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). Another point that must be noted is the tetracatakleismic on 3.4.5.14.

	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 (9L 6s⟨3/1⟩, sLsLLsLsLLsLsLL)
• 2\3edt-equivalent Muddles:
  • sLsLLsLsLL parent with 2112121 target gives LsmLmLm minor? heptatonic
  • sLLsLsLLsL parent with 2112121 target gives LmsLmLm major? heptatonic

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
	11/4	875.659
	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
	15/13	881.726
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