User:BudjarnLambeth/Table of n-comma meantone generators: Difference between revisions

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{| class="wikitable"
{{DISPLAYTITLE:User:BudjarnLambeth/Table of ''n''-comma meantone generators}}{{Editable user page}}
 
Here are all [[meantone]] tunings that can be written in the form "n-comma meantone", where the syntonic comma ([[81/80]]) is being divided and n is a fraction between -1 and 1 with a denominator 22 or smaller.
 
== Scope of table ==
 
=== Characteristics included ===
Some of the characteristics this table mentions for each temperament include:
* Whether it saw historical (pre-1950) use
* Whether it is close to (i.e. within 1/2 a degree of closing of) an [[edo]] smaller than 100
* Whether it is the closest on the table to the optimal [[CTE]] or [[POTE]] tuning of meantone or [[superpyth]] in any JI [[limit]]
* Whether it approximates a very simple n-[[Pythagorean comma]] meantone
* Whether it is about equally sharp of [[3/2]] as some other listed temperament is flat
* Whether it is close to exactly one [[just-noticeable difference]] away from 3/2
 
Occasional other comments may be included as well.
 
=== Characteristics omitted ===
Dozens of tunings on this table are significant to [[negative harmony temperaments|negative harmony temperament theory]], enough that labelling them all individually would clutter the table.
 
Every tuning on this table is close to some arbitrarily large edo, but labeling them beyond [[100edo]] would clutter the table.
 
=== Special cases included ===
A small number of additional temperaments are included. Not too many, to avoid clutter, just the bare minimum:
* {{EDOs|7, 12, 17 and 5}} edos (to delineate small [[MOS]] shapes and boundaries of [[diamond monotone]])
* any tunings listed under "[[historical temperaments]]" (e.g. 4/25-comma), ''but only the ones of the form "n-comma"''.
 
== Cautions ==
=== Preservation of meantone behavior ===
Temperaments that fall outside of the "[[Historical temperaments|historically-defined meantone]]" range will not possess most of the musical properties that meantone usually possesses, but they are included for completeness.
 
Temperaments that fall outside of the "diamond monotone" range preserve even fewer meantone properties, but they are also included for completeness.
 
== The table ==
 
=== Flatter than flattest historically-defined meantone ===
{| class="wikitable mw-collapsible"
|+Spectrum of meantone tunings 1/1-comma to 1/2-comma
!Meantone Temperament!!Generator (cents)!!Comments
|-
|[[1/1-comma meantone|1/1-comma]] ||680.449||Close to [[30edo]]
|-
|[[21/22-comma meantone|21/22-comma]]
|681.426
|Close to [[37edo]].
|-
|[[20/21-comma meantone|20/21-comma]]
|681.473
|
|-
|[[19/20-comma meantone|19/20-comma]]
|681.524
|
|-
|[[18/19-comma meantone|18/19-comma]]
|681.581
|
|-
|[[17/18-comma meantone|17/18-comma]]
|681.644
|
|-
|[[16/17-comma meantone|16/17-comma]]
|681.713
|
|-
|[[15/16-comma meantone|15/16-comma]] ||681.793|| Close to [[44edo]].
|-
|[[14/15-comma meantone|14/15-comma]] ||681.883||
|-
|[[13/14-comma meantone|13/14-comma]] ||681.985||
|-
|[[12/13-comma meantone|12/13-comma]] ||682.103||
|-
|[[11/12-comma meantone|11/12-comma]] ||682.241||
|-
|[[10/11-comma meantone|10/11-comma]] ||682.404||Close to [[51edo]].
|-
|[[19/21-comma meantone|19/21-comma]]
|682.497
|
|-
|[[9/10-comma meantone|9/10-comma]]||682.599||
|-
|[[17/19-comma meantone|17/19-comma]]
|682.713
|
|-
|[[8/9-comma meantone|8/9-comma]] ||682.838||Close to [[58edo]].
|-
|[[15/17-comma meantone|15/17-comma]]
|682.979
|
|-
|[[7/8-comma meantone|7/8-comma]] ||683.137||Close to [[65edo]].
|-
|[[13/15-comma meantone|13/15-comma]] ||683.316||Close to [[72edo]].
|-
|[[19/22-comma meantone|19/22-comma]]
|683.381
|
|-
|[[6/7-comma meantone|6/7-comma]] ||683.521||Close to [[79edo]].
|-
|[[17/20-comma meantone|17/20-comma]]
|683.675
|
|-
|[[11/13-comma meantone|11/13-comma]] ||683.757||Close to [[86edo]].
|-
|[[16/19-comma meantone|16/19-comma]]
|683.844
|
|-
|[[21/25-comma meantone|21/25-comma]]
|683.890
|Close to [[93edo]]
|-
|[[5/6-comma meantone|5/6-comma]] ||684.033|| Close to [[100edo]].
|-
|[[14/17-comma meantone|14/17-comma]]
|684.244
|
|-
|[[9/11-comma meantone|9/11-comma]] ||684.359||
|-
|[[13/16-comma meantone|13/16-comma]] ||684.481||
|-
|[[17/21-comma meantone|17/21-comma]]
|684.545
|
|-
|[[4/5-comma meantone|4/5-comma]] ||684.750||
|-
|[[15/19-comma meantone|15/19-comma]]
|684.976
|
|-
|[[11/14-comma meantone|11/14-comma]] ||685.057||
|-
|[[7/9-comma meantone|7/9-comma]] ||685.228||
|-
|[[17/22-comma meantone|17/22-comma]]
|685.337
|
|-
|[[10/13-comma meantone|10/13-comma]] ||685.412||
|-
|[[13/17-comma meantone|13/17-comma]]
|685.509
|
|-
|[[16/21-comma meantone|16/21-comma]]
|685.569
|Everything up to this point generates 9 and 16 tone MOS scales.
|-
|[[7edo]]||685.714||The largest MOS scale this can generate is 7 tone. '''Lower boundary of 5-limit diamond monotone.'''
|-
|[[3/4-comma meantone|3/4-comma]] ||685.825||Everything from this point onwards generates 12 and 19 tone MOS scales.
|-
|[[14/19-comma meantone|14/19-comma]]
|686.108
|
|-
|[[11/15-comma meantone|11/15-comma]] ||686.184||
|-
|[[19/26-comma meantone|19/26-comma]]
|686.239
|
|-
|[[8/11-comma meantone|8/11-comma]]||686.314||
|-
|[[13/18-comma meantone|13/18-comma]]
|686.423
|
|-
|[[5/7-comma meantone|5/7-comma]] ||686.593||
|-
|[[17/24-comma meantone|17/24-comma]]
|686.721
|
|-
|[[12/17-comma meantone|12/17-comma]]
|686.774
|
|-
|[[7/10-comma meantone|7/10-comma]] ||686.901||
|-
|[[9/13-comma meantone|9/13-comma]] ||687.066||
|-
|[[11/16-comma meantone|11/16-comma]] ||687.169||
|-
|[[13/19-comma meantone|13/19-comma]]
|687.240
|
|-
|[[15/22-comma meantone|15/22-comma]] 
|687.292
|
|-
|[[17/25-comma meantone|17/25-comma]]
|687.331
|
|-
|[[19/28-comma]]
|687.361
|
|-
|[[2/3-comma meantone|2/3-comma]] ||687.617||Close to [[89edo]].
|-
|[[17/26-comma meantone|17/26-comma]]
|687.893
|Close to [[82edo]].
|-
|[[15/23-comma meantone|15/23-comma]]
|687.929
|
|-
|[[13/20-comma meantone|13/20-comma]]
|687.976
|
|-
|[[11/17-comma  meantone|11/17-comma]] 
|688.039
|Close to [[75edo]]
|-
|[[9/14-comma meantone|9/14-comma]] ||688.129||
|-
|[[7/11-comma meantone|7/11-comma]] ||688.269||Close to [[68edo]].
|-
|[[12/19-comma meantone|12/19-comma]]
|688.372
|
|-
|[[5/8-comma meantone|5/8-comma]] ||688.514||Close to [[61edo]] and [[43/32]].
|-
|[[13/21-comma meantone|13/21-comma]]
|688.641
|
|-
|[[1/φ-comma meantone|1/ϕ-comma]]
|688.663
|
|-
|[[8/13-comma meantone|8/13-comma]] ||688.720||
|-
|[[11/18-comma meantone|11/18-comma]]
|688.812
|Close to [[54edo]].
|-
|[[14/23-comma meantone|14/23-comma]] 
|688.864
|
|-
|[[3/5-comma meantone|3/5-comma]] ||689.051||
|-
|[[13/22-comma meantone|13/22-comma]]
|689.247
|
|-
|[[10/17-comma meantone|10/17-comma]]
|689.304
|
|-
|[[7/12-comma meantone|7/12-comma]] ||689.410||Close to [[47edo]].
|-
|[[11/19-comma meantone|11/19-comma]]
|689.504
|
|-
|[[4/7-comma meantone|4/7-comma]] ||689.666||Close to [[87edo]].
|-
|[[9/16-comma meantone|9/16-comma]] ||689.858||
|-
|[[5/9-comma meantone|5/9-comma]] ||690.007||Close to [[40edo]].
|-
|[[11/20-comma meantone|11/20-comma]]
|690.127
|
|-
|[[6/11-comma meantone|6/11-comma]] ||690.224||
|-
|[[7/13-comma meantone|7/13-comma]] ||690.375||Close to [[73edo]].
|-
|[[8/15-comma meantone|8/15-comma]] ||690.485||
|-
|[[9/17-comma meantone|9/17-comma]]
|690.569
|
|-
|[[10/19-comma meantone|10/19-comma]]
|690.636
|
|-
|[[11/21-comma meantone|11/21-comma]]
|690.690
|Close to [[33edo]]
|-
|[[1/2-comma meantone|1/2-comma]] ||691.202||Close to [[92edo]], [[59edo]]. Historically significant (see [[historical temperaments]]). Everything up to this point does not have a whole tone between 10/9 and 9/8.
|}
 
