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

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m Remove “the largest this can generate is n tone” because that’s self evident and doesn’t need to be said, remove all the “important to negative harmony” because that was on almost every entry and cluttered the table making it hard to read, remove the unsourced claims about history that contradict everything I’ve ever read, remove extra added EDOs because this is about n-comma, not about EDOs, EDOs can have their own different table, other similar cleanup
Line 1: Line 1:
{{Editable user page}}
{{Editable user page}}


Here are all [[meantone]] tunings that can be written in the form "n-comma meantone", where n is a fraction between 0 and 1 with a denominator 18 or smaller. Also included are 5, 7, 12, 17 and 19edo (to delineate MOS shapes), as well as a few other notable meantone tunings (e.g. 4/25-comma).
Here are all [[meantone]] tunings that can be written in the form "n-comma meantone", where n is a fraction between 0 and 1 with a denominator 18 or smaller. Also included are 5, 7 and 12edo (to delineate MOS shapes), as well as a few other notable meantone tunings (e.g. 4/25-comma).


The comma being divided here is the syntonic comma ([[81/80]]).
The comma being divided here is the syntonic comma ([[81/80]]).


Temperaments that fall outside of the diamond monotone range will not provide most of the advantages that meantone usually provides, but they are included for completeness.
Temperaments that fall outside of the diamond monotone range will not provide most of the advantages that meantone usually provides, but they are included for completeness.
Dozens of tunings on the table are significant to [[negative harmony temperaments|negative harmony temperament theory]], enough that labelling them all individually would clutter the table.


