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 |
| # | 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
| 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 ↩↩ |
| 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 |