=== Historically-defined meantone ===
{| class="wikitable mw-collapsible"
|+Spectrum of meantone tunings 10/21-comma to 1/22-comma
!Temperament!!Generator (cents)!!Comments
!Temperament!!Generator (cents)!!Comments
|-
|-
|1/1-comma “meantone” (actually [[mavila]])||680.449||Not meantone but included for reference
|[[10/21-comma meantone|10/21-comma]]
|691.714
|
|-
|[[9/19-comma meantone|9/19-comma]]
|691.768
|Close to [[85edo]].
|-
|[[8/17-comma meantone|8/17-comma]]
|691.834
|
|-
|[[7/15-comma meantone|7/15-comma]] ||691.919||
|-
|[[6/13-comma meantone|6/13-comma]] ||692.029||
|-
|[[5/11-comma meantone|5/11-comma]] ||692.179||
|-
|[[9/20-comma meantone|9/20-comma]]
|692.277
|Close to [[26edo]].
|-
|[[4/9-comma meantone|4/9-comma]] ||692.397||
|-
|[[7/16-comma meantone|7/16-comma]] ||692.546||
|-
|[[3/7-comma meantone|3/7-comma]] ||692.738||
|-
|[[8/19-comma meantone|8/19-comma]]
|692.899
|
|-
|[[5/12-comma meantone|5/12-comma]] ||692.994||Close to [[71edo]].
|-
|[[7/17-comma meantone|7/17-comma]] ||693.099||
|-
|[[9/22-comma meantone|9/22-comma]]
|693.157
|
|-
|[[2/5-comma meantone|2/5-comma]] ||693.352||Close to [[45edo]].
|-
|[[9/23-comma meantone|9/23-comma]]
|693.539
|
|-
|[[7/18-comma meantone|7/18-comma]] ||693.591||
|-
|[[5/13-comma meantone|5/13-comma]] ||693.683||
|-
|[[1/(φ+1)-comma meantone|1/(ϕ+1)-comma]]
|693.740
|Close to [[64edo]].
|-
|[[8/21-comma meantone|8/21-comma]]
|693.762
|
|-
|[[3/8-comma meantone|3/8-comma]] ||693.890||Close to [[83edo]].
|-
|[[7/19-comma meantone|7/19-comma]]
|694.032
|
|-
|[[4/11-comma meantone|4/11-comma]] ||694.134||Almost exactly 1/3-''Pythagorean'' comma meantone.
|-
|[[5/14-comma meantone|5/14-comma]] ||694.274||
|-
|[[6/17-comma meantone|6/17-comma]] ||694.365||
|-
|[[7/20-comma meantone|7/20-comma]]
|694.428
|
|-
|[[8/23-comma meantone|8/23-comma]]
|694.475
|
|-
|[[9/26-comma meantone|9/26-comma]]
|694.511
|
|-
|[[1/3-comma meantone|1/3-comma]] ||694.786||Close to [[19edo]]. Historically significant (see [[historical temperaments]]).
|-
|[[9/28-comma meantone|9/28-comma]] 
|695.042
|
|-
|[[8/25-comma meantone|8/25-comma]] 
|695.073
|
|-
|[[7/22-comma meantone|7/22-comma]] 
|695.112
|
|-
|[[6/19-comma meantone|6/19-comma]]
|695.164
|
|-
|[[5/16-comma meantone|5/16-comma]] ||695.234||
|-
|[[4/13-comma meantone|4/13-comma]] ||695.338||
|-
|[[3/10-comma meantone|3/10-comma]] ||695.503||Close to [[88edo]] and [[Lucy tuning]]. Historically significant (see [[historical temperaments]]).
|-
|[[5/17-comma meantone|5/17-comma]] ||695.630||Close to [[69edo]].
|-
|[[7/24-comma meantone|7/24-comma]]
|695.682
|
|-
|[[2/7-comma meantone|2/7-comma]] ||695.810||Historically significant (see [[historical temperaments]]).
|-
|[[5/18-comma meantone|5/18-comma]] ||695.981||Close to [[50edo]].
|-
|[[3/11-comma meantone|3/11-comma]] ||696.090||
|-
|[[7/26-comma meantone|7/26-comma]] ||696.165||Close to [[golden meantone]]. Historically significant (see [[historical temperaments]]).
|-
|[[4/15-comma meantone|4/15-comma]] ||696.220||Close to [[5-limit]] meantone [[POTE]] tuning.
|-
|[[5/19-comma meantone|5/19-comma]]
|696.295
|Close to [[81edo]].
|-
|[[Quarter-comma meantone|1/4-comma]] ||696.578||Close to [[7-limit|septimal]] and [[tridecimal]] meantone POTE tunings. Historically significant (see [[historical temperaments]]).
|-
|[[5/21-comma meantone|5/21-comma]]
|696.834
|Close to [[31edo]].
|-
|[[4/17-comma meantone|4/17-comma]] ||696.895||
|-
|[[3/13-comma meantone|3/13-comma]] ||696.992||Close to [[7-limit|septimal]] & [[tridecimal]] meantone [[CTE]] tunings. Close to [[undecimal]] meantone POTE tuning.
|-
|[[5/22-comma meantone|5/22-comma]] 
|697.067
|
|-
|[[2/9-comma meantone|2/9-comma]] ||697.176||Close to [[5-limit]] and [[undecimal]] meantone CTE tunings. Historically significant (see [[historical temperaments]]).
|-
|[[3/14-comma meantone|3/14-comma]] ||697.346||Close to [[74edo]]. Historically significant (see [[historical temperaments]]).
|-
|[[4/19-comma meantone|4/19-comma]]
|697.427
|
|-
|[[1/5-comma meantone|1/5-comma]] ||697.654||Close to [[43edo]]. Historically significant (see [[historical temperaments]]).
|-
|[[4/21-comma meantone|4/21-comma]]
|697.859
|
|-
|[[3/16-comma meantone|3/16-comma]] ||697.923||
|-
|[[2/11-comma meantone|2/11-comma]] ||698.045||Close to [[55edo]]. Historically significant (see [[historical temperaments]]).
|-
|[[3/17-comma meantone|3/17-comma]] ||698.159||
|-
|[[4/23-comma meantone|4/23-comma]] 
|698.215
|
|-
|[[1/6-comma meantone|1/6-comma]] ||698.371||Historically significant (see [[historical temperaments]]). Everything up to this point has a fifth which is flat of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
|[[4/25-comma meantone|4/25-comma]] ||698.514||Close to [[67edo]].
|-
|[[3/19-comma meantone|3/19-comma]]
|698.559
|
|-
|[[2/13-comma meantone|2/13-comma]] ||698.646|| Close to [[79edo]].
|-
|[[3/20-comma meantone|3/20-comma]]
|698.729
|
|-
|[[1/7-comma meantone|1/7-comma]] ||698.883||Close to [[91edo]]. Historically significant (see [[historical temperaments]]).
|-
|[[3/22-comma meantone|3/22-comma]] 
|699.022
|
|-
|[[2/15-comma meantone|2/15-comma]] ||699.088||
|-
|[[1/8-comma meantone|1/8-comma]] ||699.267||
|-
|[[2/17-comma meantone|2/17-comma]] ||699.425||
|-
|[[1/9-comma meantone|1/9-comma]] ||699.565||
|-
|[[2/19-comma meantone|2/19-comma]]
|699.691
|
|-
|[[1/10-comma meantone|1/10-comma]] ||699.804||
|-
|[[2/21-comma meantone|2/21-comma]]
|699.907
|
|-
|[[1/11-comma meantone|1/11-comma]] ||700.000||Everything up to this point generates 12 and 19 tone MOS scales.
|-
|[[12edo]]||700.000||The largest MOS scale this can generate is 12 tone. Historically significant (see [[historical temperaments]].)
|-
|[[1/12-comma meantone|1/12-comma]] ||700.163||Everything from this point onwards generates 12 and 17 tone MOS scales.
|-
|[[1/13-comma meantone|1/13-comma]] ||700.301||
|-
|[[1/14-comma meantone|1/14-comma]] ||700.419||
|-
|[[1/15-comma meantone|1/15-comma]] ||700.521||
|-
|[[1/16-comma meantone|1/16-comma]] ||700.611||
|-
|[[1/17-comma meantone|1/17-comma]] ||700.690||
|-
|[[1/18-comma meantone|1/18-comma]] ||700.760||
|-
|[[1/19-comma meantone|1/19-comma]]
|700.823
|
|-
|[[1/20-comma meantone|1/20-comma]]
|700.879
|
|-
|[[1/21-comma meantone|1/21-comma]]
|700.931
|
|-
|[[1/22-comma meantone|1/22-comma]]
|700.977
|
|}
 
