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.
__TOC__
{{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'''.  

Revision as of 08:24, 28 January 2026

Lua error in Module:Infobox_regtemp at line 138: attempt to perform arithmetic on local 'generator_size' (a nil value). 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 1113.1 15/8, 40/21 313.1 6/5 713.1 3/2
2 1026.3 9/5, 20/11 226.3 8/7, 9/8 626.3 10/7, 16/11
3 939.4 12/7 139.4 12/11, 15/14 539.4 15/11
4 852.6 18/11 52.6 45/44 452.6 9/7
5 765.7 54/35 1165.7 27/14, 96/49 365.7 27/22

* In 11-odd-limit minimax tuning, octave reduced

Tunings

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 53bce 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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