Dominant (temperament): Difference between revisions

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'''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]].  
{{Infobox regtemp
| Title = Dominant
| Subgroups = 2.3.5.7
| Comma basis = [[36/35]], [[64/63]]
| Edo join 1 = 12 | Edo join 2 = 17c
| Generators = 3/2 | Generators tuning = 701.1 | Optimization method = CWE
| MOS scales = [[2L 3s]], [[5L 2s]], [[5L 7s]]
| Mapping = 1; 1 4 -2
| Pergen = (P8, P5)
| Odd limit 1 = 7 | Mistuning 1 = 25.3 | Complexity 1 = 7
| 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]].


Other possible tunings include [[17edo]] (17c val), [[29edo]] (29cd val), [[41edo]] (41cd val), [[53edo]] (53cdd val), as well as [[Pythagorean tuning]].  
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<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10437.html Yahoo! Tuning Group | ''The meantone family'']</ref><ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10864.html#10870 Yahoo! Tuning Group | ''names and definitions: meantone'']</ref>.  


See [[Meantone family #Dominant]] for technical data.  
See [[Meantone family #Dominant]] for technical data.  
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{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
|-
|-
! &#35; !! Cents* !! Approximate Ratios
! # !! Cents* !! Approximate Ratios
|-
|-
| 0 || 0.0 || '''1/1'''
| 0 || 0.0 || '''1/1'''
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| 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 ==
Much of 12edo harmony can be used. Dominant enables chords of [[didymic chords|didymic]] and [[archytas chords|archytas]].  
Much of 12edo harmony can be used. Dominant enables chords of [[didymic chords|didymic]] and [[archytas chords|archytas]]. The dominant seventh chord represents the harmonic seventh chord, whereas the [[German sixth chord|(German) augmented sixth chord]] is more or less equivalent to meantone's dominant seventh chord, as a tuning of [[28:35:42:50|1–5/4–3/2–25/14]].


== Tunings ==
== Tunings ==
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 7-limit norm-based tunings
|-
! rowspan="2" |
! colspan="3" | Euclidean
|-
! Constrained
! Constrained & skewed
! Destretched
|-
! Tenney
| CTE: ~3/2 = 699.6218{{c}}
| CWE: ~3/2 = 701.1125{{c}}
| POTE: ~3/2 = 701.5732{{c}}
|}
=== Tuning spectrum ===
=== Tuning spectrum ===
{{todo|complete table|comment=add the missing 11- and 13-limit eigenmonzo tunings}}
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
|-
|-
! Edo<br />Generator
! Edo<br>generator
! [[Eigenmonzo|Eigenmonzo<br />(Unchanged-interval)]]*
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]*
! Generator (¢)
! Generator (¢)
! Comments
! Comments
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| [[5/4]]
| [[5/4]]
| 696.578
| 696.578
| [[Quarter-comma meantone|1/4 syntonic comma]], 5-odd-limit minimax
| [[Quarter-comma meantone|1/4 syntonic comma]], [[5-odd-limit]] minimax
|-
|-
|
|
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| [[1/5-comma meantone|1/5 syntonic comma]]
| [[1/5-comma meantone|1/5 syntonic comma]]
|-
|-
| [[12edo|7\12]]
| '''[[12edo|7\12]]'''
|
|
| 700.000
| '''700.000'''
| Lower bound of 7- and 9-odd-limit diamond monotone
| '''Lower bound of 7- and 9-odd-limit diamond monotone'''
|-
|-
|
|
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| [[7/5]]
| [[7/5]]
| 702.915
| 702.915
| 7 & 9-odd limit minimax tuning
| 7- & 9-odd-limit minimax tuning
|-
|-
|
|
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|  
|  
| 703.448
| 703.448
| 29cd val
| 29cdef val
|-
|-
|  
|  
| 11/10
| [[11/10]]
| 703.499
| 703.499
| 11-odd limit minimax tuning
| [[11-odd-limit]] minimax tuning
|-
|-
|  
|  
| 13/10
| [[13/10]]
| 703.522
| 703.522
| 13-odd limit minimax tuning
| [[13-odd-limit]] minimax tuning
|-
|-
| [[17edo|10\17]]
| [[17edo|10\17]]
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| 1/2 septimal comma
| 1/2 septimal comma
|-
|-
| [[5edo|3\5]]
| '''[[5edo|3\5]]'''
|
|
| 720.000
| '''720.000'''
| Upper bound of 7- and 9-odd-limit diamond monotone
| '''Upper bound of 7- and 9-odd-limit diamond monotone'''
|-
|-
|
|
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| Full septimal comma
| Full septimal comma
|}
|}
<nowiki />* Besides the octave
<nowiki/>* Besides the octave
 
== References ==


[[Category:Temperaments]]
[[Category:Dominant| ]] <!-- Main article -->
[[Category:Dominant| ]] <!-- ain article -->
[[Category:Rank-2 temperaments]]
[[Category:Meantone family]]
[[Category:Meantone family]]
[[Category:Archytas clan]]
[[Category:Archytas clan]]
[[Category:Mint temperaments]]
[[Category:Mint temperaments]]

Latest revision as of 09:07, 9 February 2026

Dominant
Subgroups 2.3.5.7
Comma basis 36/35, 64/63
Reduced mapping ⟨1; 1 4 -2]
ET join 12 & 17c
Generators (CWE) ~3/2 = 701.1 ¢
MOS scales 2L 3s, 5L 2s, 5L 7s
Ploidacot monocot
Pergen (P8, P5)
Minimax error 7-odd-limit: 25.3 ¢;
(7-limit) 27-odd-limit: 26.3 ¢
Target scale size 7-odd-limit: 7 notes;
(7-limit) 27-odd-limit: 12 notes

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. The dominant seventh chord represents the harmonic seventh chord, whereas the (German) augmented sixth chord is more or less equivalent to meantone's dominant seventh chord, as a tuning of 1–5/4–3/2–25/14.

Tunings

7-limit norm-based 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

References

  1. ↑ Yahoo! Tuning Group | The meantone family
  2. ↑ Yahoo! Tuning Group | names and definitions: meantone
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