=== Negative harmony theory-defined meantone (most often approached as [[superpyth]]) ===
{| class="wikitable mw-collapsible"
|+Spectrum of meantone tunings 0/1-comma to -10/21-comma
!Meantone Temperament!!Generator (cents)!!Comments
|-
|[[Pythagorean tuning]]
|701.955||Historically significant (see [[historical temperaments]].)  Everything from this point onwards does not have a whole tone between 10/9 and 9/8.
|-
|[[-1/22-comma meantone|-1/22-comma]]
|702.933
|
|-
|[[-1/21-comma meantone|-1/21-comma]]
|702.979
|
|-
|[[-1/20-comma meantone|-1/20-comma]]
|703.030
|
|-
|[[-1/19-comma meantone|-1/19-comma]]
|703.087
|
|-
|[[-1/18-comma meantone|-1/18-comma]]
|703.150
|
|-
|[[-1/17-comma meantone|-1/17-comma]]
|703.220
|
|-
|[[-1/16-comma meantone|-1/16-comma]]
|703.299
|
|-
|[[-1/15-comma meantone|-1/15-comma]]
|703.389
|Close to 11/13 third-[[kleisma]] temperament.
|-
|[[-1/14-comma meantone|-1/14-comma]]
|703.491
|Close to [[29edo]].
|-
|[[-1/13-comma meantone|-1/13-comma]]
|703.609
|
|-
|[[-1/12-comma meantone|-1/12-comma]]
|703.747
|
|-
|[[-1/11-comma meantone|-1/11-comma]]
|703.910
|About as sharp of [[Pythagorean tuning]] as [[12edo]] is flat.
|-
|[[-2/21-comma meantone|-2/21-comma]]
|704.003
|Close to [[75edo]].
|-
|[[-1/10-comma meantone|-1/10-comma]]
|704.105
|
|-
|[[-2/19-comma meantone|-2/19-comma]]
|704.219
|
|-
|[[-1/9-comma meantone|-1/9-comma]]
|704.344
|Close to [[46edo]], 11/7 quarter-kleisma temperament.
|-
|[[-2/17-comma meantone|-2/17-comma]]
|704.483
|
|-
|[[-1/8-comma meantone|-1/8-comma]]
|704.643
|
|-
|[[-2/15-comma meantone|-2/15-comma]]
|704.823
|Close to [[63edo]].
|-
|[[-3/22-comma meantone|-3/22-comma]]
|704.888
|
|-
|[[-1/7-comma meantone|-1/7-comma]]
|705.027
|Close to [[80edo]].
|-
|[[-3/20-comma meantone|-3/20-comma]]
|705.181
|
|-
|[[-2/13-comma meantone|-2/13-comma]]
|705.350
|
|-
|[[-3/19-comma meantone|-3/19-comma]]
|705.350
|
|-
|[[-4/25-comma meantone|-4/25-comma]]
|705.396
|
|-
|[[-1/6-comma meantone|-1/6-comma]]
|705.538
| Everything from this point onwards has a fifth which is sharp of [[Pythagorean tuning]] by at least the [[just-noticeable difference]].
|-
|[[-4/23-comma meantone|-4/23-comma]] 
|705.695
|
|-
|[[-3/17-comma meantone|-3/17-comma]]
|705.750
|About as sharp of [[Pythagorean tuning]] as [[55edo]] is flat.
|-
|[[-2/11-comma meantone|-2/11-comma]]
|705.865
|Everything up to this point generates 17 and 29 tone MOS scales.
|-
|[[17edo]]
|705.882
|The largest MOS scale this can generate is 17 tone. Vaguely resembles Middle Eastern [[neutral third scale]]s.
|-
|[[-3/16-comma meantone|-3/16-comma]]
|705.987
|Everything from this point onwards generates 17 and 22 tone MOS scales.
|-
|[[-4/21-comma meantone|-4/21-comma]]
|706.051
|
|-
|[[-1/5-comma meantone|-1/5-comma]]
|706.256
|About as sharp of [[Pythagorean tuning]] as [[43edo]] is flat.
|-
|[[-4/19 comma meantone|-4/19 comma]]
|706.483
|
|-
|[[-3/14-comma meantone|-3/14-comma]]
|706.563
| About as sharp of [[Pythagorean tuning]] as [[74edo]] is flat.
|-
|[[-2/9-comma meantone|-2/9-comma]]
|706.734
|
|-
|[[-5/22-comma meantone|-5/22-comma]] 
|706.843
|
|-
|[[-3/13-comma meantone|-3/13-comma]]
|706.918
|Close to [[39edo]].
|-
|[[-4/17-comma meantone|-4/17-comma]]
|707.015
|
|-
|[[-5/21-comma meantone|-5/21-comma]]
|707.076
|About as sharp of [[Pythagorean tuning]] as [[31edo]] is flat.
|-
|[[-1/4-comma meantone|-1/4-comma]]
|707.332
|
|-
|[[-5/19-comma meantone|-5/19-comma]]
|707.615
|
|-
|[[-4/15-comma meantone|-4/15-comma]]
|707.690
|About as sharp of [[Pythagorean tuning]] as [[golden meantone]] is flat.
|-
|[[-7/26-comma meantone|-7/26-comma]]
|707.745
|
|-
|[[-3/11-comma meantone|-3/11-comma]]
|707.820
|Almost exactly -1/4-''Pythagorean'' comma meantone
|-
|[[-5/18-comma meantone|-5/18-comma]]
|707.930
|About as sharp of [[Pythagorean tuning]] as [[50edo]] is flat. Close to [[100edo]].
|-
|[[-2/7-comma meantone|-2/7-comma]]
|708.100
|
|-
|[[-7/24-comma meantone|-7/24-comma]]
|708.227
|
|-
|[[-5/17-comma meantone|-5/17-comma]]
|708.280
|
|-
|[[-3/10-comma meantone|-3/10-comma]]
|708.407
|Nearly as sharp of [[Pythagorean tuning]] as [[Lucy tuning]] is flat. Nearly as sharp of [[Pythagorean tuning]] as [[88edo]] is flat.
|-
|[[-4/13-comma meantone|-4/13-comma]]
|708.572
|
|-
|[[-5/16-comma meantone|-5/16-comma]]
|708.675
|
|-
|[[-6/19-comma meantone|-6/19-comma]]
|708.746
|
|-
|[[-7/22-comma meantone|-7/22-comma]]
|708.800
|
|-
|[[-8/25-comma meantone|-8/25-comma]]
|708.837
|
|-
|[[-9/28-comma meantone|-9/28-comma]]
|708.867
|
|-
|[[-1/3-comma meantone|-1/3-comma]]
|709.124
|Close to [[22edo]]. About as sharp of [[Pythagorean tuning]] as [[19edo]] is flat.
|-
|[[-9/26-comma meantone|-9/26-comma]]
|709.399
|Close to [[2.3.7-limit]] superpyth [[POTE]] tuning.
|-
|[[-8/23-comma meantone|-8/23-comma]]
|709.435
|
|-
|[[-7/20-comma meantone|-7/20-comma]]
|709.482
|
|-
|[[-6/17-comma meantone|-6/17-comma]]
|709.545
|Close to [[11-limit]] superpyth [[CTE]] tuning.
|-
|[[-5/14-comma meantone|-5/14-comma]]
|709.636
|Close to [[93edo]]. Close to [[2.3.7-limit]] and [[7-limit]] superpyth CTE tunings.
|-
|[[-4/11-comma meantone|-4/11-comma]]
|709.775
|Almost exactly -1/3-''Pythagorean'' comma meantone.
|-
|[[-7/19-comma meantone|-7/19-comma]]
|709.878
|Close to [[13-limit]] superpyth CTE tuning.
|-
|[[-3/8-comma meantone|-3/8-comma]]
|710.019
|
|-
|[[-8/21-comma meantone|-8/21-comma]]
|710.148
|
|-
|[[-1/(φ+1)-comma meantone|-1/(ϕ+1)-comma]]
|710.170
|Close to [[11-limit]] superpyth POTE tuning.
|-
|[[-5/13-comma meantone|-5/13-comma]]
|710.227
|Close to [[49edo]]. Close to [[7-limit]] superpyth POTE tuning.
|-
|[[-7/18-comma meantone|-7/18-comma]]
|710.319
|
|-
|[[-9/23-comma meantone|-9/23-comma]] 
|710.371
|
|-
|[[-2/5-comma meantone|-2/5-comma]]
|710.558
|Close to [[13-limit]] superpyth POTE tuning.
|-
|[[-9/22-comma meantone|-9/22-comma]]
|710.753
|
|-
|[[-7/17-comma meantone|-7/17-comma]]
|710.810
|
|-
|[[-5/12-comma meantone|-5/12-comma]]
|710.915
|
|-
|[[-8/19-comma meantone|-8/19-comma]]
|711.010
|
|-
|[[-3/7-comma meantone|-3/7-comma]]
|711.172
|Close to [[27edo]].
|-
|[[-7/16-comma meantone|-7/16-comma]]
|711.364
|
|-
|[[-4/9-comma meantone|-4/9-comma]]
|711.513
|
|-
|[[-9/20-comma meantone|-9/20-comma]]
|711.633
|
|-
|[[-5/11-comma meantone|-5/11-comma]]
|711.731
|
|-
|[[-6/13-comma meantone|-6/13-comma]]
|711.880
|Close to [[59edo]].
|-
|[[-7/15-comma meantone|-7/15-comma]]
|711.991
|
|-
|[[-8/17-comma meantone|-8/17-comma]]
|712.075
|
|-
|[[-9/19-comma meantone|-9/19-comma]]
|712.142
|
|-
|[[-10/21-comma meantone|-10/21-comma]]
|712.196
|
|}
 
