Tetrahanson

The tetrahanson temperament is a nonoctave kleismic temperament, tempering out the kleisma in the 4.3.5 subgroup and repeating at the double octave 4/1. It is generated by 5/3 and, like in normal hanson temperament, 6 of them make a 4/3. Tetrahanson does not contain any 5-limit major or minor triads, but it does have different voicings of them (3:4:5 and 12:15:20), which, to a 12edo-accustomed listener, can make it sound like the root is the real root and the perfect fifth above it at the same time.

For technical information see Subgroup temperaments#Tetrahanson.

Interval chain

Generators	Cents (CTE)	Approximate ratios
-7	1019.413	9/5
-6	1902.354	3/1
-5	385.295	5/4
-4	1268.236	25/12
-3	2151.177	125/36
-2	634.118	36/25
-1	1517.059	12/5
0	0.000	1/1
1	882.941	5/3
2	1765.882	25/9
3	248.823	144/125
4	1131.764	48/25
5	2014.705	16/5
6	497.646	4/3
7	1380.587	20/9

Tetrahanson on tritave

In tritave-repeating tetrahanson (3.4.5 subgroup), 36/25 actually represents 1\3edt, which makes the 3rd-tritave period.

b39 & b15

This is restriction of catalan extension. This can maintain the structure of the 3rd-octave period in 3.4.5, 3.5.11, and 3.5.13. Well, 3.4.5 and 3.5.13, which do not include 7 nor 11 in their basis, should simply be called tetrahanson (and no-twos cata, respectively).

	3.4.5	3.5.11	3.5.13
CWE	883.071	879.416	883.808
Badness (Dirichlet)	0.155	2.708	0.069
	3.4.5.11	3.4.5.13	3.5.11.13
CWE	880.672	882.854	879.352
Badness (Dirichlet)	0.384	0.066	0.44

Interval chain

#	Period 0		Period 1		Period 2
	Cents*	Approximate ratios	Cents*	Approximate ratios	Cents*	Approximate ratios
-1	1656.6	125/48, 13/5	388.6	5/4	1022.6	9/5
0	0.0	1/1	634.0	36/25, 13/9	1268.0	25/12, 27/13, 52/25
1	245.3	125/108, 15/13	879.3	5/3, 33/20	1513.3	12/5
2	490.7	4/3, 33/25	1124.7	48/25, 25/13	1758.7	25/9, 36/13, 11/4
3	736.0	20/13, 55/36	1370.0	20/9, 11/5	102.1	16/15, 55/52
4	981.4	16/9, 44/25	1615.4	64/25, 33/13	347.4	100/81, 16/13, 11/9

* In 3.5.11.13-subgroup 13-throdd-limit minimax tuning

• [1] – 15-note scale (Template:Sl, sLsLLsLsLLsLsLL)

Chords

Mossteps	Tonic and then every +2 mossteps
0-3-6	36:44:55 → 36:45:55 → 16:20:25 ↩↩
0-3-7	9:11:15 → 12:15:20 ↩↩ → 16:20:27
0-4-7	3:4:5 ↩↩ → 20:27:33 → 25:44:60
0-5-7	9:13:15 ↩↩↩ → 44:64:75
0-6-10	13:20:27 ↩ → 16:25:33 ↩↩

Tuning spectrum

ET generator	Eigenmonzo (unchanged-interval)	Generator (¢)	Comments
	11/5	877.658
18\39edt		877.825
	33/13	878.675
	144/125	878.954
	11/9	879.333	3.5.11.13-subgroup 13-throdd-limit minimax
43\93edt		879.399
	44/15	879.798
25\54edt		880.534
	5/4	881.656
	36/13	881.691
32\69edt		882.066
	16/13	882.349
	16/15	882.557
	4/3	883.007	3.4.5-subgroup 5-throdd-limit minimax
	5/3	884.359
	125/108	887.061
7\15edt		887.579