Father
- This page is about the regular temperament. For the scale structure sometimes associated with it, see 5L 3s.
| Father |
16/15, 28/27 (7-limit)
7-odd-limit: 68.1 ¢
7-odd-limit: 5 notes
Father is a very coarse, simplistic, and inaccurate exotemperament. It tempers out 16/15, the classical diatonic semitone. This means the classical major third (5/4) is conflated with the perfect fourth (4/3), making it one that challenges the very notion of JI approximation, and playing harmony in it, it sounds only remotely reminiscent of the 5-limit no matter how it is tuned. If one could get their head around this way of hearing intervals, they may as well take a look at the 7-limit interpretation, where it tempers out 28/27 and 36/35.
The main interest in this temperament is its mos scales, as antipentic (3L 2s) and oneirotonic (5L 3s) are often chosen first, and only later is each step associated with a ratio consistent with this temperament. Another potential reason to choose this temperament is to equate suspended chords and more conventional tertian chords (though options like trienstonian (4/3~9/7), blackwood (4/3~81/64), and fendo (4/3~13/10) are more accurate).
As an exotemperament, it has a large range of acceptable tunings, from roughly 3\5 (720 ¢) to 2\3 (800 ¢). However, only tunings between 3\5 and 5\8 (750 ¢) generate oneirotonic scales, which are the primary structure it represents as an exotemperament.
See Father family #Father for technical details.
Interval chain
In the following table, odd harmonics 1–9 are labeled in bold.
| # | Cents* | Approximate ratios |
|---|---|---|
| 0 | 0.0 | 1/1 |
| 1 | 738.4 | 3/2, 8/5, 14/9 |
| 2 | 276.9 | 6/5, 7/6, 9/8 |
| 3 | 1015.3 | 7/4, 9/5 |
| 4 | 553.8 | 7/5 |
| 5 | 92.2 | 21/20 |
* In 7-limit CWE tuning
Tunings
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Tenney | CTE: ~3/2 = 737.469 ¢ | CWE: ~3/2 = 742.290 ¢ | POTE: ~3/2 = 743.986 ¢ |
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Tenney | CTE: ~3/2 = 727.855 ¢ | CWE: ~3/2 = 738.443 ¢ | POTE: ~3/2 = 742.002 ¢ |
Tuning spectrum
| Edo generator |
Eigenmonzo (Unchanged-interval)* |
Generator (¢) | Comments |
|---|---|---|---|
| 1\2 | 600.0 | Lower bound of 5-odd-limit diamond monotone | |
| 3/2 | 702.0 | Pythagorean tuning | |
| 3\5 | 720.0 | Lower bound of 7-odd-limit diamond monotone 9-odd-limit diamond monotone (singleton) | |
| 7/4 | 722.9 | ||
| 7/6 | 733.4 | ||
| 8\13 | 738.5 | ||
| 9/5 | 739.2 | 1/3-comma | |
| 7/5 | 745.6 | 7-odd-limit minimax | |
| 5\8 | 750.0 | Upper bound of 7-odd-limit diamond monotone | |
| 5/3 | 757.8 | 1/2-comma, 5-odd-limit minimax, 5-limit CEE & CSEE | |
| 9/7 | 764.9 | 9-odd-limit minimax | |
| 2\3 | 800.0 | Upper bound of 5-odd-limit diamond monotone | |
| 5/4 | 813.7 | Full-comma |
* Besides the octave