=== Sharper than sharpest negative harmonic-defined meantone ===
{| class="wikitable mw-collapsible"
|+ Spectrum of meantone tunings -1/2-comma to -1/1-comma
!Meantone Temperament!!Generator (cents)!!Comments
|-
|[[-1/2-comma meantone|-1/2-comma]]
|712.708
|Close to [[32edo]]. Everything from this point onwards does not have a whole tone being between 9/8 and 729/640.
|-
|[[-11/21-comma meantone|-11/21-comma]]
|713.220
|
|-
|[[-10/19-comma meantone|-10/19-comma]]
|713.274
|
|-
|[[-9/17-comma meantone|-9/17-comma]]
|713.340
|
|-
|[[-8/15-comma meantone|-8/15-comma]]
|713.425
|
|-
|[[-7/13-comma meantone|-7/13-comma]]
|713.535
|Close to [[37edo]].
|-
|[[-6/11-comma meantone|-6/11-comma]]
|713.686
|
|-
|[[-11/20-comma meantone|-11/20-comma]]
|713.783
|
|-
|[[-5/9-comma meantone|-5/9-comma]]
|713.903
|
|-
|[[-9/16-comma meantone|-9/16-comma]]
|714.052
|
|-
|[[-4/7-comma meantone|-4/7-comma]]
|714.244
|Close to [[42edo]].
|-
|[[-11/19-comma meantone|-11/19-comma]]
|714.406
|
|-
|[[-7/12-comma meantone|-7/12-comma]]
|714.500
|
|-
|[[-10/17-comma meantone|-10/17-comma]]
|714.606
|
|-
|[[-13/22-comma meantone|-13/22-comma]]
|714.663
|
|-
|[[-3/5-comma meantone|-3/5-comma]]
|714.859
|Close to [[47edo]].
|-
|[[-14/23-comma meantone|-14/23-comma]]
|715.046
|
|-
|[[-11/18-comma meantone|-11/18-comma]]
|715.098
|
|-
|[[-8/13-comma meantone|-8/13-comma]]
|715.190
|
|-
|[[-1/φ-comma meantone|-1/ϕ-comma]]
|715.247
|
|-
|[[-13/21-comma meantone|-13/21-comma]]
|715.268
|
|-
|[[-5/8-comma meantone|-5/8-comma]]
|715.396
|Close to [[52edo]] and 387/256.
|-
|[[-12/19-comma meantone|-12/19-comma]]
|715.538
|
|-
|-
|[[3/4-comma meantone]]||685.825||Whole tone is not between 10/9 and 9/8, so whether it counts as meantone is very questionable
|[[-7/11-comma meantone|-7/11-comma]]  
|715.641
|
|-
|-
|[[11/15-comma meantone]]||686.184||Whole tone not between 10/9 and 9/8
|[[-9/14-comma meantone|-9/14-comma]]  
|715.780
|Close to [[57edo]].
|-
|-
|[[8/11-comma meantone]]||686.314||Whole tone not between 10/9 and 9/8
|[[-11/17-comma meantone|-11/17-comma]]  
|715.871
|
|-
|-
|[[5/7-comma meantone]]||686.593||Whole tone not between 10/9 and 9/8
|[[-13/20-comma meantone|-13/20-comma]]
|715.934
|
|-
|-
|[[7/10-comma meantone]]||686.901||Whole tone not between 10/9 and 9/8
|[[-2/3-comma meantone|-2/3-comma]]  
|716.293
|Close to [[62edo]].
|-
|-
|[[9/13-comma meantone]]||687.066||Whole tone not between 10/9 and 9/8
|[[-15/22 comma meantone|-15/22 comma]]  
|716.618
|Close to [[67edo]].
|-
|-
|[[11/16-comma meantone]]||687.169||Whole tone not between 10/9 and 9/8
|[[-13/19 comma meantone|-13/19 comma]]  
|716.669
|Close to [[72edo]].
|-
|-
|[[2/3-comma meantone]]||687.617||Whole tone not between 10/9 and 9/8
|[[-11/16-comma meantone|-11/16-comma]]  
|716.741
|
|-
|-
|[[9/14-comma meantone]]||688.129||Whole tone not between 10/9 and 9/8
|[[-9/13-comma meantone|-9/13-comma]]  
|716.844
|Close to [[77edo]].
|-
|-
|[[7/11-comma meantone]]||688.269||Whole tone not between 10/9 and 9/8
|[[-7/10-comma meantone|-7/10-comma]]  
|717.009
|
|-
|-
|[[5/8-comma meantone]]||688.514||Whole tone not between 10/9 and 9/8
|[[-12/17-comma meantone|-12/17-comma]]  
|717.136
|Close to [[82edo]].
|-
|-
|[[8/13-comma meantone]]||688.72||Whole tone not between 10/9 and 9/8
|[[-17/24-comma meantone|-17/24-comma]]  
|717.188
|Close to [[87edo]].
|-
|-
|[[3/5-comma meantone]]||689.051||Whole tone not between 10/9 and 9/8
|[[-5/7-comma meantone|-5/7-comma]]  
|717.317
|Close to [[92edo]].
|-
|-
|[[7/12-comma meantone]]||689.41||Whole tone not between 10/9 and 9/8
|[[-13/18-comma meantone|-13/18-comma]]  
|717.487
|Close to [[97edo]].
|-
|-
|[[4/7-comma meantone]]||689.666||Whole tone not between 10/9 and 9/8
|[[-8/11-comma meantone|-8/11-comma]]  
|717.596
|
|-
|-
|[[9/16-comma meantone]]||689.858||Whole tone not between 10/9 and 9/8
|[[-19/26-comma meantone|-19/26-comma]]  
|717.671
|
|-
|-
|[[5/9-comma meantone]]||690.007||Whole tone not between 10/9 and 9/8
|[[-11/15-comma meantone|-11/15-comma]]  
|717.726
|
|-
|-
|[[6/11-comma meantone]]||690.224||Whole tone not between 10/9 and 9/8
|[[-14/19-comma meantone|-14/19-comma]]  
|717.802
|
|-
|-
|[[7/13-comma meantone]]||690.375||Whole tone not between 10/9 and 9/8
|[[-3/4-comma meantone|-3/4-comma]]  
|718.085
|About as sharp of [[Pythagorean tuning]] as [[7edo]] is flat.
|-
|-
|[[8/15-comma meantone]]||690.485||Whole tone not between 10/9 and 9/8
|[[-21/26-comma meantone|-21/26-comma]]
|718.325
|
|-
|-
|[[1/2-comma meantone]]||691.202||Whole tone not between 10/9 and 9/8 (it’s exactly 10/9)
