Meantone: Difference between revisions

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Line 38: Line 38:
|690.225
|690.225
|1/2 Pythagorean comma, Pythagorean dilimma
|1/2 Pythagorean comma, Pythagorean dilimma
|-
|76/51
|690.603
|
|-
|-
|<nowiki>| -19 9 0 2 ></nowiki>
|<nowiki>| -19 9 0 2 ></nowiki>
Line 74: Line 78:
|694.165
|694.165
|2/7 septimal comma
|2/7 septimal comma
|-
|[[14/13]]
|694.34
|Tridecimal Meantone
|-
|-
| [[56/45]]
| [[56/45]]
Line 91: Line 99:
|
|
|-
|-
| [[6/5]]
| [[6/5]], [[25/18]]
| 694.786
| 694.786
| 1/3 comma
| 1/3 comma
|-
|[[18/13]]
|695.124
|Tridecimal Meantone
|-
|-
|5103/4095
|5103/4095
|695.139
|695.139
|1/4 septimal comma
|1/4 septimal comma
|-
|[[15/13]]
|695.226
|Tridecimal Meantone
|-
|-
|<nowiki>| 27 -17 ></nowiki>
|<nowiki>| 27 -17 ></nowiki>
Line 114: Line 130:
| 695.493
| 695.493
| Lucy Tuning
| Lucy Tuning
|-
|[[13/12]]
|695.612
|Tridecimal Meantone
|-
|-
| [[9/7]]
| [[9/7]]
Line 120: Line 140:
|-
|-
| f^4 = 2f + 2
| f^4 = 2f + 2
| 695.630
| 695.63
| Wilson fifth
| Wilson fifth
|-
|-
Line 128: Line 148:
|-
|-
| [[25/24]]
| [[25/24]]
| 695.810
| 695.81
| 2/7 comma
| 2/7 comma
|-
|-
| [[13/10]]
| [[13/10]]
| 695.838
| 695.838
| ratwolf fifth, meanpop eigenmonzo
| ratwolf fifth, Tridecimal Meantone and meanpop eigenmonzo
|-
|[[81/80]]
|695.869
|
|-
|-
| [[36/35]]
| [[36/35]]
Line 146: Line 170:
| 696
| 696
|
|
|-
|16/13
|696.035
|Tridecimal Meantone
|-
|-
|8192/6561
|8192/6561
Line 166: Line 194:
| 696.245
| 696.245
|
|
|-
|19/17
|696.279
|Classical meantone
|-
|-
| [[81edo|47\81]]
| [[81edo|47\81]]
Line 177: Line 209:
| [[48/35]]
| [[48/35]]
| 696.399
| 696.399
|
|-
|39/32
|696.405
|
|
|-
|-
Line 189: Line 225:
| [[5/4]]
| [[5/4]]
| 696.578
| 696.578
| 5- 7- and 9-limit minimax, 1/4 comma
| 5-, 7-, 9- and 11- (Meanpop) limit minimax, 1/4 comma
|-
|-
| 49/48
| 49/48
Line 202: Line 238:
| 696.648
| 696.648
| [[7-limit]] least squares
| [[7-limit]] least squares
|-
|[[11/9]]
|696.713
|11-, 13- and 15- limit (Tridecimal Meantone) minimax
|-
|-
| [[31edo|18\31]]
| [[31edo|18\31]]
Line 217: Line 257:
| [[49/40]]
| [[49/40]]
| 696.959
| 696.959
|
|-
|[[12/11]]
|697.021
|
|
|-
|-
| [[7/5]]
| [[7/5]]
| 697.085
| 697.085
|
|-
|[[15/11]]
|697.158
|
|-
|[[27/22]]
|697.159
|
|
|-
|-
Line 226: Line 278:
| 697.176
| 697.176
|
|
|-
|14/13
|697.242
|13, 15 limit minimax (Grosstone)
|-
|-
|[[2187/2048]]
|[[2187/2048]]
|697.263
|697.263
|1/5 Pythagorean comma, Pythagorean aptome
|1/5 Pythagorean comma, Pythagorean apotome
|-
|13/10
|697.289
|Grosstone
|-
|[[11/8]]
|697.295
|
|-
|-
| [[74edo|43\74]]
| [[74edo|43\74]]
Line 238: Line 302:
| 697.344
| 697.344
|
|
|-
|[[13/11]]
|697.376
|Meridetone
|-
|-
|45927/32768
|45927/32768
|697.411
|697.411
|1/6 septimal comma
|1/6 septimal comma
|-
|18/13
|697.465
|13, 15 limit minimax (Meridetone)
|-
|[[16/13]]
|696.467
|Grosstone
|-
|[[11/10]]
|697.5
|
|-
|15/13
|697.511
|Grosstone
|-
|13/12
|697.637
|Meridetone
|-
|-
| [[16/15]]
| [[16/15]]
Line 258: Line 346:
| 697.781
| 697.781
|
|
|-
|16/13
|697.797
|Meridetone
|-
|[[14/11]]
|697.812
|
|-
|15/13
|697.83
|Meridetone
|-
|[[18/13]]
|697.966
|Grosstone
|-
|13/10
|698.009
|Meridetone
|-
|-
|[[1024/729]]
|[[1024/729]]
|698.045
|698.045
|1/6 Pythagorean comma, Pythagorean tritone
|1/6 Pythagorean comma, lesser Pythagorean tritone
|-
|-
|<nowiki>| - 17 9 0 1 ></nowiki>
|<nowiki>| - 17 9 0 1 ></nowiki>
Line 273: Line 381:
| [[55edo|32\55]]
| [[55edo|32\55]]
| 698.182
| 698.182
|
|-
|33/28
|698.272
|
|
|-
|-
| [[80/63]]
| [[80/63]]
| 698.303
| 698.303
|
|-
|17/15
|698.331
|
|
|-
|-
Line 290: Line 406:
|698.604
|698.604
|1/7 Pythagorean comma, Pythagorean limma
|1/7 Pythagorean comma, Pythagorean limma
|-
|45/34
|698.661
|
|-
|-
| [[79edo|46\79]]
| [[79edo|46\79]]
| 698.734
| 698.734
|
|
|-
|13/11
|698.801
|Meridetone
|-
|-
|[[135/128]]
|[[135/128]]
Line 311: Line 435:
|
|
|-
|-
|[[18/17]]
|[[18/17|''18/17'']]
|700.209
|''700.209''
|
|-
|''[[19/16]]''
|''700.829''
|
|
|-
|-
| [[53edo|31\53]]
|[[81/80|''81/80'']]
| 701.887
|''701.792''
|
|
|-
|-
| [[3/2]]
| [[53edo|''31\53'']]
| 701.955
| ''701.887''
| [[Pythagorean tuning]]
|
|-
| [[3/2|''3/2'']]
| ''701.955''
| [[Pythagorean tuning|''Pythagorean tuning'']]
|-
|[[64/63|''64/63'']]
|''702.272''
|
|-
|''256/189''
|''702.301''
|
|-
|''33/26''
|''703.186''
|''Tridecimal Meantone''
|-
|''13/11''
|''703.597''
|''Tridecimal Meantone''
|-
|''88/81''
|''710.4335''
|
|}
|}


