Dominant (temperament): Difference between revisions
m I've consistently put breadcrumbs on *all* pages describing extensions, e.g. "luna and hemithirds" < "didacus" and "rodan" < "slendric" because I deem this to be helpful Tag: Undo |
Verbally note its relation to meantone and superpyth/archy cuz we're documenting this temp as a coherent 7-limit structure |
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'''Dominant''' is a [[regular temperament|temperament]] which is an [[extension]] of both [[meantone]] and [[archy]]. It is defined by [[tempering out]] the [[81/80|syntonic comma (81/80)]] and [[64/63|septimal comma (64/63)]] in the 7-limit. It also tempers out the [[36/35|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]]. | |||
'''Dominant''' is a [[regular temperament|temperament]] [[tempering out]] the [[81/80|syntonic comma (81/80)]] and [[64/63|septimal comma (64/63)]] in the 7-limit. It also tempers out the [[36/35|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]]. | |||
Other possible tunings include [[17edo]] (17c val), [[29edo]] (29cd val), [[41edo]] (41cd val), [[53edo]] (53cdd val), as well as [[Pythagorean tuning]]. | Other possible tunings include [[17edo]] (17c val), [[29edo]] (29cd val), [[41edo]] (41cd val), [[53edo]] (53cdd val), as well as [[Pythagorean tuning]]. | ||
Revision as of 13:56, 21 April 2025
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.
Other possible tunings include 17edo (17c val), 29edo (29cd val), 41edo (41cd val), 53edo (53cdd val), as well as Pythagorean tuning.
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