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]].
+'''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 "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<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.
 == Interval chain ==
@@ Line 26: / Line 32: @@
 | 7 || 107.8 || 15/14
 |}
-<nowiki>*</nowiki> in 7-limit [[CWE]] tuning
+<nowiki />* In 7-limit [[CWE]] tuning
 == Chords and harmony ==
 Much of 12edo harmony can be used. Dominant enables chords of [[didymic chords|didymic]] and [[archytas chords|archytas]].
-[[Category:Temperaments]]
+== Tunings ==
-[[Category:Dominant (temperament)| ]] <!-- main article -->
+=== Tuning spectrum ===
+{| class="wikitable center-all left-4"
+|-
+! Edo<br>generator
+! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]*
+! Generator (¢)
+! Comments
+|-
+|
+| [[9/5]]
+| 691.202
+| [[1/2-comma meantone|1/2 syntonic comma]]
+|-
+|
+| [[5/3]]
+| 694.786
+| [[1/3-comma meantone|1/3 syntonic comma]]
+|-
+|
+| [[5/4]]
+| 696.578
+| [[Quarter-comma meantone|1/4 syntonic comma]], [[5-odd-limit]] minimax
+|-
+|
+| [[15/8]]
+| 697.654
+| [[1/5-comma meantone|1/5 syntonic comma]]
+|-
+| [[12edo|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
+|
+|-
+| [[29edo|17\29]]
+|
+| 703.448
+| 29cd val
+|-
+|
+| [[11/10]]
+| 703.499
+| [[11-odd-limit]] minimax tuning
+|-
+|
+| [[13/10]]
+| 703.522
+| [[13-odd-limit]] minimax tuning
+|-
+| [[17edo|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
+|-
+| [[5edo|3\5]]
+|
+| 720.000
+| Upper bound of 7- and 9-odd-limit diamond monotone
+|-
+|
+| [[21/16]]
+| 729.219
+| Full septimal comma
+|}
+<nowiki />* Besides the octave
+== References ==
+[[Category:Dominant| ]] <!-- Main article -->
+[[Category:Rank-2 temperaments]]
 [[Category:Meantone family]]
 [[Category:Archytas clan]]
 [[Category:Mint temperaments]]