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Dominant (temperament): Difference between revisions

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**Imported revision 258794370 - Original comment: **
 
m Cleanup on infobox
 
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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Infobox regtemp
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| Title = Dominant
: This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2011-09-27 16:21:16 UTC</tt>.<br>
| Subgroups = 2.3.5.7
: The original revision id was <tt>258794370</tt>.<br>
| Comma basis = [[36/35]], [[64/63]]
: The revision comment was: <tt></tt><br>
| Edo join 1 = 12 | Edo join 2 = 17c
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
| Generators = 3/2 | Generators tuning = 701.1 | Optimization method = CWE
<h4>Original Wikitext content:</h4>
| MOS scales = [[2L 3s]], [[5L 2s]], [[5L 7s]]
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">See [[Meantone family#Dominant]].</pre></div>
| Mapping = 1; 1 4 -2
<h4>Original HTML content:</h4>
| Pergen = (P8, P5)
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;dominant&lt;/title&gt;&lt;/head&gt;&lt;body&gt;See &lt;a class="wiki_link" href="/Meantone%20family#Dominant"&gt;Meantone family&lt;/a&gt;.&lt;/body&gt;&lt;/html&gt;</pre></div>
| 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]]. 
 
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.
 
== Interval chain ==
In the following table, odd harmonics 1–9 are in '''bold'''.  
 
{| class="wikitable center-1 right-2"
|-
! # !! 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
|}
<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]]. 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 ==
{| 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 ===
{{todo|complete table|comment=add the missing 11- and 13-limit eigenmonzo tunings}}
{| 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
| 29cdef 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]]
Retrieved from "https://en.xen.wiki/w/Dominant_(temperament)"