Editing Meantone temperament (section)

==Meantone temperaments==
[[File:Meantone_spectrum.svg|225px|thumb|right|For a tuning to be meantone, its fifth must be between {{sfrac|685|5|7}} and 700&nbsp;¢ in size. Note that {{nobr|[[7 equal temperament|7 {{sc|TET}}]]}} is on the flatmost extreme, {{nobr|[[12 equal temperament|12 {{sc|TET}}]]}} is on the sharpmost extreme, and {{nobr|[[19 equal temperament|19 {{sc|TET}}]]}} forms the midpoint of the spectrum.]]

A meantone temperament is a [[regular temperament]], distinguished by the fact that the correction factor to the Pythagorean perfect fifths, given usually as a specific fraction of the syntonic comma, is chosen to make the whole tone intervals equal, as closely as possible, to the geometric mean of the major tone and the minor tone. Historically, commonly used meantone temperaments, discussed below, occupy a narrow portion of this tuning continuum, with fifths ranging from approximately 695&nbsp;to 699&nbsp;cents.

Meantone temperaments can be specified in various ways: By what fraction of a syntonic comma the fifth is being flattened (as above), the width of the tempered perfect fifth in cents, or the ratio of the whole tone (in cents) to the diatonic [[semitone]]. This last ratio was termed {{nobr|"{{mvar|R}}"}} by American composer, pianist and theoretician [[Easley Blackwood Jr.|Easley Blackwood]]. If {{mvar|R}} happens to be a rational number {{mvar|R}} = <math>{\scriptstyle\frac{N}{D}}</math>, then  <math>2^{\frac{3R+1}{5R+2}}</math> is the  closest approximation to the corresponding meantone tempered fifth within the equitempered division of the octave into <math>5N +2D</math> equal parts. Such divisions of the octave into a number of small parts greater than 12 are sometimes refererred to as [[microtonality]], and the smallest intervals called [[microtones]].

In these terms, some historically notable meantone tunings are listed below, and compared with the closest equitempered microtonal tuning. The first column gives the fraction of the syntonic comma by which the perfect fifths are tempered in the meantone system. The second lists [[Five-limit tuning|5-limit]] rational intervals that occur within this tuning. The third gives the fraction of an octave, within the corresponding equitempered microinterval system, that best approximates the meantone fifth.  The fourth gives the difference between the two, in [[Cent (music)|cents]]. The fifth is the corresponding value of the fraction {{mvar|R}} = <math>\scriptstyle{\frac{N}{D}}</math>, and the fifth is the number <math>5N+2D</math> of equitempered ({{sc|ET}} ) microtones in an octave.

{| class="wikitable" style="text-align: center;"
|+ Meantone vs. Equitempered tunings
|- style="vertical-align:bottom;"
! Meantone fraction of <br/>(syntonic) comma
! [[Five-limit tuning|5-limit]] rational intervals
! Size of {{sc|ET}} fifths <br> as fractions of an octave
! Error between <br> meantone fifths <br> and {{sc|ET}} fifths <br>(in cents)
! Blackwood’s<br/>ratio<br/> {{mvar|R}} =<math>\scriptstyle{\frac{N}{D}}</math>
! Number of [[equal temperament|{{sc|ET}}]] microtones <br><math>5N +2D</math>
|- style="vertical-align:middle"
|
{{big| {{sfrac| 1 | 315 |font-size=1em}} }}

({{small|very nearly<br/> Pythagorean tuning}})
|{{small| For all practical purposes,}}

{{small|the fifth is a "perfect" {{nobr|{{sfrac| 3 | 2 }} .}} }}
| {{sfrac| 31 | 53 }}
| +0.000066

({{small|+6.55227×{{10^|−5}}}})
| {{sfrac| 9 | 4 }} {{=}} 2.25
| {{big|[[53 equal temperament|53]]}}
|- style="vertical-align:middle"
|
{{big| {{sfrac| 1 | 11 |font-size=1em}} }}

({{small| or {{sfrac| 1 | 12 |font-size=1em}} Pythagorean comma}})
|
{{sfrac| 16384 | 10935 }} ={{sfrac| 2{{sup|14}} | 3{{sup|7}} × 5 }}

({{small| Kirnberger fifth: a just fifth flattened by a [[schisma]].
<br> Equals meantone to 6 significant figures.)}}
| {{sfrac| 7 | 12}}
| +0.000116

({{small|+1.16371×{{10^|−4}}}})
| {{sfrac| 2 | 1 }} {{=}} 2.00
| {{big| [[12 equal temperament|12]]}}
|- style="vertical-align:middle"
| {{big| {{sfrac| 1 | 6 |font-size=1em}} }}
| {{small |{{sfrac| 45 | 32 |font-size=1em}} and {{sfrac| 64 | 45 |font-size=1em}} }}

({{small|tritones}})
| {{sfrac| 32 | 55 }}
| −0.188801
| {{sfrac| 9 | 5 }} {{=}} 1.80
| {{big|55}}
|- style="vertical-align:middle"
| {{big| {{sfrac| 1 | 5 |font-size=1em}} }}
|
{{small | {{sfrac| 16 | 15 |font-size=1em}} and {{sfrac| 15 | 8 |font-size=1em}}}}

