Xenharmonic Wiki

Meantone: Difference between revisions

← Older editNewer edit →
VisualWikitext
Xenwolf (talk | contribs)
autopatrolled, Bureaucrats, confirmaccount-notify, Interface administrators, Sysops
8,705 edits
newline between article contents and categories
m General cleanup
Line 2: Line 2:


== History ==
== 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 [[12edo|12-tone Equal]] Temperament won in popularity.
Meantone was the dominant tuning used in Europe from around late 15th century to around early 18th century, after which various [[Well Temperament]]s and eventually [[12edo|12-tone equal temperament]] won in popularity.


== Theory and Classification ==
== 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]] <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]] {{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 ==
== Meantone temperaments ==


The meantone temperaments (''recte'' tunings) include:
The common meantone temperaments (''recte'' tunings) include:
* [[1/2_syntonic_comma_meantone]]
* [[1/2 syntonic comma meantone]]
* [[33edo]]
* [[33edo]]
* [[26edo]]
* [[26edo]]
* [[2/5_syntonic_comma_meantone]]
* [[2/5 syntonic comma meantone]]
* [[45edo]]
* [[45edo]]
* [[1/3 syntonic comma meantone]]
* [[1/3 syntonic comma meantone]]
Line 33: Line 33:
* [[Tungsten meantone]]
* [[Tungsten meantone]]


== Spectrum of Meantone Tunings by Eigenmonzos ==
== Spectrum of meantone tunings by eigenmonzos ==