Line 327: Line 479:


== 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 23:33, 4 June 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
567/512 688.323 1/2 septimal comma
| 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
15\26 692.308
| 31 -19 > 692.571 2/5 Pythagorean comma
2048/1701 692.867 1/3 septimal comma
26\45 693.333
27/25 693.352 2/5 comma
19683/16384 694.135 1/3 Pythagorean comma, Pythagorean augmented second
| -23 11 0 2 > 694.165 2/7 septimal comma
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
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
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
13/10 695.838 ratwolf fifth, Tridecimal Meantone and meanpop eigenmonzo
81/80 695.869
36/35 695.936
54/49 695.987
29\50 696
16/13 696.035 Tridecimal Meantone
8192/6561 696.09 1/4 Pythagorean comma, Pythagorean diminished fourth
15/14 696.111
78125/73728 696.165 5-limit least squares
(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
48/35 696.399
39/32 696.405
[19 9 -1 -11⟩ 696.436 9-limit least squares
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
8/7 696.883
49/40 696.959
12/11 697.021
7/5 697.085
15/11 697.158
27/22 697.159
75/64 697.176
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
43\74 697.297
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
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
16/13 697.797 Meridetone
14/11 697.812
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
80/63 698.303
17/15 698.331
45/32 698.371 1/6 comma
39\67 698.507
256/243 698.604 1/7 Pythagorean comma, Pythagorean limma
45/34 698.661
46\79 698.734
13/11 698.801 Meridetone
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
13/11 703.597 Tridecimal Meantone
88/81 710.4335

[5/4 7] eigenmonzos: meanwoo12, meanwoo19

Links

Retrieved from "https://en.xen.wiki/index.php?title=Meantone&oldid=41903"