|[[-16/21-comma meantone|-16/21-comma]]  
|718.341
|
|-
|-
|[[7/15-comma meantone]]||691.919||This is the first tuning on this list which is unambiguously meantone, since the whole tone is between 10/9 and 9/8
|[[-13/17-comma meantone|-13/17-comma]]  
|718.401
|
|-
|-
|[[6/13-comma meantone]]||692.029||  
|[[-10/13-comma meantone|-10/13-comma]]  
|718.498
|
|-
|-
|[[5/11-comma meantone]]||692.179||  
|[[-17/22-comma meantone|-17/22-comma]]  
|718.574
|
|-
|-
|[[4/9-comma meantone]]||692.397||  
|[[-7/9-comma meantone|-7/9-comma]]  
|718.682
|
|-
|-
|[[7/16-comma meantone]]||692.546||  
|[[-11/14-comma meantone|-11/14-comma]]  
|718.853
|
|-
|-
|[[3/7-comma meantone]]||692.738||  
|[[-15/19-comma meantone|-15/19-comma]]  
|718.934
|
|-
|-
|[[5/12-comma meantone]]||692.994||  
|[[-4/5-comma meantone|-4/5-comma]]  
|719.160
|
|-
|-
|[[2/5-comma meantone]]||693.352||  
|[[-17/21-comma meantone|-17/21-comma]]  
|719.365
|
|-
|-
|[[5/13-comma meantone]]||693.683||  
|[[-13/16-comma meantone|-13/16-comma]]  
|719.429
|
|-
|-
|[[3/8-comma meantone]]||693.89||  
|[[-9/11-comma meantone|-9/11-comma]]  
|719.551
|
|-
|-
|[[4/11-comma meantone]]||694.134||  
|[[-14/17-comma meantone|-14/17-comma]]  
|719.666
|
|-
|-
|[[5/14-comma meantone]]||694.274||
|[[-5/6-comma meantone|-5/6-comma]]  
|719.877
|Everything up to this point generates 12 and 17 tone MOS scales.
|-
|-
|[[1/3-comma meantone]]||694.786||  
|[[5edo]]||720.000||The largest MOS scale this can generate is 5 tone. '''Upper boundary of 5-limit diamond monotone.'''
|-
|-
|[[5/16-comma meantone]]||695.234||
|[[-21/25-comma meantone|-21/25-comma]]  
|720.020
|Everything from this point onwards generates 13 and 18 tone MOS scales.
|-
|-
|[[4/13-comma meantone]]||695.338||  
|[[-16/19-comma meantone|-16/19-comma]]  
|720.066
|
|-
|-
|[[3/10-comma meantone]]||695.503||  
|[[-11/13-comma meantone|-11/13-comma]]  
|720.153
|
|-
|-
|[[2/7-comma meantone]]||695.81||  
|[[-17/20-comma meantone|-17/20-comma]]  
|720.235
|
|-
|-
|[[3/11-comma meantone]]||696.09||  
|[[-6/7-comma meantone|-6/7-comma]]  
|720.399
|
|-
|-
|[[4/15-comma meantone]]||696.22||  
|[[-19/22-comma meantone|-19/22-comma]]  
|720.529
|
|-
|-
|[[Quarter-comma meantone|1/4-comma meantone]]||696.578||  
|[[-13/15-comma meantone|-13/15-comma]]  
|720.594
|
|-
|-
|[[3/13-comma meantone]]||696.992||  
| -[[7/8-comma meantone|7/8-comma]]  
|720.773
|
|-
|-
|[[2/9-comma meantone]]||697.176||  
|[[-15/17-comma meantone|-15/17-comma]]  
|720.931
|
|-
|-
|[[3/14-comma meantone]]||697.346||  
|[[-8/9-comma meantone|-8/9-comma]]  
|721.017
|
|-
|-
|[[1/5-comma meantone]]||697.654||  
|[[-17/19-comma meantone|-17/19-comma]]  
|721.197
|
|-
|-
|[[3/16-comma meantone]]||697.923||  
|[[-9/10-comma meantone|-9/10-comma]]  
|721.311
|
|-
|-
|[[2/11-comma meantone]]||698.045||  
|[[-19/21-comma meantone|-19/21-comma]]  
|721.413
|
|-
|-
|[[1/6-comma meantone]]||698.371||  
|[[-10/11-comma meantone|-10/11-comma]]  
|721.506
|
|-
|-
|[[2/13-comma meantone]]||698.646||  
|[[-11/12-comma meantone|-11/12-comma]]  
|721.669
|
|-
|-
|[[1/7-comma meantone]]||698.883||  
|[[-12/13-comma meantone|-12/13-comma]]  
|721.807
|
|-
|-
|[[2/15-comma meantone]]||699.088||  
|[[-13/14-comma meantone|-13/14-comma]]  
|721.925
|
|-
|-
|[[1/8-comma meantone]]||699.267||  
|[[-14/15-comma meantone|-14/15-comma]]  
|722.028
|
|-
|-
|[[1/9-comma meantone]]||699.565||  
|[[-15/16-comma meantone|-15/16-comma]]  
|722.117
|
|-
|-
|[[1/10-comma meantone]]||699.804||  
|[[-16/17-comma meantone|-16/17-comma]]  
|722.196
|
|-
|-
|[[1/11-comma meantone]]||700.000||Audibly indistinguishable from [[12edo]], does not generate a 19-tone MOS scale, so whether it counts as meantone is very questionable
|[[-17/18-comma meantone|-17/18-comma]]  
|722.266
|
|-
|-
|[[1/12-comma meantone]]||700.163||No 19-tone MOS
|[[-18/19-comma meantone|-18/19-comma]]  
|722.329
|
|-
|-
|[[1/13-comma meantone]]||700.301||No 19-tone MOS
|[[-19/20-comma meantone|-19/20-comma]]  
|722.386
|
|-
|-
|[[1/14-comma meantone]]||700.419||No 19-tone MOS
|[[-20/21-comma meantone|-20/21-comma]]  
|722.437
|
|-
|-
|[[1/15-comma meantone]]||700.521||No 19-tone MOS
|[[-21/22-comma meantone|-21/22-comma]]  
|722.484
|
|-
|-
|[[1/16-comma meantone]]||700.611||No 19-tone MOS
|[[-1/1-comma meantone|-1/1-comma]]  
|723.461
|Close to [[68edo]].
|-
|-
|0/1-comma “meantone” ([[Pythagorean tuning]])||701.955||Not meantone but included for reference.
|}
|}
[[Category:Tables]][[Category:Meantone]]
 
