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== 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]]

Revision as of 10:28, 2 February 2021

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

The common meantone temperaments (recte tunings) include:

Spectrum of meantone tunings by eigenmonzos

Eigenmonzo Fifth size Comments
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
33/26 693.178 Grosstone
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
14/13 694.340 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
f4 = 2f + 2 695.630 Wilson fifth
(40\69) 695.652
25/24 695.810 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) 696.000
16/13 696.035 Tridecimal meantone, tridecimal meanpop
13/11 696.043 13- and 15-odd-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-odd-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-odd-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-odd-limit (meanpop) minimax, 1/4 comma
49/48 696.616
60/49 696.626
[-55 -11 1 25⟩ 696.648 7-odd-limit least squares
11/9 696.713 11-, 13- and 15-odd-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
22/21 697.220 Undecimal meantone
14/13 697.242 13- and 15-odd-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-odd-limit minimax (meridetone)
16/13 696.467 Grosstone
11/10 697.500 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
15/13 697.830 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⟩ 698.060 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
112/99 698.640 Undecimal meantone
45/34 698.661 [clarification needed]
(46\79) 698.734
13/11 698.801 Meridetone, meanplop
135/128 698.883 1/7 comma
17/16 699.009 [clarification needed]
25/21 699.384
(7\12) 700.000
18/17 700.209 [clarification needed]
19/16 700.829 [clarification needed]
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
88/81 710.4335 Undecimal meantone

[5/4 7] eigenmonzos: meanwoo12, meanwoo19

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