{| class="wikitable"
{| class="wikitable center-1 right-2"
|-
|-
! [[Eigenmonzo]]
! [[Eigenmonzo]]
! Fifth size
! Fifth size
! Usual name
! Comments
|-
|-
|567/512
| 567/512
|688.323
| 688.323
|1/2 septimal comma
| 1/2 septimal comma
|-
|-
|16/13
| 16/13
|689.868
| 689.868
|Meanplop
| Meanplop
|-
|-
|<nowiki>| 16 -10 ></nowiki>
| {{monzo| 16 -10 }}
|690.225
| 690.225
|1/2 Pythagorean comma, Pythagorean dilimma
| 1/2 Pythagorean comma, Pythagorean dilimma
|-
|-
|76/51
| 76/51
|690.603
| 690.603
|
|  
|-
|-
|<nowiki>| -19 9 0 2 ></nowiki>
| {{monzo| -19 9 0 2 }}
|691.049
| 691.049
|2/5 septimal comma
| 2/5 septimal comma
|-
|-
| [[10/9]]
| [[10/9]]
Line 65: Line 65:
| 1/2 comma
| 1/2 comma
|-
|-
|13/12
| 13/12
|692.285
| 692.285
|Meanplop
| Meanplop
|-
|-
| [[26edo|15\26]]
| [[26edo|(15\26)]]
| 692.308
| 692.308
|
|  
|-
|-
|<nowiki>| 31 -19 ></nowiki>
| {{monzo| 31 -19 }}
|692.571
| 692.571
|2/5 Pythagorean comma
| 2/5 Pythagorean comma
|-
|-
|2048/1701
| 2048/1701
|692.867
| 692.867
|1/3 septimal comma
| 1/3 septimal comma
|-
|-
|33/26
| 33/26
|693.178
| 693.178
|Grosstone
| Grosstone
|-
|-
|13/10
| 13/10
|693.223
| 693.223
|Meanplop
| Meanplop
|-
|-
| [[45edo|26\45]]
| [[45edo|(26\45)]]
| 693.333
| 693.333
|
|
Line 97: Line 97:
| 2/5 comma
| 2/5 comma
|-
|-
|18/13
| 18/13
|693.897
| 693.897
|Meanplop
| Meanplop
|-
|-
|19683/16384
| 19683/16384
|694.135
| 694.135
|1/3 Pythagorean comma, Pythagorean augmented second
| 1/3 Pythagorean comma, Pythagorean augmented second
|-
|-
|<nowiki>| -23 11 0 2 ></nowiki>
| {{monzo| -23 11 0 2 }}
|694.165
| 694.165
|2/7 septimal comma
| 2/7 septimal comma
|-
|-
|15/13
| 15/13
|694.193
| 694.193
|Meanplop
| Meanplop
|-
|-
|[[14/13]]
| [[14/13]]
|694.34
| 694.340
|Tridecimal Meantone
| Tridecimal meantone
|-
|-
| [[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
|
|
Line 137: Line 137:
| 1/3 comma
| 1/3 comma
|-
|-
|14/13
| 14/13
|694.878
| 694.878
|Meanplop
| Meanplop
|-
|-
|[[18/13]]
| [[18/13]]
|695.124
| 695.124
|Tridecimal Meantone
| Tridecimal meantone
|-
|-
|5103/4095
| 5103/4095
|695.139
| 695.139
|1/4 septimal comma
| 1/4 septimal comma
|-
|-
|[[15/13]]
| [[15/13]]
|695.226
| 695.226
|Tridecimal Meantone
| Tridecimal Meantone
|-
|-
|<nowiki>| 27 -17 ></nowiki>
| {{monzo| 27 -17 }}
|695.252
| 695.252
|2/7 Pythagorean comma, [[17-comma]]
| 2/7 Pythagorean comma, [[17-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)
Line 169: Line 169:
| Lucy Tuning
| Lucy Tuning
|-
|-
|39/28
| 39/28
|695.6095
| 695.6095
|Tridecimal Meantone, Tridecimal Meanpop
| Tridecimal meantone, tridecimal meanpop
|-
|-
|[[13/12]]
| [[13/12]]
|695.612
| 695.612
|Tridecimal Meantone
| Tridecimal meantone
|-
|-
| [[9/7]]
| [[9/7]]
Line 181: Line 181:
|
|
|-
|-
| f^4 = 2f + 2
| ''f''<sup>4</sup> = 2''f'' + 2
| 695.63
| 695.630
| Wilson fifth
| Wilson fifth
|-
|-
| [[69edo|40\69]]
| [[69edo|(40\69)]]
| 695.652
| 695.652
|
|
|-
|-
| [[25/24]]
| [[25/24]]
| 695.81
| 695.810
| 2/7 comma
| 2/7 comma
|-
|-
|40/33
| 40/33
|695.815
| 695.815
|Meanpop
| Meanpop
|-
|-
| [[13/10]]
| [[13/10]]
| 695.838
| 695.838
| ratwolf fifth, Tridecimal Meantone and meanpop eigenmonzo
| Ratwolf fifth, tridecimal meantone and meanpop eigenmonzo
|-
|-
|[[81/80]]
| [[81/80]]
|695.869
| 695.869
|
|
|-
|-
|112/99
| 112/99
|695.886
| 695.886
|Meanpop
| Meanpop
|-
|-
| [[36/35]]
| [[36/35]]
Line 217: Line 217:
|
|
|-
|-
| [[50edo|29\50]]
| [[50edo|(29\50)]]
| 696
| 696.000
|
|
|-
|-
|16/13
| 16/13
|696.035
| 696.035
|Tridecimal Meantone, Tridecimal Meanpop
| Tridecimal meantone, tridecimal meanpop
|-
|-
|13/11
| 13/11
|696.043
| 696.043
|13, 15 limit minimax (Tridecimal Meanpop)
| 13- and 15-odd-limit minimax (tridecimal meanpop)