[[Category:Tables]]
[[Category:Meantone]]

Latest revision as of 14:29, 8 June 2025

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Here are all meantone tunings that can be written in the form "n-comma meantone", where the syntonic comma (81/80) is being divided and n is a fraction between -1 and 1 with a denominator 22 or smaller.

Scope of table

Characteristics included

Some of the characteristics this table mentions for each temperament include:

  • Whether it saw historical (pre-1950) use
  • Whether it is close to (i.e. within 1/2 a degree of closing of) an edo smaller than 100
  • Whether it is the closest on the table to the optimal CTE or POTE tuning of meantone or superpyth in any JI limit
  • Whether it approximates a very simple n-Pythagorean comma meantone
  • Whether it is about equally sharp of 3/2 as some other listed temperament is flat
  • Whether it is close to exactly one just-noticeable difference away from 3/2

Occasional other comments may be included as well.

Characteristics omitted

Dozens of tunings on this table are significant to negative harmony temperament theory, enough that labelling them all individually would clutter the table.

Every tuning on this table is close to some arbitrarily large edo, but labeling them beyond 100edo would clutter the table.

Special cases included

A small number of additional temperaments are included. Not too many, to avoid clutter, just the bare minimum:

Cautions

Preservation of meantone behavior

Temperaments that fall outside of the "historically-defined meantone" range will not possess most of the musical properties that meantone usually possesses, but they are included for completeness.

Temperaments that fall outside of the "diamond monotone" range preserve even fewer meantone properties, but they are also included for completeness.

