Augmented (temperament): Difference between revisions

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Augene is a regular temperament of the [[Augmented_family#x-Augene|augmented family]], which means that 128/125 is tempered out and three 5/4s make a 2/1. Augene is distinguished from its relative [[august|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 MOS 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]].  


The first few augene MOSes are [[3L_3s|3L 3s]], [[3L_6s|3L 6s]], [[3L_9s|3L 9s]], [[12L_3s|12L 3s]]... and the first EDOs that reasonably support augene are [[12edo|12edo]], [[15edo|15edo]], and especially [[27edo|27edo]].
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]].  


== Tuning spectrum ==
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]].


{| class="wikitable center-all"
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 ==
In the following table, odd harmonics 1–11 are in '''bold'''.
 
{| class="wikitable center-1 right-2 right-4 right-6"
|-
|-
! | ET<br>generator
! rowspan="2" | #
! | [[eigenmonzo|eigenmonzo<br>(unchanged interval]])
! colspan="2" | Period 0
! | fifth<br>(¢)
! colspan="2" | Period 1
! | comments
! colspan="2" | Period 2
|-
|-
| |
! Cents*
| | 14/13
! Approx. ratios
| | 671.702
! Cents*
| |
! Approx. ratios
! Cents*
! Approx. ratios
|-
|-
| | 12\28
| 0
| |  
| 0.0
| | 685.714
| '''1/1'''
| |
| 400.0
| '''5/4''', 14/11
| 800.0
| '''8/5''', 11/7
|-
|-
| |  
| 1
| | 16/15
| 711.6
| | 688.269
| '''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
| | 13/11
| 934.8
| | 689.210
| 12/7
| |
| 134.8
| 12/11, 15/14
| 534.8
| 15/11
|-
|-
| | 7\12
| 4
| |  
| 446.4
| | 700.000
| 9/7
| |
| 846.4
| 18/11
| 46.4
| 45/44
|-
|-
| | 31\53
| 5
| |  
| 1158.0
| | 701.887
| 27/14, 96/49
| |  
| 358.0
| 27/22
| 758.0
| 54/35
|}
<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" |  
| | 4/3
! colspan="3" | Euclidean
| | 701.955
| |  
|-
|-
| | 30\51
! Constrained
| |
! Constrained & skewed
| | 705.882
! Destretched
| |
|-
|-
| |  
! Tenney
| | 15/14
| CTE: ~3/2 = 701.9550{{c}}
| | 706.481
| 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
|-
|-
| | 23\39
! rowspan="2" |  
| |
! colspan="3" | Euclidean
| | 707.692
| |
|-
|-
| |
! Constrained
| | 7/5
! Constrained & skewed
| | 708.744
! Destretched
| |
|-
|-
| |  
! Tenney
| | 9/7
| CTE: ~3/2 = 709.5949{{c}}
| | 708.771
| 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" |  
| | 10/9
! colspan="3" | Euclidean
| | 708.798
| | 9 limit minimax
|-
|-
| | 39\66
! Constrained
| |
! Constrained & skewed
| | 709.091
! Destretched
| |
|-
|-
| |  
! Tenney
| | 7/6
| CTE: ~3/2 = 713.5701{{c}}
| | 711.043
| CWE: ~3/2 = 711.6031{{c}}
| |  
| POTE: ~3/2 = 711.1766{{c}}
|}
 
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
|-
! Edo<br>generator
! [[Eigenmonzo|Eigenmonzo<br>(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
|  
|-
|-
| | 16\27
| 23\39
| |  
|  
| | 711.111
| 707.692
| |
| 39dee val
|-
|-
| |  
|  
| | 15/11
| 7/5
| | 712.317
| 708.744
| |  
|  
|-
|-
| | 57\96
|  
| |
| 9/7
| | 712.5
| 708.771
| |  
|  
|-
|-
| |  
|  
| | 11/9
| 9/5
| | 713.148
| 708.798
| | 11 limit minimax
| 1/6 comma, 9-odd-limit minimax
|-
|-
| | 25\42
| 39\66
| |  
|  
| | 714.286
| 709.091
| |
| 66cdeee val
|-
|-
| |  
|  
| | 8/7
| 7/6
| | 715.587
| 711.043
| | 7, 13 and 15 limit minimax
|  
|-
|-
| |
| 16\27
| | 6/5
|  
| | 715.641
| 711.111
| | 5 limit minimax
| 27e val
|-
|-
| | 34\57
|  
| |
| 15/11
| | 715.789
| 712.317
| |  
|  
|-
|-
| | 25\72
| 41\69
| |  
|  
| | 716.667
| 713.043
| |
| 69bcee val
|-
|-
| |  
|  
| | 12/11
| 11/9
| | 716.879
| 713.148
| |
| 11-odd-limit minimax
|-
|-
| |
| 25\42
| | 11/10
|  
| | 717.498
| 714.286
| |
| 42e val
|-
|-
| | 9\15
|  
| |
| 7/4
| | 720.000
| 715.587
| |
| 7-odd-limit minimax
|-
|-
| |  
|  
| | 18/13
| 5/3
| | 721.127
| 715.641
| |
| 1/3 comma, 5-odd-limit minimax
|-
|-
| |
| 34\57
| | 15/13
|  
| | 723.871
| 715.789
| |
| 57bce val
|-
|-
| |  
|  
| | 11/8
| 11/6
| | 724.341
| 716.879
| |  
|  
|-
|-
| |  
|  
| | 13/12
| 11/10
| | 730.714
| 717.498
| |  
|  
|-
|-
| |
| '''9\15'''
| | 13/10
|  
| | 745.786
| '''720.000'''
| |
| '''Upper bound of 7- to 11-odd-limit diamond monotone'''
|-
|-
| |  
|  
| | 16/13
| 11/8
| | 759.472
| 724.341
| |  
|  
|}
|}
<nowiki/>* Besides the octave
== Music ==
; [[Igliashon Jones]]
* [https://web.archive.org/web/20201127012539/http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Sad%20Like%20Winter%20Leaves.mp3 ''Sad Like Winter Leaves''] &ndash; in Augene[12] tuned to 27edo


=Music=
; [[Joel Grant Taylor]]
''[http://micro.soonlabel.com/gene_ward_smith/Others/Igs/Sad%20Like%20Winter%20Leaves.mp3 Sad Like Winter Leaves]'' by [http://soundcloud.com/cityoftheasleep/sad-like-winter-trees Igliashon Jones] 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


''[http://micro.soonlabel.com/gene_ward_smith/Others/Taylor/12of27sonatina.mp3 Galticeran Sonatina]'' by [http://soundcloud.com/joelgranttaylor/galticeran_sonatina Joel Grant Taylor] in Augene[12] tuned to 27edo
[[Category:Augmented| ]] <!-- Main article -->
[[Category:Rank-2 temperaments]]
[[Category:Augmented family]]
[[Category:Archytas clan]]
[[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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