Augmented (temperament)

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

12-27 – 27edo tuning

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

Sad Like Winter Leaves – in Augene[12] tuned to 27edo

Joel Grant Taylor

Galticeran Sonatina – in Augene[12] tuned to 27edo