Meantone: Difference between revisions

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Line 30: Line 30:
! Fifth size
! Fifth size
! usual name
! usual name
|-
|16/13
|689.868
|Meanplop
|-
|-
|567/512
|567/512
|688.323
|688.323
|1/2 septimal comma
|1/2 septimal comma
|-
|16/13
|689.868
|Meanplop
|-
|-
|<nowiki>| 16 -10 ></nowiki>
|<nowiki>| 16 -10 ></nowiki>
Line 50: Line 58:
| 691.202
| 691.202
| 1/2 comma
| 1/2 comma
|-
|13/12
|692.285
|Meanplop
|-
|-
| [[26edo|15\26]]
| [[26edo|15\26]]
Line 62: Line 74:
|692.867
|692.867
|1/3 septimal comma
|1/3 septimal comma
|-
|13/10
|693.223
|Meanplop
|-
|-
| [[45edo|26\45]]
| [[45edo|26\45]]
Line 70: Line 86:
| 693.352
| 693.352
| 2/5 comma
| 2/5 comma
|-
|18/13
|693.897
|Meanplop
|-
|-
|19683/16384
|19683/16384
Line 78: Line 98:
|694.165
|694.165
|2/7 septimal comma
|2/7 septimal comma
|-
|15/13
|694.193
|Meanplop
|-
|-
|[[14/13]]
|[[14/13]]
Line 102: Line 126:
| 694.786
| 694.786
| 1/3 comma
| 1/3 comma
|-
|14/13
|694.878
|Meanplop
|-
|-
|[[18/13]]
|[[18/13]]
Line 130: Line 158:
| 695.493
| 695.493
| Lucy Tuning
| Lucy Tuning
|-
|39/28
|695.6095
|Tridecimal Meantone, Tridecimal Meanpop
|-
|-
|[[13/12]]
|[[13/12]]
Line 150: Line 182:
| 695.81
| 695.81
| 2/7 comma
| 2/7 comma
|-
|40/33
|695.815
|Meanpop
|-
|-
| [[13/10]]
| [[13/10]]
Line 158: Line 194:
|695.869
|695.869
|
|
|-
|112/99
|695.886
|Meanpop
|-
|-
| [[36/35]]
| [[36/35]]
Line 173: Line 213:
|16/13
|16/13
|696.035
|696.035
|Tridecimal Meantone
|Tridecimal Meantone, Tridecimal Meanpop
|-
|13/11
|696.043
|13, 15 limit minimax (Tridecimal Meanpop)
|-
|11/8
|696.052
|Meanpop
|-
|-
|8192/6561
|8192/6561
Line 186: Line 234:
| 696.165
| 696.165
| [[5-limit]] least squares
| [[5-limit]] least squares
|-
|11/10
|696.176
|Meanpop
|-
|-
| (8 - φ)\11
| (8 - φ)\11
Line 206: Line 258:
| 696.319
| 696.319
|
|
|-
|27/22
|696.3635
|Meanpop
|-
|-
| [[48/35]]
| [[48/35]]
Line 213: Line 269:
|39/32
|39/32
|696.405
|696.405
|
|Tridecimal Meantone, Tridecimal Meanpop
|-
|14/11
|696.413
|Meanpop
|-
|-
| {{Monzo| 19 9 -1 -11 }}