The table

Flatter than flattest historically-defined meantone

Spectrum of meantone tunings 1/1-comma to 1/2-comma
Meantone Temperament Generator (cents) Comments
1/1-comma 680.449 Close to 30edo
21/22-comma 681.426 Close to 37edo.
20/21-comma 681.473
19/20-comma 681.524
18/19-comma 681.581
17/18-comma 681.644
16/17-comma 681.713
15/16-comma 681.793 Close to 44edo.
14/15-comma 681.883
13/14-comma 681.985
12/13-comma 682.103
11/12-comma 682.241
10/11-comma 682.404 Close to 51edo.
19/21-comma 682.497
9/10-comma 682.599
17/19-comma 682.713
8/9-comma 682.838 Close to 58edo.
15/17-comma 682.979
7/8-comma 683.137 Close to 65edo.
13/15-comma 683.316 Close to 72edo.
19/22-comma 683.381
6/7-comma 683.521 Close to 79edo.
17/20-comma 683.675
11/13-comma 683.757 Close to 86edo.
16/19-comma 683.844
21/25-comma 683.890 Close to 93edo
5/6-comma 684.033 Close to 100edo.
14/17-comma 684.244
9/11-comma 684.359
13/16-comma 684.481
17/21-comma 684.545
4/5-comma 684.750
15/19-comma 684.976
11/14-comma 685.057
7/9-comma 685.228
17/22-comma 685.337
10/13-comma 685.412
13/17-comma 685.509
16/21-comma 685.569 Everything up to this point generates 9 and 16 tone MOS scales.
7edo 685.714 The largest MOS scale this can generate is 7 tone. Lower boundary of 5-limit diamond monotone.
3/4-comma 685.825 Everything from this point onwards generates 12 and 19 tone MOS scales.
14/19-comma 686.108
11/15-comma 686.184
19/26-comma 686.239
8/11-comma 686.314
13/18-comma 686.423
5/7-comma 686.593
17/24-comma 686.721
12/17-comma 686.774
7/10-comma 686.901
9/13-comma 687.066
11/16-comma 687.169
13/19-comma 687.240
15/22-comma 687.292
17/25-comma 687.331
19/28-comma 687.361
2/3-comma 687.617 Close to 89edo.
17/26-comma 687.893 Close to 82edo.
15/23-comma 687.929
13/20-comma 687.976
11/17-comma 688.039 Close to 75edo
9/14-comma 688.129
7/11-comma 688.269 Close to 68edo.
12/19-comma 688.372
5/8-comma 688.514 Close to 61edo and 43/32.
13/21-comma 688.641
1/ϕ-comma 688.663
8/13-comma 688.720
11/18-comma 688.812 Close to 54edo.
14/23-comma 688.864
3/5-comma 689.051
13/22-comma 689.247
10/17-comma 689.304
7/12-comma 689.410 Close to 47edo.
11/19-comma 689.504
4/7-comma 689.666 Close to 87edo.
9/16-comma 689.858
5/9-comma 690.007 Close to 40edo.
11/20-comma 690.127
6/11-comma 690.224
7/13-comma 690.375 Close to 73edo.
8/15-comma 690.485
9/17-comma 690.569
10/19-comma 690.636
11/21-comma 690.690 Close to 33edo
1/2-comma 691.202 Close to 92edo, 59edo. Historically significant (see historical temperaments). Everything up to this point does not have a whole tone between 10/9 and 9/8.

Historically-defined meantone

Spectrum of meantone tunings 10/21-comma to 1/22-comma
Temperament Generator (cents) Comments
10/21-comma 691.714
9/19-comma 691.768 Close to 85edo.
8/17-comma 691.834
7/15-comma 691.919
6/13-comma 692.029
5/11-comma 692.179
9/20-comma 692.277 Close to 26edo.
4/9-comma 692.397
7/16-comma 692.546
3/7-comma 692.738
8/19-comma 692.899
5/12-comma 692.994 Close to 71edo.
7/17-comma 693.099
9/22-comma 693.157
2/5-comma 693.352 Close to 45edo.
9/23-comma 693.539
7/18-comma 693.591
5/13-comma 693.683
1/(ϕ+1)-comma 693.740 Close to 64edo.
8/21-comma 693.762
3/8-comma 693.890 Close to 83edo.
7/19-comma 694.032
4/11-comma 694.134 Almost exactly 1/3-Pythagorean comma meantone.
5/14-comma 694.274
6/17-comma 694.365
7/20-comma 694.428
8/23-comma 694.475
9/26-comma 694.511
1/3-comma 694.786 Close to 19edo. Historically significant (see historical temperaments).
9/28-comma 695.042
8/25-comma 695.073
7/22-comma 695.112
6/19-comma 695.164
5/16-comma 695.234
4/13-comma 695.338
3/10-comma 695.503 Close to 88edo and Lucy tuning. Historically significant (see historical temperaments).
5/17-comma 695.630 Close to 69edo.
7/24-comma 695.682
2/7-comma 695.810 Historically significant (see historical temperaments).
5/18-comma 695.981 Close to 50edo.
3/11-comma 696.090
7/26-comma 696.165 Close to golden meantone. Historically significant (see historical temperaments).
4/15-comma 696.220 Close to 5-limit meantone POTE tuning.
5/19-comma 696.295 Close to 81edo.
1/4-comma 696.578 Close to septimal and tridecimal meantone POTE tunings. Historically significant (see historical temperaments).
5/21-comma 696.834 Close to 31edo.
4/17-comma 696.895
3/13-comma 696.992 Close to septimal & tridecimal meantone CTE tunings. Close to undecimal meantone POTE tuning.
5/22-comma 697.067
2/9-comma 697.176 Close to 5-limit and undecimal meantone CTE tunings. Historically significant (see historical temperaments).
3/14-comma 697.346 Close to 74edo. Historically significant (see historical temperaments).
4/19-comma 697.427
1/5-comma 697.654 Close to 43edo. Historically significant (see historical temperaments).
4/21-comma 697.859
3/16-comma 697.923
2/11-comma 698.045 Close to 55edo. Historically significant (see historical temperaments).
3/17-comma 698.159
4/23-comma 698.215
1/6-comma 698.371 Historically significant (see historical temperaments). Everything up to this point has a fifth which is flat of Pythagorean tuning by at least the just-noticeable difference.
4/25-comma 698.514 Close to 67edo.
3/19-comma 698.559
2/13-comma 698.646 Close to 79edo.
3/20-comma 698.729
1/7-comma 698.883 Close to 91edo. Historically significant (see historical temperaments).
3/22-comma 699.022
2/15-comma 699.088
1/8-comma 699.267
2/17-comma 699.425
1/9-comma 699.565
2/19-comma 699.691
1/10-comma 699.804
2/21-comma 699.907
1/11-comma 700.000 Everything up to this point generates 12 and 19 tone MOS scales.
12edo 700.000 The largest MOS scale this can generate is 12 tone. Historically significant (see historical temperaments.)
1/12-comma 700.163 Everything from this point onwards generates 12 and 17 tone MOS scales.
1/13-comma 700.301
1/14-comma 700.419
1/15-comma 700.521
1/16-comma 700.611
1/17-comma 700.690
1/18-comma 700.760
1/19-comma 700.823
1/20-comma 700.879
1/21-comma 700.931
1/22-comma 700.977

Negative harmony theory-defined meantone (most often approached as superpyth)

