Dominant (temperament)
Dominant is a temperament which is an extension of both meantone and archy. It is defined by tempering out the syntonic comma (81/80) and septimal comma (64/63) in the 7-limit. It also tempers out the septimal quartertone (36/35), as 36/35 = (64/63)(81/80). It is the unique temperament that identifies the harmonic seventh chord with the dominant seventh chord, which is a familiar feature from 12edo.
However, it is not very accurate for the same reason that 12edo is inaccurate in the 7-limit, as either 5/4 or 7/4 must be tuned very sharply (with 5/4 reaching over 462 cents in the best tuning of 7/4, and likewise 7/4 reaching over 1006 cents in the best tuning of 5/4). Thus, the "best tuning" is a compromise between the two, tuning 3/2 basically just.
Other possible tunings include 17edo (17c val), 29edo (29cd val), 41edo (41cd val), 53edo (53cdd val), as well as Pythagorean tuning.
Dominant was known as dominant seventh in 2003, but the seventh part was dropped shortly after[1][2].
See Meantone family #Dominant for technical data.
Interval chain
In the following table, odd harmonics 1–9 are in bold.
| # | Cents* | Approximate Ratios |
|---|---|---|
| 0 | 0.0 | 1/1 |
| 1 | 701.1 | 3/2 |
| 2 | 202.2 | 8/7, 9/8, 10/9 |
| 3 | 903.3 | 5/3, 12/7 |
| 4 | 404.5 | 5/4, 9/7 |
| 5 | 1105.6 | 15/8, 27/14, 40/21 |
| 6 | 606.7 | 10/7 |
| 7 | 107.8 | 15/14 |
* In 7-limit CWE tuning
Chords and harmony
Much of 12edo harmony can be used. Dominant enables chords of didymic and archytas.
Tunings
Tuning spectrum
| Edo generator |
Eigenmonzo (unchanged-interval)* |
Generator (¢) | Comments |
|---|---|---|---|
| 9/5 | 691.202 | 1/2 syntonic comma | |
| 5/3 | 694.786 | 1/3 syntonic comma | |
| 5/4 | 696.578 | 1/4 syntonic comma, 5-odd-limit minimax | |
| 15/8 | 697.654 | 1/5 syntonic comma | |
| 7\12 | 700.000 | Lower bound of 7- and 9-odd-limit diamond monotone | |
| 3/2 | 701.955 | Pythagorean tuning | |
| 15/14 | 702.778 | ||
| 7/5 | 702.915 | 7 & 9-odd limit minimax tuning | |
| 21/20 | 703.107 | ||
| 17\29 | 703.448 | 29cd val | |
| 11/10 | 703.499 | 11-odd-limit minimax tuning | |
| 13/10 | 703.522 | 13-odd-limit minimax tuning | |
| 10\17 | 705.882 | 17c val | |
| 9/7 | 708.771 | 1/4 septimal comma | |
| 7/6 | 711.043 | 1/3 septimal comma | |
| 7/4 | 715.587 | 1/2 septimal comma | |
| 3\5 | 720.000 | Upper bound of 7- and 9-odd-limit diamond monotone | |
| 21/16 | 729.219 | Full septimal comma |
* Besides the octave