Trisected

Trisected
Subgroups	2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.13
Comma basis	128/125, 1029/1000 (7-limit);; 56/55, 128/125, 1029/1000 (11-limit);; 56/55, 91/90, 128/125, 1029/1000 (13-limit)
Reduced mapping	⟨3; 3 0 -1 -1 7]
ET join	15 & 36
Generators (CWE)	~10/7 = 635.0 ¢
MOS scales	6L 9s, 15L 6s, 15L 21s
Ploidacot	triploid tricot
Pergen	(P8/3, P5/3)
Minimax error	9-odd-limit: 16.1 ¢; ; 13-limit 21-odd-limit: 17.5 ¢
Target scale size	9-odd-limit: 36 notes; ; 13-limit 21-odd-limit: 36 notes

Trisected is the rank-2 temperament tempering out 128/125, 1029/1000, and 1029/1024 in the 7-limit, making it a member of the augmented family, keegic temperaments, and gamelismic clan. Since it tempers out 128/125, the octave is split into 3 ~5/4's, each tuned to 400 ¢ if the octave is pure. Since it tempers out 1029/1024, the perfect fifth is split into three intervals of ~8/7.Since it tempers out 1029/1000, the tritave is split into three intervals of 10/7. This means that every Pythagorean interval is split into three equal parts.

In the 11-limit, the perfect fourth is split into three ~11/10's, thus tempering out 4000/3993. Additionally, the 1/3-octave period represents 14/11, tempering out 56/55 and 176/175. The 13-limit extension equates the ~10/7 with 13/9, tempering out 91/90 and 2197/2187.

The 2.3.7.11/5 subgroup restriction, known as trisect, removes the individual mappings for 5 and 11 while still tempering out 1029/1024 and 4000/3993, and is much more accurate.

For technical data, see Augmented family #Trisected.

Intervals

In the following table, odd harmonics 1–21 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	235.0	8/7	635.0	10/7, 16/11, 13/9	1035.0	20/11
2	470.0	21/16	870.0	33/20	70.0	21/20, 33/32
3	705.0	3/2	1105.0	15/8, 21/11	305.0	6/5
4	940.0	12/7, 26/15	140.0	12/11, 13/12, 15/14	540.0	15/11
5	1175.0	63/32	375.0	26/21	775.0	52/33
6	209.9	9/8	609.9	45/32, 63/44	1009.9	9/5
7	444.9	9/7, 13/10	844.9	18/11, 13/8	44.9	36/35
8	679.9	52/35, 72/49	1079.9	13/7	279.9	13/11

* In 13-limit CWE tuning, octave reduced

Tunings

Norm-based tunings

7-limit norm-based tunings
	Euclidean
	Constrained	Constrained & skewed	Destretched
Tenney	CTE: ~10/7 = 633.889 ¢	CWE: ~10/7 = 634.339 ¢	POTE: ~10/7 = 634.476 ¢

11-limit norm-based tunings
	Euclidean
	Constrained	Constrained & skewed	Destretched
Tenney	CTE: ~10/7 = 634.215 ¢	CWE: ~10/7 = 634.769 ¢	POTE: ~10/7 = 634.893 ¢

13-limit norm-based tunings
	Euclidean
	Constrained	Constrained & skewed	Destretched
Tenney	CTE: ~10/7 = 634.286 ¢	CWE: ~10/7 = 634.991 ¢	POTE: ~10/7 = 635.144 ¢

Target tunings

Todo: complete section

Tuning spectrum

Todo: complete section