789
-| 53bce val
+| 57bce val
 |-
 |
@@ Line 188: / Line 255: @@
 |
 |-
-| 9\15
+| '''9\15'''
-|
-| 720.000
 |
+| '''720.000'''
+| '''Upper bound of 7- to 11-odd-limit diamond monotone'''
 |-
 |
@@ Line 207: / Line 274: @@
 * [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:Augmented family]]
 [[Category:Archytas clan]]
 [[Category:Starling temperaments]]