({{small|diatonic semitone and major seventh}})
| {{sfrac| 25 | 43 }}
| +0.0206757
| {{sfrac| 7 | 4 }} {{=}} 1.75
| {{big|43}}
|- style="vertical-align:middle"
| {{big| {{sfrac| 1 | 4 |font-size=1em}} }}
|
{{small | {{sfrac| 5 | 4 |font-size=1em}} and {{sfrac| 8 | 5 |font-size=1em}}}}

({{small|just major third and minor sixth}})
| {{sfrac| 18 | 31 }}
| +0.195765
| {{sfrac| 5 | 3 }} {{=}} 1.66
| {{big|[[31 equal temperament|31]]}}
|- style="vertical-align:middle"
| {{big| {{sfrac| 2 | 7 |font-size=1em}} }}
|
{{small |{{sfrac| 25 | 24 |font-size=1em}} and {{sfrac| 48 | 25 |font-size=1em}}}}

({{small|chromatic semitone and major seventh }})
| {{sfrac| 29 | 50 }}
| +0.189653
| {{sfrac| 8 | 5 }} {{=}} 1.60
| {{big|50}}
|- style="vertical-align:middle"
| {{big| {{sfrac| 1 | 3 |font-size=1em}} }}
|
{{small |{{sfrac| 6 | 5 |font-size=1em}}  and {{sfrac| 5 | 3 |font-size=1em}}}}

({{small|just minor third and major sixth}})
| {{sfrac| 11 | 19 }}
| −0.0493956
| {{sfrac| 3 | 2 }} {{=}} 1.50
| {{big|[[19 equal temperament|19]]}}
|- style="vertical-align:middle"
| {{big| {{sfrac|&nbsp;2&nbsp;|&nbsp;5&nbsp;|font-size=1em}} }}
|{{sfrac| 27 | 25 }}

({{small|large limma}})
| {{sfrac| 26 | 45 }}
| +0.0958
| {{sfrac| 7 | 5 }} {{=}} 1.40
| {{big|[[15 equal temperament#Further subdivisions|45]]}}
|- style="vertical-align:middle"
| {{big| {{sfrac| 1 | 2 |font-size=1em}} }}
|{{small | {{sfrac| 10 | 9 |font-size=1em}} and {{sfrac| 9 | 5 |font-size=1em}}}} 

({{small|just minor tone and diminished seventh}})
| {{sfrac| 19 | 33 }}
| −0.292765
| {{sfrac| 5 | 4 }} {{=}} 1.25
| {{big|33}}
|}

===Equal temperaments===

In neither the twelve tone equitemperament nor the quarter-comma meantone is the fifth a [[rational number|rational]] fraction of the octave, but several tunings exist which approximate the fifth by such an interval; these are a subset of the [[equal temperament]]s {{nobr|( "{{mvar|N}} {{sc|TET}}" ),}} in which the octave is divided into some number ({{mvar|N}}) of equally wide intervals.

Equal temperaments that are useful as approximations to meantone tunings include (in order of increasing [[Generated collection|generator]] width) {{nobr|[[19 equal temperament|19 {{sc|TET}}]]}} {{nobr|({{sfrac| ~ | 1 | 3 }} comma),}} {{nobr|50 {{sc|TET}}}} {{nobr|({{sfrac| ~ | 2 | 7 }} comma),}} {{nobr|[[31 equal temperament|31 {{sc|TET}}]]}} {{nobr|({{sfrac| ~ | 1 | 4 }} comma),}} {{nobr|43 {{sc|TET}}}} {{nobr|({{sfrac| ~ | 1 | 5 }} comma),}} and {{nobr|55 {{sc|TET}}}} {{nobr|({{sfrac| ~ | 1 | 6 }} comma).}} The farther the tuning gets away from quarter-comma meantone, however, the less related the tuning is to harmonic ratios. This can be overcome by [[dynamic tonality|tempering the partials]] to match the tuning, which is possible, however, only on electronic synthesizers.<ref>
{{cite journal
 | last1 = Sethares | first1 = W.A. | author1-link = William Sethares
 | last2=Milne      | first2 = A.       | last3=Tiedje    | first3 = S.
 | last4=Prechtl    | first4 = A.       | last5=Plamondon | first5 = J.
 | year = 2009
 | title = Spectral tools for dynamic tonality and audio morphing
 | journal = [[Computer Music Journal]]
 | volume = 33  | issue = 2  | pages = 71–84
 | citeseerx = 10.1.1.159.838  | s2cid = 216636537
 | doi = 10.1162/comj.2009.33.2.71
 | id = {{Project MUSE|266411}}
}}
</ref>

[[File:Meantone_fifths_vs_thirds_comparison.svg|800px|thumb|center|Comparison of perfect fifths, major thirds, and minor thirds in various meantone tunings with just intonation]]
<br />
{{Center | '''Approximation of just intervals in equal temperaments'''<br />{{Gallery items | itemalign="center" | captionalign="center"
| [[File:12ed2-5Limit.svg|150px|thumb|center]] | 12-ET
| [[File:19ed2.svg|150px|thumb|center]] | 19-ET
| [[File:31ed2.svg|150px|thumb|center]] | 31-ET
| [[File:43ed2.svg|150px|thumb|center]] | 43-ET
| [[File:50ed2.svg|150px|thumb|center]] | 50-ET
| [[File:55ed2.svg|150px|thumb|center]] | 55-ET
}}}}