| {{Monzo| 19 9 -1 -11 }}
| 696.436
| 696.436
| 9-limit least squares
| 9-limit least squares
|-
|12/11
|696.474
|Meanpop
|-
|-
|16384/15309
|16384/15309
Line 250: Line 314:
| 696.796
| 696.796
|
|
|-
|11/9
|696.839
|Meanpop
|-
|-
| [[8/7]]
| [[8/7]]
Line 261: Line 329:
|[[12/11]]
|[[12/11]]
|697.021
|697.021
|
|Undecimal Meantone
|-
|-
| [[7/5]]
| [[7/5]]
Line 269: Line 337:
|[[15/11]]
|[[15/11]]
|697.158
|697.158
|
| rowspan="2" |Undecimal Meantone
|-
|-
|[[27/22]]
|[[27/22]]
|697.159
|697.159
|
|-
|39/32
|697.168
|Grosstone
|-
|-
| [[75/64]]
| [[75/64]]
| 697.176
| 697.176
|
|
|-
|22/21
|697.22
|Undecimal Meantone
|-
|-
|14/13
|14/13
Line 293: Line 368:
|[[11/8]]
|[[11/8]]
|697.295
|697.295
|
|Undecimal Meantone
|-
|-
| [[74edo|43\74]]
| [[74edo|43\74]]
| 697.297
| 697.297
|
|-
|[5/4 7]
|697.339
|
|
|-
|-
Line 321: Line 400:
|[[11/10]]
|[[11/10]]
|697.5
|697.5
|
|Undecimal Meantone
|-
|-
|15/13
|15/13
Line 347: Line 426:
|
|
|-
|-
|16/13
|40/33, 16/13
|697.797
|697.797
|Meridetone
|Undecimal Meantone, Meridetone
|-
|-
|[[14/11]]
|[[14/11]]
|697.812
|697.812
|
|Undecimal Meantone
|-
|-
|15/13
|15/13
Line 385: Line 464:
|33/28
|33/28
|698.272
|698.272
|
|Undecimal Meantone
|-
|-
| [[80/63]]
| [[80/63]]
Line 398: Line 477:
| 698.371
| 698.371
| 1/6 comma
| 1/6 comma
|-
|33/26
|698.407
|Meanplop, Meridetone
|-
|-
| [[67edo|39\67]]
| [[67edo|39\67]]
Line 406: Line 489:
|698.604
|698.604
|1/7 Pythagorean comma, Pythagorean limma
|1/7 Pythagorean comma, Pythagorean limma
|-
|112/99
|698.64
|Undecimal Meantone
|-
|-
|45/34
|45/34
Line 417: Line 504:
|13/11
|13/11
|698.801
|698.801
|Meridetone
|Meridetone, Meanplop
|-
|-
|[[135/128]]
|[[135/128]]
Line 466: Line 553:
|''703.186''
|''703.186''
|''Tridecimal Meantone''
|''Tridecimal Meantone''
|-
|''22/21''
|''703.356''
|''Meanpop''
|-
|-
|''13/11''
|''13/11''
Line 471: Line 562:
|''Tridecimal Meantone''
|''Tridecimal Meantone''
|-
|-
|''88/81''
| rowspan="2" |''88/81''
|''707.946''
|''Meanpop''
|-
|''710.4335''
|''710.4335''
|
|''Undecimal Meantone''
|}
|}