Spectrum of meantone tunings 0/1-comma to -10/21-comma
Meantone Temperament Generator (cents) Comments
Pythagorean tuning 701.955 Historically significant (see historical temperaments.) Everything from this point onwards does not have a whole tone between 10/9 and 9/8.
-1/22-comma 702.933
-1/21-comma 702.979
-1/20-comma 703.030
-1/19-comma 703.087
-1/18-comma 703.150
-1/17-comma 703.220
-1/16-comma 703.299
-1/15-comma 703.389 Close to 11/13 third-kleisma temperament.
-1/14-comma 703.491 Close to 29edo.
-1/13-comma 703.609
-1/12-comma 703.747
-1/11-comma 703.910 About as sharp of Pythagorean tuning as 12edo is flat.
-2/21-comma 704.003 Close to 75edo.
-1/10-comma 704.105
-2/19-comma 704.219
-1/9-comma 704.344 Close to 46edo, 11/7 quarter-kleisma temperament.
-2/17-comma 704.483
-1/8-comma 704.643
-2/15-comma 704.823 Close to 63edo.
-3/22-comma 704.888
-1/7-comma 705.027 Close to 80edo.
-3/20-comma 705.181
-2/13-comma 705.350
-3/19-comma 705.350
-4/25-comma 705.396
-1/6-comma 705.538 Everything from this point onwards has a fifth which is sharp of Pythagorean tuning by at least the just-noticeable difference.
-4/23-comma 705.695
-3/17-comma 705.750 About as sharp of Pythagorean tuning as 55edo is flat.
-2/11-comma 705.865 Everything up to this point generates 17 and 29 tone MOS scales.
17edo 705.882 The largest MOS scale this can generate is 17 tone. Vaguely resembles Middle Eastern neutral third scales.
-3/16-comma 705.987 Everything from this point onwards generates 17 and 22 tone MOS scales.
-4/21-comma 706.051
-1/5-comma 706.256 About as sharp of Pythagorean tuning as 43edo is flat.
-4/19 comma 706.483
-3/14-comma 706.563 About as sharp of Pythagorean tuning as 74edo is flat.
-2/9-comma 706.734
-5/22-comma 706.843
-3/13-comma 706.918 Close to 39edo.
-4/17-comma 707.015
-5/21-comma 707.076 About as sharp of Pythagorean tuning as 31edo is flat.
-1/4-comma 707.332
-5/19-comma 707.615
-4/15-comma 707.690 About as sharp of Pythagorean tuning as golden meantone is flat.
-7/26-comma 707.745
-3/11-comma 707.820 Almost exactly -1/4-Pythagorean comma meantone
-5/18-comma 707.930 About as sharp of Pythagorean tuning as 50edo is flat. Close to 100edo.
-2/7-comma 708.100
-7/24-comma 708.227
-5/17-comma 708.280
-3/10-comma 708.407 Nearly as sharp of Pythagorean tuning as Lucy tuning is flat. Nearly as sharp of Pythagorean tuning as 88edo is flat.
-4/13-comma 708.572
-5/16-comma 708.675
-6/19-comma 708.746
-7/22-comma 708.800
-8/25-comma 708.837
-9/28-comma 708.867
-1/3-comma 709.124 Close to 22edo. About as sharp of Pythagorean tuning as 19edo is flat.
-9/26-comma 709.399 Close to 2.3.7-limit superpyth POTE tuning.
-8/23-comma 709.435
-7/20-comma 709.482
-6/17-comma 709.545 Close to 11-limit superpyth CTE tuning.
-5/14-comma 709.636 Close to 93edo. Close to 2.3.7-limit and 7-limit superpyth CTE tunings.
-4/11-comma 709.775 Almost exactly -1/3-Pythagorean comma meantone.
-7/19-comma 709.878 Close to 13-limit superpyth CTE tuning.
-3/8-comma 710.019
-8/21-comma 710.148
-1/(ϕ+1)-comma 710.170 Close to 11-limit superpyth POTE tuning.
-5/13-comma 710.227 Close to 49edo. Close to 7-limit superpyth POTE tuning.
-7/18-comma 710.319
-9/23-comma 710.371
-2/5-comma 710.558 Close to 13-limit superpyth POTE tuning.
-9/22-comma 710.753
-7/17-comma 710.810
-5/12-comma 710.915
-8/19-comma 711.010
-3/7-comma 711.172 Close to 27edo.
-7/16-comma 711.364
-4/9-comma 711.513
-9/20-comma 711.633
-5/11-comma 711.731
-6/13-comma 711.880 Close to 59edo.
-7/15-comma 711.991
-8/17-comma 712.075
-9/19-comma 712.142
-10/21-comma 712.196

Sharper than sharpest negative harmonic-defined meantone

Spectrum of meantone tunings -1/2-comma to -1/1-comma
Meantone Temperament Generator (cents) Comments
-1/2-comma 712.708 Close to 32edo. Everything from this point onwards does not have a whole tone being between 9/8 and 729/640.
-11/21-comma 713.220
-10/19-comma 713.274
-9/17-comma 713.340
-8/15-comma 713.425
-7/13-comma 713.535 Close to 37edo.
-6/11-comma 713.686
-11/20-comma 713.783
-5/9-comma 713.903
-9/16-comma 714.052
-4/7-comma 714.244 Close to 42edo.
-11/19-comma 714.406
-7/12-comma 714.500
-10/17-comma 714.606
-13/22-comma 714.663
-3/5-comma 714.859 Close to 47edo.
-14/23-comma 715.046
-11/18-comma 715.098
-8/13-comma 715.190
-1/ϕ-comma 715.247
-13/21-comma 715.268
-5/8-comma 715.396 Close to 52edo and 387/256.
-12/19-comma 715.538
-7/11-comma 715.641
-9/14-comma 715.780 Close to 57edo.
-11/17-comma 715.871
-13/20-comma 715.934
-2/3-comma 716.293 Close to 62edo.
-15/22 comma 716.618 Close to 67edo.
-13/19 comma 716.669 Close to 72edo.
-11/16-comma 716.741
-9/13-comma 716.844 Close to 77edo.
-7/10-comma 717.009
-12/17-comma 717.136 Close to 82edo.
-17/24-comma 717.188 Close to 87edo.
-5/7-comma 717.317 Close to 92edo.
-13/18-comma 717.487 Close to 97edo.
-8/11-comma 717.596
-19/26-comma 717.671
-11/15-comma 717.726
-14/19-comma 717.802
-3/4-comma 718.085 About as sharp of Pythagorean tuning as 7edo is flat.
-21/26-comma 718.325
-16/21-comma 718.341
-13/17-comma 718.401
-10/13-comma 718.498
-17/22-comma 718.574
-7/9-comma 718.682
-11/14-comma 718.853
-15/19-comma 718.934
-4/5-comma 719.160
-17/21-comma 719.365
-13/16-comma 719.429
-9/11-comma 719.551
-14/17-comma 719.666
-5/6-comma 719.877 Everything up to this point generates 12 and 17 tone MOS scales.
5edo 720.000 The largest MOS scale this can generate is 5 tone. Upper boundary of 5-limit diamond monotone.
-21/25-comma 720.020 Everything from this point onwards generates 13 and 18 tone MOS scales.
-16/19-comma 720.066
-11/13-comma 720.153
-17/20-comma 720.235
-6/7-comma 720.399
-19/22-comma 720.529
-13/15-comma 720.594
-7/8-comma 720.773
-15/17-comma 720.931
-8/9-comma 721.017
-17/19-comma 721.197
-9/10-comma 721.311
-19/21-comma 721.413
-10/11-comma 721.506
-11/12-comma 721.669
-12/13-comma 721.807
-13/14-comma 721.925
-14/15-comma 722.028
-15/16-comma 722.117
-16/17-comma 722.196
-17/18-comma 722.266
-18/19-comma 722.329
-19/20-comma 722.386
-20/21-comma 722.437
-21/22-comma 722.484
-1/1-comma 723.461 Close to 68edo.
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