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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 ==
== Interval chain ==
Line 26: Line 43:
| 0
| 0
| 0.0
| 0.0
| 1/1
| '''1/1'''
| 400.0
| 400.0
| '''5/4''', 14/11
| '''5/4''', 14/11
Line 33: Line 50:
|-
|-
| 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 ==
=== Scala files ===
* [[12-27]] – 27edo tuning


== 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"
|+ 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 ===
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
Line 88: Line 155:
| -1/3 comma
| -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
| 705.882
|  
| 51cdeee val
|-
|-
|  
|  
Line 176: Line 243:
|  
|  
| 715.789
| 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
* [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]]

Latest revision as of 10:18, 9 February 2026

Augmented
Subgroups 2.3.5, 2.3.5.7, 2.3.5.7.11
Comma basis 128/125 (5-limit);
64/63, 126/125 (7-limit);
56/55, 64/63, 100/99 (11-limit)
Reduced mapping ⟨3; 1 0 -2 -2]
ET join 12 & 15
Generators (CWE) ~3/2 = 711.6 ¢
MOS scales 3L 3s, 3L 6s, 3L 9s, 12L 3s
Ploidacot triploid monocot
Pergen (P8/3, P5)
Minimax error 7-odd-limit: 13.7 ¢;
11-odd-limit: 22.4 ¢
Target scale size 7-odd-limit: 12 notes;
11-odd-limit: 15 notes

Augmented is a temperament that sets 5/4 to one third of an octave, tempering out the diesis, 128/125, and has a generator of a 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.

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 augmented mos scales 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 #Septimal augmented (augene) for technical data. See Augmented extensions for a discussion on 13-limit extensions.

Interval chain

In the following table, odd harmonics 1–11 are in bold.

# Period 0 Period 1 Period 2
Cents* Approx. ratios Cents* Approx. ratios Cents* Approx. ratios
0 0.0 1/1 400.0 5/4, 14/11 800.0 8/5, 11/7
1 711.6 3/2 1111.6 15/8, 40/21 311.6 6/5
2 223.2 8/7, 9/8 623.2 10/7, 16/11 1023.2 9/5, 20/11
3 934.8 12/7 134.8 12/11, 15/14 534.8 15/11
4 446.4 9/7 846.4 18/11 46.4 45/44
5 1158.0 27/14, 96/49 358.0 27/22 758.0 54/35

* In 11-limit CWE tuning, octave reduced

Scales

Scala files

Tunings

5-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~3/2 = 701.9550 ¢ CWE: ~3/2 = 705.0691 ¢ POTE: ~3/2 = 706.6376 ¢
7-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~3/2 = 709.5949 ¢ CWE: ~3/2 = 709.3249 ¢ POTE: ~3/2 = 709.2568 ¢
11-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~3/2 = 713.5701 ¢ CWE: ~3/2 = 711.6031 ¢ POTE: ~3/2 = 711.1766 ¢

Tuning spectrum

Edo
generator 		Eigenmonzo
(unchanged-interval)* 		Generator (¢) Comments
15/8 688.269 -1/3 comma
7\12 700.000 Lower bound of 7- to 11-odd-limit diamond monotone
3/2 701.955 Untempered
30\51 705.882 51cdeee val
15/14 706.481
23\39 707.692 39dee val
7/5 708.744
9/7 708.771
9/5 708.798 1/6 comma, 9-odd-limit minimax
39\66 709.091 66cdeee val
7/6 711.043
16\27 711.111 27e val
15/11 712.317
41\69 713.043 69bcee val
11/9 713.148 11-odd-limit minimax
25\42 714.286 42e val
7/4 715.587 7-odd-limit minimax
5/3 715.641 1/3 comma, 5-odd-limit minimax
34\57 715.789 57bce val
11/6 716.879
11/10 717.498
9\15 720.000 Upper bound of 7- to 11-odd-limit diamond monotone
11/8 724.341

* Besides the octave

Music

Igliashon Jones
Joel Grant Taylor
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