Line 479: Line 573:


== Links ==
== Links ==
* http://www.kylegann.com/histune.html -- An Introduction to Historical Tunings, by [[Kyle Gann]]   [[Category:Meantone| ]] <!-- main article -->
* http://www.kylegann.com/histune.html -- An Introduction to Historical Tunings, by [[Kyle Gann]]   [[Category:Meantone| ]] <!-- main article -->
[[Category:Temperament]]
[[Category:Temperament]]
[[Category:Theory]]
[[Category:Theory]]

Revision as of 02:21, 3 July 2019

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
16/13 689.868 Meanplop
567/512 688.323 1/2 septimal comma
16/13 689.868 Meanplop
| 16 -10 > 690.225 1/2 Pythagorean comma, Pythagorean dilimma
76/51 690.603
| -19 9 0 2 > 691.049 2/5 septimal comma
10/9 691.202 1/2 comma
13/12 692.285 Meanplop
15\26 692.308
| 31 -19 > 692.571 2/5 Pythagorean comma
2048/1701 692.867 1/3 septimal comma
13/10 693.223 Meanplop
26\45 693.333
27/25 693.352 2/5 comma
18/13 693.897 Meanplop
19683/16384 694.135 1/3 Pythagorean comma, Pythagorean augmented second
| -23 11 0 2 > 694.165 2/7 septimal comma
15/13 694.193 Meanplop
14/13 694.34 Tridecimal Meantone
56/45 694.651
28/27 694.709
81/70 694.732
11\19 694.737
6/5, 25/18 694.786 1/3 comma
14/13 694.878 Meanplop
18/13 695.124 Tridecimal Meantone
5103/4095 695.139 1/4 septimal comma
15/13 695.226 Tridecimal Meantone
| 27 -17 > 695.252 2/7 Pythagorean comma, 17-comma
35/27 695.389
51\88 695.455
1\2 + 1\(4π) 695.493 Lucy Tuning
39/28 695.6095 Tridecimal Meantone, Tridecimal Meanpop
13/12 695.612 Tridecimal Meantone
9/7 695.614
f^4 = 2f + 2 695.63 Wilson fifth
40\69 695.652
25/24 695.81 2/7 comma
40/33 695.815 Meanpop
13/10 695.838 ratwolf fifth, Tridecimal Meantone and meanpop eigenmonzo
81/80 695.869
112/99 695.886 Meanpop
36/35 695.936
54/49 695.987
29\50 696
16/13 696.035 Tridecimal Meantone, Tridecimal Meanpop
13/11 696.043 13, 15 limit minimax (Tridecimal Meanpop)
11/8 696.052 Meanpop
8192/6561 696.09 1/4 Pythagorean comma, Pythagorean diminished fourth
15/14 696.111
78125/73728 696.165 5-limit least squares
11/10 696.176 Meanpop
(8 - φ)\11 696.214 Golden meantone
49/45 696.245
19/17 696.279 Classical meantone
47\81 696.296
7/6 696.319
27/22 696.3635 Meanpop
48/35 696.399
39/32 696.405 Tridecimal Meantone, Tridecimal Meanpop
14/11 696.413 Meanpop
[19 9 -1 -11⟩ 696.436 9-limit least squares
12/11 696.474 Meanpop
16384/15309 696.502 1/5 septimal comma
5/4 696.578 5-, 7-, 9- and 11- (Meanpop) limit minimax, 1/4 comma
49/48 696.616
60/49 696.626
[-55 -11 1 25⟩ 696.648 7-limit least squares
11/9 696.713 11-, 13- and 15- limit (Tridecimal Meantone) minimax
18\31 696.774
35/32 696.796
11/9 696.839 Meanpop
8/7 696.883
49/40 696.959
12/11 697.021 Undecimal Meantone
7/5 697.085
15/11 697.158 Undecimal Meantone
27/22 697.159
39/32 697.168 Grosstone
75/64 697.176
22/21 697.22 Undecimal Meantone
14/13 697.242 13, 15 limit minimax (Grosstone)
2187/2048 697.263 1/5 Pythagorean comma, Pythagorean apotome
13/10 697.289 Grosstone
11/8 697.295 Undecimal Meantone
43\74 697.297
[5/4 7] 697.339
21/16 697.344
13/11 697.376 Meridetone
45927/32768 697.411 1/6 septimal comma
18/13 697.465 13, 15 limit minimax (Meridetone)
16/13 696.467 Grosstone
11/10 697.5 Undecimal Meantone
15/13 697.511 Grosstone
13/12 697.637 Meridetone
16/15 697.654 1/5 comma
25\43 697.674
64/63 697.728
21/20 697.781
40/33, 16/13 697.797 Undecimal Meantone, Meridetone
14/11 697.812 Undecimal Meantone
15/13 697.83 Meridetone
18/13 697.966 Grosstone
13/10 698.009 Meridetone
1024/729 698.045 1/6 Pythagorean comma, lesser Pythagorean tritone
| - 17 9 0 1 > 698.06 1/7 septimal comma
28/25 698.099
32\55 698.182
33/28 698.272 Undecimal Meantone
80/63 698.303
17/15 698.331
45/32 698.371 1/6 comma
33/26 698.407 Meanplop, Meridetone
39\67 698.507
256/243 698.604 1/7 Pythagorean comma, Pythagorean limma
112/99 698.64 Undecimal Meantone
45/34 698.661
46\79 698.734
13/11 698.801 Meridetone, Meanplop
135/128 698.883 1/7 comma
17/16 699.009
25/21 699.384
7\12 700
18/17 700.209
19/16 700.829
81/80 701.792
31\53 701.887
3/2 701.955 Pythagorean tuning
64/63 702.272
256/189 702.301
33/26 703.186 Tridecimal Meantone
22/21 703.356 Meanpop
13/11 703.597 Tridecimal Meantone
88/81 707.946 Meanpop
710.4335 Undecimal Meantone

[5/4 7] eigenmonzos: meanwoo12, meanwoo19

Links

Retrieved from "https://en.xen.wiki/index.php?title=Meantone&oldid=42080"