Dominant (temperament): Difference between revisions
→Tunings: + basic norm-based tunings |
m Better equal temp join, Misc. cleanup |
||
| Line 3: | Line 3: | ||
| Subgroups = 2.3.5.7 | | Subgroups = 2.3.5.7 | ||
| Comma basis = [[36/35]], [[64/63]] | | Comma basis = [[36/35]], [[64/63]] | ||
| Edo join 1 = | | Edo join 1 = 12 | Edo join 2 = 17c | ||
| Generator = 3/2 | Generator tuning = 701.1 | Optimization method = CWE | | Generator = 3/2 | Generator tuning = 701.1 | Optimization method = CWE | ||
| MOS scales = [[2L 3s]], [[5L 2s]], [[5L 7s]] | | MOS scales = [[2L 3s]], [[5L 2s]], [[5L 7s]] | ||
| Line 11: | Line 11: | ||
| Odd limit 2 = (7-limit) 27 | Mistuning 2 = 26.3 | Complexity 2 = 12 | | Odd limit 2 = (7-limit) 27 | Mistuning 2 = 26.3 | Complexity 2 = 12 | ||
}} | }} | ||
'''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]] 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]]. | ||
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 | 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. | ||
The most obvious extension to the 11 and 13-limit is treating the major and minor thirds as [[14/11]] and [[13/11]] as well as 5/4 and 6/5, tempering out [[56/55]] and [[66/65]]. This favors even sharper fifths on the edge of the [[gentle region]]. 29edo tunes this about as well as possible, albeit using the second best approximation of most harmonics. | The most obvious extension to the 11- and 13-limit is treating the major and minor thirds as [[14/11]] and [[13/11]] as well as 5/4 and 6/5, tempering out [[56/55]] and [[66/65]]. This favors even sharper fifths on the edge of the [[gentle region]]. [[29edo]] tunes this about as well as possible, albeit using the second best approximation of most harmonics. | ||
Other possible tunings include [[17edo]] (17c val), [[41edo]] (41cd val), [[53edo]] (53cdd val), as well as [[Pythagorean tuning]]. | Other possible tunings include [[17edo]] (17c val), [[41edo]] (41cd val), [[53edo]] (53cdd val), as well as [[Pythagorean tuning]]. | ||
| Line 46: | Line 46: | ||
| 7 || 107.8 || 15/14 | | 7 || 107.8 || 15/14 | ||
|} | |} | ||
<nowiki />* In 7-limit [[CWE]] tuning | <nowiki/>* In 7-limit [[CWE]] tuning | ||
== Chords and harmony == | == Chords and harmony == | ||
| Line 114: | Line 114: | ||
| [[7/5]] | | [[7/5]] | ||
| 702.915 | | 702.915 | ||
| 7 & 9-odd limit minimax tuning | | 7- & 9-odd-limit minimax tuning | ||
|- | |- | ||
| | | | ||
| Line 166: | Line 166: | ||
| Full septimal comma | | Full septimal comma | ||
|} | |} | ||
<nowiki />* Besides the octave | <nowiki/>* Besides the octave | ||
== References == | == References == | ||
Revision as of 12:56, 21 December 2025
Lua error in Module:Infobox_regtemp at line 138: attempt to perform arithmetic on local 'generator_size' (a nil value). 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.
The most obvious extension to the 11- and 13-limit is treating the major and minor thirds as 14/11 and 13/11 as well as 5/4 and 6/5, tempering out 56/55 and 66/65. This favors even sharper fifths on the edge of the gentle region. 29edo tunes this about as well as possible, albeit using the second best approximation of most harmonics.
Other possible tunings include 17edo (17c 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
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Tenney | CTE: ~3/2 = 699.6218 ¢ | CWE: ~3/2 = 701.1125 ¢ | POTE: ~3/2 = 701.5732 ¢ |
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 | 29cdef 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