|-
|-
|11/8
| 11/8
|696.052
| 696.052
|Meanpop
| Meanpop
|-
|-
|8192/6561
| 8192/6561
|696.09
| 696.09
|1/4 Pythagorean comma, Pythagorean diminished fourth
| 1/4 Pythagorean comma, Pythagorean diminished fourth
|-
|-
| [[15/14]]
| [[15/14]]
Line 243: Line 243:
| 78125/73728
| 78125/73728
| 696.165
| 696.165
| [[5-limit]] least squares
| [[5-odd-limit]] least squares
|-
|-
|11/10
| 11/10
|696.176
| 696.176
|Meanpop
| Meanpop
|-
|-
| (8 - φ)\11
| (8 - φ)\11
Line 257: Line 257:
|
|
|-
|-
|19/17
| 19/17
|696.279
| 696.279
|Classical meantone
| Classical meantone
|-
|-
| [[81edo|47\81]]
| [[81edo|(47\81)]]
| 696.296
| 696.296
|
|
Line 269: Line 269:
|
|
|-
|-
|27/22
| 27/22
|696.3635
| 696.3635
|Meanpop
| Meanpop
|-
|-
| [[48/35]]
| [[48/35]]
Line 277: Line 277:
|
|
|-
|-
|39/32
| 39/32
|696.405
| 696.405
|Tridecimal Meantone, Tridecimal Meanpop
| Tridecimal meantone, tridecimal meanpop
|-
|-
|14/11
| 14/11
|696.413
| 696.413
|Meanpop
| Meanpop
|-
|-
| {{Monzo| 19 9 -1 -11 }}
| {{Monzo| 19 9 -1 -11 }}
| 696.436
| 696.436
| 9-limit least squares
| 9-odd-limit least squares
|-
|-
|12/11
| 12/11
|696.474
| 696.474
|Meanpop
| Meanpop
|-
|-
|16384/15309
| 16384/15309
|696.502
| 696.502
|1/5 septimal comma
| 1/5 septimal comma
|-
|-
| [[5/4]]
| [[5/4]]
| 696.578
| 696.578
| 5-, 7-, 9- and 11- (Meanpop) limit minimax, 1/4 comma
| 5-, 7-, 9- and 11-odd-limit (meanpop) minimax, 1/4 comma
|-
|-
| 49/48
| 49/48
Line 309: Line 309:
|
|
|-
|-
| {{Monzo| -55 -11 1 25 }}
| {{monzo| -55 -11 1 25 }}
| 696.648
| 696.648
| [[7-limit]] least squares
| [[7-odd-limit]] least squares
|-
|-
|[[11/9]]
| [[11/9]]
|696.713
| 696.713
|11-, 13- and 15- limit (Tridecimal Meantone) minimax
| 11-, 13- and 15-odd-limit (tridecimal meantone) minimax
|-
|-
| [[31edo|18\31]]
| [[31edo|(18\31)]]
| 696.774
| 696.774
|
|
Line 325: Line 325:
|
|
|-
|-
|11/9
| 11/9
|696.839
| 696.839
|Meanpop
| Meanpop
|-
|-
| [[8/7]]
| [[8/7]]
Line 337: Line 337:
|
|
|-
|-
|[[12/11]]
| [[12/11]]
|697.021
| 697.021
|Undecimal Meantone
| Undecimal meantone
|-
|-
| [[7/5]]
| [[7/5]]
Line 345: Line 345:
|
|
|-
|-
|[[15/11]]
| [[15/11]]
|697.158
| 697.158
| rowspan="2" |Undecimal Meantone
| rowspan="2" | Undecimal meantone
|-
|-
|[[27/22]]
| [[27/22]]
|697.159
| 697.159
|-
|-
|39/32
| 39/32
|697.168
| 697.168
|Grosstone
| Grosstone
|-
|-
| [[75/64]]
| [[75/64]]
Line 360: Line 360:
|
|
|-
|-
|22/21
| 22/21
|697.22
| 697.220
|Undecimal Meantone
| Undecimal meantone
|-
|-
|14/13
| 14/13
|697.242
| 697.242
|13, 15 limit minimax (Grosstone)
| 13- and 15-odd-limit minimax (grosstone)
|-
|-
|[[2187/2048]]
| [[2187/2048]]
|697.263
| 697.263
|1/5 Pythagorean comma, Pythagorean apotome
| 1/5 Pythagorean comma, Pythagorean apotome
|-
|-
|13/10
| 13/10
|697.289
| 697.289
|Grosstone
| Grosstone
|-
|-
|[[11/8]]
| [[11/8]]
|697.295
| 697.295
|Undecimal Meantone
| Undecimal meantone
|-
|-
| [[74edo|43\74]]
| [[74edo|(43\74)]]
| 697.297
| 697.297
|
|
|-
|-
|[5/4 7]
| [5/4 7]
|697.339
| 697.339
|
|
|-
|-
Line 392: Line 392:
|
|
|-
|-
|[[13/11]]
| [[13/11]]
|697.376
| 697.376
|Meridetone
| Meridetone
|-
|-
|45927/32768
| 45927/32768
|697.411
| 697.411
|1/6 septimal comma
| 1/6 septimal comma
|-
|-
|18/13
| 18/13
|697.465  
| 697.465  
|13, 15 limit minimax (Meridetone)
| 13-, 15-odd-limit minimax (meridetone)
|-
|-
|[[16/13]]
| [[16/13]]
|696.467
| 696.467
|Grosstone
| Grosstone
|-
|-
|[[11/10]]
| [[11/10]]
|697.5
| 697.500
|Undecimal Meantone
| Undecimal meantone
|-
|-
|15/13
| 15/13
|697.511
| 697.511
|Grosstone
| Grosstone
|-
|-
|13/12
| 13/12
|697.637
| 697.637
|Meridetone
| Meridetone
|-
|-
| [[16/15]]
| [[16/15]]
Line 424: Line 424:
| 1/5 comma
