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<span style="display: block; text-align: right;">[[de:mitteltönig|Deutsch]]</span>
'''Meantone''' is a familar historical [[temperament|temperament]] based on a chain of fifths (or fourths), which is discussed in [[meantone family]] in the context of the associated family of temperaments, and in [[meantone vs meanpop]] in terms of 11-limit extensions.


Meantone is a familar historical [[temperament|temperament]] based on a chain of fifths (or fourths), which is discussed [[Meantone_family|here]] in the context of the associated family of temperaments, and [[Meantone_vs_meanpop|here]] in terms of 11-limit extensions.
== History ==
Meantone was the dominant tuning used in Europe from around late 15th century to around early 18th century, after which various [[Well Temperament|Well Temperaments]] and eventually 12-tone [[Equal Temperament]] won in popularity.


=History=
== Theory and Classification ==
Meantone was the dominant tuning used in Europe from around late 15th century to around early 18th century, after which various [[Well_Temperament|Well Temperaments]] and eventually 12-tone [[Equal_Temperament|Equal Temperament]] won in popularity.
 
=Theory and Classification=
Meantone temperaments are based on two generating intervals; the octave and the fifth, from which all pitches are composed. This qualifies it as a [[Regular_Temperaments|rank-2 temperament]]. The octave is typically pure or close to pure, and the fifth is a few cents narrower than pure. The rationale for narrowing the fifth is to temper out the syntonic comma. This means that stacking four fifths (such as C-G-D-A-E) results in a major third (C-E) that is close to just.
Meantone temperaments are based on two generating intervals; the octave and the fifth, from which all pitches are composed. This qualifies it as a [[Regular_Temperaments|rank-2 temperament]]. The octave is typically pure or close to pure, and the fifth is a few cents narrower than pure. The rationale for narrowing the fifth is to temper out the syntonic comma. This means that stacking four fifths (such as C-G-D-A-E) results in a major third (C-E) that is close to just.


[[Meantone_intervals|Intervals in meantone]] have standard names based on the number of steps of the diatonic scale they span (this corresponds to the [[val|val]] &lt;7 11 16|), with a modifier {..."double diminished", "diminished", "minor", "major", "augmented", "double augmented"...} that tells you the specific interval in increments of a chromatic semitone. Note that in a general meantone system, all of these intervals are distinct. For example, a diminished fourth is a different interval from a major third.
[[Meantone intervals|Intervals in meantone]] have standard names based on the number of steps of the diatonic scale they span (this corresponds to the [[val]] &lt;7 11 16|), with a modifier {..."double diminished", "diminished", "minor", "major", "augmented", "double augmented"...} that tells you the specific interval in increments of a chromatic semitone. Note that in a general meantone system, all of these intervals are distinct. For example, a diminished fourth is a different interval from a major third.


=Meantone Temperaments (ie, tunings)=
== Meantone Temperaments (ie, tunings) ==
<ul><li>[[19edo|19-edo]]</li><li>[[1-3_Syntonic_Comma_Meantone|1/3 Syntonic Comma Meantone]]</li><li>[[Golden_Meantone|Golden Meantone]]</li><li>[[Quarter-comma_meantone|1/4 Syntonic Comma Meantone]]</li><li>[[31edo|31-edo]]</li><li>[[1-5_Syntonic_Comma_Meantone|1/5 Syntonic Comma Meantone]]</li><li>[[1-6_Syntonic_Comma_Meantone|1/6 Syntonic Comma Meantone]]</li><li>[[12edo|12-edo]]</li><li>[[Lucy_Tuning|Lucy tuning]]</li><li>[[50edo|50-edo]]</li><li>[[55edo|55-edo]]</li><li>[[Tungsten_meantone|Tungsten meantone]]</li></ul>
* [[19edo|19-edo]]
* [[1-3_Syntonic_Comma_Meantone|1/3 Syntonic Comma Meantone]]
* [[Golden Meantone]]
* [[Quarter-comma meantone|1/4 Syntonic Comma Meantone]]
* [[31edo|31-edo]]
* [[1-5_Syntonic_Comma_Meantone|1/5 Syntonic Comma Meantone]]
* [[1-6_Syntonic_Comma_Meantone|1/6 Syntonic Comma Meantone]]
* [[12edo|12-edo]]
* [[Lucy Tuning]]
* [[50edo|50-edo]]
* [[55edo|55-edo]]
* [[Tungsten meantone]]


=Spectrum of Meantone Tunings by Eigenmonzos=
== Spectrum of Meantone Tunings by Eigenmonzos ==


