Meantone: Difference between revisions

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== Meantone Temperaments (ie, tunings) ==
== Meantone Temperaments (ie, tunings) ==
* [[19edo|19-edo]]
* [[19edo]]
* [[1-3_Syntonic_Comma_Meantone|1/3 Syntonic Comma Meantone]]
* [[1/3 Syntonic Comma Meantone]]
* [[Golden Meantone]]
* [[Golden Meantone]]
* [[Quarter-comma meantone|1/4 Syntonic Comma Meantone]]
* [[Quarter-comma meantone|1/4 Syntonic Comma Meantone]]
* [[31edo|31-edo]]
* [[31edo]]
* [[1-5_Syntonic_Comma_Meantone|1/5 Syntonic Comma Meantone]]
* [[1/5 Syntonic Comma Meantone]]
* [[1-6_Syntonic_Comma_Meantone|1/6 Syntonic Comma Meantone]]
* [[1/6 Syntonic Comma Meantone]]
* [[12edo|12-edo]]
* [[12edo]]
* [[Lucy Tuning]]
* [[Lucy Tuning]]
* [[50edo|50-edo]]
* [[50edo]]
* [[55edo|55-edo]]
* [[55edo]]
* [[Tungsten meantone]]
* [[Tungsten meantone]]


Line 28: Line 28:
|-
|-
! [[Eigenmonzo]]
! [[Eigenmonzo]]
! Fifth size (usual name)
! Fifth size
! usual name
|-
|-
| [[10/9]]
| [[10/9]]
| 691.202 (1/2 comma)
| 691.202
| 1/2 comma
|-
|-
| [[26edo|15\26]]
| [[26edo|15\26]]
| 692.308
| 692.308
|
|-
|-
| [[45edo|26\45]]
| [[45edo|26\45]]
| 693.333
| 693.333
|
|-
|-
| [[27/25]]
| [[27/25]]
| 693.352 (2/5 comma)
| 693.352
| 2/5 comma
|-
|-
| [[56/45]]
| [[56/45]]
| 694.651
| 694.651
|
|-
|-
| [[28/27]]
| [[28/27]]
| 694.709
| 694.709
|
|-
|-
| 81/70
| 81/70
| 694.732
| 694.732
|
|-
|-
| [[19edo|11\19]]
| [[19edo|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
|
|-
|-
| [[88edo|51\88]]
| [[88edo|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
|-
|-
| [[69edo|40\69]]
| [[69edo|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
|
|-
|-
| [[50edo|29\50]]
| [[50edo|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
|
|-
|-
| [[81edo|47\81]]
| [[81edo|47\81]]
| 696.296
| 696.296
|
|-
|-
| [[7/6]]
| [[7/6]]
| 696.319
| 696.319
|
|-
|-
| [[48/35]]
| [[48/35]]
| 696.399
| 696.399
|
|-
|-
| {{Monzo| 19 9 -1 -11 }}
| {{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
|
|-
|-
| {{Monzo| -55 -11 1 25 }}
| {{Monzo| -55 -11 1 25 }}
| 696.648 ([[7-limit]] least squares)
| 696.648
| [[7-limit]] least squares
|-
|-
| [[31edo|18\31]]
| [[31edo|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
|
|-
|-
| [[75/64]]
| [[75/64]]
| 697.176
| 697.176
|
|-
|-
| [[74edo|43\74]]
| [[74edo|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
|-
|-
| [[43edo|25\43]]
| [[43edo|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
|
|-
|-
| [[55edo|32\55]]
| [[55edo|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
|-
|-
| [[67edo|39\67]]
| [[67edo|39\67]]
| 698.507
| 698.507
|
|-
|-
| [[79edo|46\79]]
| [[79edo|46\79]]
| 698.734
| 698.734
|
|-
|-
| [[25/21]]
| [[25/21]]
| 699.384
| 699.384
|
|-
|-
| [[12edo|7\12]]
| [[12edo|7\12]]
| 700.000
| 700.000
|
|-
|-
| [[53edo|31\53]]
| [[53edo|31\53]]
| 701.887
| 701.887
|
|-
|-
| [[3/2]]
| [[3/2]]
| 701.955 ([[Pythagorean tuning]])
| 701.955
| [[Pythagorean tuning]]
|}
|}
[5/4 7] eigenmonos: [[meanwoo12]], [[meanwoo19]]
 
[5/4 7] eigenmonzos: [[meanwoo12]], [[meanwoo19]]


== Links ==
== Links ==

Revision as of 03:54, 6 December 2018

Meantone is a familar historical 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.

History

Meantone was the dominant tuning used in Europe from around late 15th century to around early 18th century, after which various Well Temperaments and eventually 12-tone 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 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.

Intervals in meantone have standard names based on the number of steps of the diatonic scale they span (this corresponds to the val <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)

Spectrum of Meantone Tunings by Eigenmonzos

Eigenmonzo Fifth size usual name
10/9 691.202 1/2 comma
15\26 692.308
26\45 693.333
27/25 693.352 2/5 comma
56/45 694.651
28/27 694.709
81/70 694.732
11\19 694.737
6/5 694.786 1/3 comma
35/27 695.389
51\88 695.455
1\2 + 1\(4π) 695.493 Lucy Tuning
9/7 695.614
f^4 = 2f + 2 695.630 Wilson fifth
40\69 695.652
25/24 695.810 2/7 comma
13/10 695.838 ratwolf fifth, meanpop eigenmonzo
36/35 695.936
54/49 695.987
29\50 696.000
15/14 696.111
78125/73728 696.165 5-limit least squares
(8 - φ)\11 696.214 Golden meantone
49/45 696.245
47\81 696.296
7/6 696.319
48/35 696.399
[19 9 -1 -11⟩ 696.436 9-limit least squares
5/4 696.578 5- 7- and 9-limit minimax, 1/4 comma
49/48 696.616
60/49 696.626
[-55 -11 1 25⟩ 696.648 7-limit least squares
18\31 696.774
35/32 696.796
8/7 696.883
49/40 696.959
7/5 697.085
75/64 697.176
43\74 697.297
21/16 697.344
16/15 697.654 1/5 comma
25\43 697.674
64/63 697.728
21/20 697.781
28/25 698.099
32\55 698.182
80/63 698.303
45/32 698.371 1/6 comma
39\67 698.507
46\79 698.734
25/21 699.384
7\12 700.000
31\53 701.887
3/2 701.955 Pythagorean tuning

[5/4 7] eigenmonzos: meanwoo12, meanwoo19

Links

Retrieved from "https://en.xen.wiki/index.php?title=Meantone&oldid=38852"