| 1/5 comma
|-
|-
| [[43edo|25\43]]
| [[43edo|(25\43)]]
| 697.674
| 697.674
|
|
Line 436: Line 436:
|
|
|-
|-
|40/33, 16/13
| 40/33, 16/13
|697.797
| 697.797
|Undecimal Meantone, Meridetone
| Undecimal meantone, meridetone
|-
|-
|[[14/11]]
| [[14/11]]
|697.812
| 697.812
|Undecimal Meantone
| Undecimal meantone
|-
|-
|15/13
| 15/13
|697.83
| 697.830
|Meridetone
| Meridetone
|-
|-
|[[18/13]]
| [[18/13]]
|697.966
| 697.966
|Grosstone
| Grosstone
|-
|-
|13/10
| 13/10
|698.009
| 698.009
|Meridetone
| Meridetone
|-
|-
|[[1024/729]]
| [[1024/729]]
|698.045
| 698.045
|1/6 Pythagorean comma, lesser Pythagorean tritone
| 1/6 Pythagorean comma, lesser Pythagorean tritone
|-
|-
|<nowiki>| - 17 9 0 1 ></nowiki>
| {{monzo| - 17 9 0 1 }}
|698.06
| 698.060
|1/7 septimal comma
| 1/7 septimal comma
|-
|-
| [[28/25]]
| [[28/25]]
Line 468: Line 468:
|
|
|-
|-
| [[55edo|32\55]]
| [[55edo|(32\55)]]
| 698.182
| 698.182
|
|
|-
|-
|33/28
| 33/28
|698.272
| 698.272
|Undecimal Meantone
| Undecimal meantone
|-
|-
| [[80/63]]
| [[80/63]]
Line 480: Line 480:
|
|
|-
|-
|17/15
| 17/15
|698.331
| 698.331
|
|
|-
|-
Line 488: Line 488:
| 1/6 comma
| 1/6 comma
|-
|-
|33/26
| 33/26
|698.407
| 698.407
|Meanplop, Meridetone
| Meanplop, meridetone
|-
|-
| [[67edo|39\67]]
| [[67edo|(39\67)]]
| 698.507
| 698.507
|
|
|-
|-
|[[256/243]]
| [[256/243]]
|698.604
| 698.604
|1/7 Pythagorean comma, Pythagorean limma
| 1/7 Pythagorean comma, Pythagorean limma
|-
|-
|112/99
| 112/99
|698.64
| 698.640
|Undecimal Meantone
| Undecimal meantone
|-
|-
|45/34
| 45/34
|698.661
| 698.661
|
| {{clarify}}
|-
|-
| [[79edo|46\79]]
| [[79edo|(46\79)]]
| 698.734
| 698.734
|
|
|-
|-
|13/11
| 13/11
|698.801
| 698.801
|Meridetone, Meanplop
| Meridetone, meanplop
|-
|-
|[[135/128]]
| [[135/128]]
|698.883
| 698.883
|1/7 comma
| 1/7 comma
|-
|-
|[[17/16]]
| [[17/16]]
|699.009
| 699.009
|
| {{clarify}}
|-
|-
| [[25/21]]
| [[25/21]]
Line 528: Line 528:
|
|
|-
|-
| [[12edo|7\12]]
| [[12edo|(7\12)]]
| 700
| 700.000
|
|
|-
|-
|[[18/17|''18/17'']]
| ''[[18/17]]''
|''700.209''
| ''700.209''
|
| {{clarify}}
|-
|-
|''[[19/16]]''
| ''[[19/16]]''
|''700.829''
| ''700.829''
|
| {{clarify}}
|-
|-
|[[81/80|''81/80'']]
| ''[[81/80]]''
|''701.792''
| ''701.792''
|
|  
|-
|-
| [[53edo|''31\53'']]
| [[53edo|(''31\53'')]]
| ''701.887''
| ''701.887''
|
|
|-
|-
| [[3/2|''3/2'']]
| ''[[3/2]]''
| ''701.955''
| ''701.955''
| [[Pythagorean tuning|''Pythagorean tuning'']]
| ''[[Pythagorean tuning]]''
|-
|-
|[[64/63|''64/63'']]
| ''[[64/63]]''
|''702.272''
| ''702.272''
|
|
|-
|-
|''256/189''
| ''256/189''
|''702.301''
| ''702.301''
|
|
|-
|-
|''33/26''
| ''33/26''
|''703.186''
| ''703.186''
|''Tridecimal Meantone''
| ''Tridecimal meantone''
|-
|-
|''22/21''
| ''22/21''
|''703.356''
| ''703.356''
|''Meanpop''
| ''Meanpop''
|-
|-
|''13/11''
| ''13/11''
|''703.597''
| ''703.597''
|''Tridecimal Meantone''
| ''Tridecimal meantone''
|-
|-
| rowspan="2" |''88/81''
| ''88/81''
|''707.946''
| ''707.946''
|''Meanpop''
| ''Meanpop''
|-
|-
|''710.4335''
| ''88/81''
|''Undecimal Meantone''
| ''710.4335''
| ''Undecimal meantone''
|}
|}


Line 583: Line 584:


== Links ==
== Links ==
* http://www.kylegann.com/histune.html -- An Introduction to Historical Tunings, by [[Kyle Gann]]
* [http://www.kylegann.com/histune.html An Introduction to Historical Tunings], by [[Kyle Gann]]


[[Category:Meantone| ]] <!-- main article -->
[[Category:Meantone| ]] <!-- main article -->
[[Category:Temperament]]
[[Category:Temperament]]
[[Category:Theory]]
[[Category:Theory]]
{{todo|unify precision}}


<!-- interwiki -->
<!-- interwiki -->
[[de:Mitteltönig]]
[[de:Mitteltönig]]
Retrieved from "https://en.xen.wiki/w/Meantone"