{| class="wikitable"
{| class="wikitable"
|-
|-
! | [[Eigenmonzo|Eigenmonzo]]
! [[Eigenmonzo]]
! | Fifth size (usual name)
! Fifth size (usual name)
|-
|-
| | 10/9
| [[10/9]]
| | <span style="">691.202 (1/2 comma)</span>
| 691.202 (1/2 comma)
|-
|-
| | 15\26
| 15\26
| | 692.308
| 692.308
|-
|-
| | 56/45
| [[56/45]]
| | 694.651
| 694.651
|-
|-
| | 28/27
| [[28/27]]
| | 694.709
| 694.709
|-
|-
| | 81/70
| 81/70
| | 694.732
| 694.732
|-
|-
| | 11\19
| 11\19
| | 694.737
| 694.737
|-
|-
| | 6/5
| [[6/5]]
| | 694.786 (1/3 comma)
| 694.786 (1/3 comma)
|-
|-
| | 35/27
| [[35/27]]
| | 695.389
| 695.389
|-
|-
| | 51\88
| 51\88
| | 695.455
| 695.455
|-
|-
| | 1\2 + 1\(4π)
| 1\2 + 1\(4π)
| | 695.493 (Lucy tuning)
| 695.493 (Lucy tuning)
|-
|-
| | 9/7
| [[9/7]]
| | 695.614
| 695.614
|-
|-
| | f^4 = 2f + 2
| f^4 = 2f + 2
| | 695.630 (Wilson fifth)
| 695.630 (Wilson fifth)
|-
|-
| | 40\69
| 40\69
| | 695.652
| 695.652
|-
|-
| | 25/24
| [[25/24]]
| | 695.810 (2/7 comma)
| 695.810 (2/7 comma)
|-
|-
| | 13/10
| [[13/10]]
| | 695.838 (ratwolf fifth, meanpop eigenmonzo)
| 695.838 (ratwolf fifth, meanpop eigenmonzo)
|-
|-
| | 36/35
| [[36/35]]
| | 695.936
| 695.936
|-
|-
| | 54/49
| [[54/49]]
| | 695.987
| 695.987
|-
|-
| | 29\50
| 29\50
| | 696.000
| 696.000
|-
|-
| | 15/14
| [[15/14]]
| | 696.111
| 696.111
|-
|-
| | 78125/73728
| 78125/73728
| | 696.165 (5-limit least squares)
| 696.165 ([[5-limit]] least squares)
|-
|-
| | (8 - φ)\11
| (8 - φ)\11
| | 696.214 (Golden meantone)
| 696.214 (Golden meantone)
|-
|-
| | 49/45
| [[49/45]]
| | 696.245
| 696.245
|-
|-
| | 47\81
| 47\81
| | 696.296
| 696.296
|-
|-
| | 7/6
| [[7/6]]
| | 696.319
| 696.319
|-
|-
| | 48/35
| [[48/35]]
| | 696.399
| 696.399
|-
|-
| | [19 9 -1 -11&gt;
| {{Monzo| 19 9 -1 -11 }}
| | 696.436 (9-limit least squares)
| 696.436 (9-limit least squares)
|-
|-
| | 5/4
| [[5/4]]
| | 696.578 (5- 7- and 9-limit minimax, 1/4 comma)
| 696.578 (5- 7- and 9-limit minimax, 1/4 comma)
|-
|-
| | 49/48
| 49/48
| | 696.616
| 696.616
|-
|-
| | 60/49
| 60/49
| | 696.626
| 696.626
|-
|-
| | [-55 -11 1 25&gt;
| {{Monzo| -55 -11 1 25 }}
| | 696.648 (7-limit least squares)
| 696.648 ([[7-limit]] least squares)
|-
|-
| | 18\31
| 18\31
| | 696.774
| 696.774
|-
|-
| | 35/32
| [[35/32]]
| | 696.796
| 696.796
|-
|-
| | 8/7
| [[8/7]]
| | 696.883
| 696.883
|-
|-
| | 49/40
| [[49/40]]
| | 696.959
| 696.959
|-
|-
| | 7/5
| [[7/5]]
| | 697.085
| 697.085
|-
|-
| | 43\74
| 43\74
| | 697.297
| 697.297
|-
|-
| | 21/16
| [[21/16]]
| | 697.344
| 697.344
|-
|-
| | 16/15
| [[16/15]]
| | 697.654 (1/5 comma)
| 697.654 (1/5 comma)
|-
|-
| | 25\43
| 25\43
| | 697.674
| 697.674
|-
|-
| | 64/63
| [[64/63]]
| | 697.728
| 697.728
|-
|-
| | 21/20
| [[21/20]]
| | 697.781
| 697.781
|-
|-
| | 28/25
| [[28/25]]
| | 698.099
| 698.099
|-
|-
| | 32\55
| 32\55
| | 698.182
| 698.182
|-
|-
| | 80/63
| [[80/63]]
| | 698.303
| 698.303
|-
|-
| | 45/32
| [[45/32]]
| | 698.371 (1/6 comma)
| 698.371 (1/6 comma)
|-
|-
| | 39\67
| 39\67
| | 698.507
| 698.507
|-
|-
| | 46\79
| 46\79
| | 698.734
| 698.734
|-
|-
| | 25/21
| [[25/21]]
| | 699.384
| 699.384
|-
|-
| | 7\12
| 7\12
| | 700.000
| 700.000
|-
|-
| | 31\53
| 31\53
| | 701.887
| 701.887
|-
|-
| | 3/2
| [[3/2]]
| | 701.955
| 701.955
|}
|}
[5/4 7] eigenmonos: [[meanwoo12|meanwoo12]], [[meanwoo19|meanwoo19]]
[5/4 7] eigenmonos: [[meanwoo12]], [[meanwoo19]]


=Links=
== Links ==
<ul><li>[http://www.kylegann.com/histune.html http://www.kylegann.com/histune.html] -- An Introduction to Historical Tunings, by [[Kyle_Gann|Kyle Gann]]</li></ul>      [[Category:meantone]]
* [http://www.kylegann.com/histune.html http://www.kylegann.com/histune.html] -- An Introduction to Historical Tunings, by [[Kyle Gann]]
 
[[Category:meantone]]
[[Category:temperament]]
[[Category:temperament]]
[[Category:theory]]
[[Category:theory]]
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