{| class="wikitable"
{| class="wikitable"
Line 12: Line 14:
|[[1/1-comma meantone]]||680.449||Close to [[30edo]]
|[[1/1-comma meantone]]||680.449||Close to [[30edo]]
|-
|-
|[[17/18-comma meantone]]
|[[15/16-comma meantone]]||681.793||  
|681.644
|
|-
|[[16/17-comma meantone]]
|681.714
|
|-
|[[15/16-comma meantone]]||681.793 ||Close to [[44edo]]
|-
|-
|[[14/15-comma meantone]]||681.883||
|[[14/15-comma meantone]]||681.883||Close to [[44edo]]
|-
|-
|[[13/14-comma meantone]]||681.985||
|[[13/14-comma meantone]]||681.985||  
|-
|-
|[[12/13-comma meantone]]||682.103||
|[[12/13-comma meantone]]||682.103||  
|-
|-
|[[11/12-comma meantone]]||682.241||
|[[11/12-comma meantone]]||682.241||  
|-
|-
|[[10/11-comma meantone]]||682.404||Close to [[51edo]]
|[[12/11-comma meantone]]||682.404||Close to [[51edo]]
|-
|-
|[[9/10-comma meantone]]||682.599||
|[[9/10-comma meantone]]||682.599||  
|-
|-
|[[8/9-comma meantone]]||682.838||Close to [[58edo]]
|[[8/9-comma meantone]]||682.838||Close to [[58edo]]
|-
|-
|[[15/17-comma meantone]]
|[[7/8-comma meantone]]||683.137||Close to [[65edo]]
|682.979
|
|-
|[[7/8-comma meantone]]||683.137|| Close to [[65edo]]
|-
|-
|[[13/15-comma meantone]]||683.316|| Close to [[72edo]]
|[[13/15-comma meantone]]||683.316||Close to [[72edo]]  
|-
|-
|[[6/7-comma meantone]]||683.521||Close to [[79edo]]
|[[6/7-comma meantone]]||683.521||Close to [[79edo]]
Line 48: Line 38:
|[[11/13-comma meantone]]||683.757||Close to [[86edo]]
|[[11/13-comma meantone]]||683.757||Close to [[86edo]]
|-
|-
|[[16/19-comma meantone]]
|[[5/6-comma meantone]]||684.033||  
|683.844
|
|-
|[[21/25-comma meantone]]
|683.890
|
|-
|-
|
|[[9/11-comma meantone]]||684.359||
|683.910
|As flat of [[Pythagorean tuning]] as [[5edo]] is sharp.
|-
|-
|[[5/6-comma meantone]]||684.033||
|[[13/16-comma meantone]]||684.481||  
|-
|-
|[[14/17-comma meantone]]
|[[4/5-comma meantone]]||684.75||  
|684.244
|
|-
|-
|[[9/11-comma meantone]]||684.359||
|[[11/14-comma meantone]]||685.057||  
|-
|-
|[[13/16-comma meantone]]||684.481||
|[[7/9-comma meantone]]||685.228||  
|-
|-
|[[4/5-comma meantone]]||684.750 ||
|[[10/13-comma meantone]]||685.412||Everything up to this point generates 9 and 16 tone MOS scales.
|-
|-
|[[11/14-comma meantone]]||685.057||
|[[7edo]]||685.714||The largest MOS scale this can generate is 7 tone. '''Lower boundary of 5-limit diamond monotone.'''
|-
|-
|[[7/9-comma meantone]]||685.228 ||
|[[3/4-comma meantone]]||685.825||Everything from this point onwards generates 12 and 19 tone MOS scales.
|-
|-
|[[10/13-comma meantone]]||685.412 ||
|[[11/15-comma meantone]]||686.184||
|-
|[[13/17-comma meantone]]
|685.509
|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]]||685.825||Everything from this point onwards generates 12 and 19 and 26 tone MOS scales.
|-
|[[11/15-comma meantone]]||686.184 ||
|-
|[[19/26-comma meantone]]
|686.239
|
|-
|-
|[[8/11-comma meantone]]||686.314||
|[[8/11-comma meantone]]||686.314||
|-
|-
|[[13/18-comma meantone]]
|[[5/7-comma meantone]]||686.593||  
|686.422
|
|-
|-
|[[5/7-comma meantone]]||686.593||
|[[7/10-comma meantone]]||686.901||  
|-
|-
|[[17/24-comma meantone|17/24- comma meantone]]
|[[9/13-comma meantone]]||687.066||  
|686.721
|
|-
|-
|[[12/17-comma meantone]]
|[[11/16-comma meantone]]||687.169||  
|686.774
|
|-
|[[7/10-comma meantone]]||686.901||
|-
|[[9/13-comma meantone]]||687.066||
|-
|[[11/16-comma meantone]]||687.169||
|-
|-
|[[2/3-comma meantone]]||687.617||Close to [[89edo]]
|[[2/3-comma meantone]]||687.617||Close to [[89edo]]
|-
|-
|[[11/17-comma meantone]]
|[[9/14-comma meantone]]||688.129||Close to [[75edo]]
|688.039
|Close to [[75edo]]
|-
|[[9/14-comma meantone]]|| 688.129||
|-
|-
|[[7/11-comma meantone]]||688.269||Close to [[68edo]]
|[[7/11-comma meantone]]||688.269||Close to [[68edo]]
Line 126: Line 76:
|[[5/8-comma meantone]]||688.514||Close to [[61edo]]
|[[5/8-comma meantone]]||688.514||Close to [[61edo]]
|-
|-
|[[Golden ratio comma meantone]]
|[[8/13-comma meantone]]||688.720||Close to [[54edo]]
|688.663
|
|-
|[[8/13-comma meantone]]||688.720||
|-
|-
|[[11/18-comma meantone]]
|[[3/5-comma meantone]]||689.051||  
|688.812
|Close to [[54edo]]
|-
|-
|[[3/5-comma meantone]]||689.051||
|[[7/12-comma meantone]]||689.410||Close to [[47edo]]
|-
|-
|[[10/17-comma meantone]]
|[[4/7-comma meantone]]||689.666||Close to [[87edo]]
|689.304
|
|-
|-
|[[7/12-comma meantone]]||689.410|| Close to [[47edo]]
|[[9/16-comma meantone]]||689.858||  
|-
|-
|[[4/7-comma meantone]]|| 689.666||Close to [[87edo]]
|[[5/9-comma meantone]]||690.007||Close to [[40edo]]
|-
|-
|[[9/16-comma meantone]]||689.858 ||
|[[6/11-comma meantone]]||690.224||  
|-
|[[5/9-comma meantone]]||690.007 ||Close to [[40edo]]
|-
|[[6/11-comma meantone]]||690.224||
|-
|-
|[[7/13-comma meantone]]||690.375||Close to [[73edo]]
|[[7/13-comma meantone]]||690.375||Close to [[73edo]]
|-
|-
|[[8/15-comma meantone]]||690.485||
|[[8/15-comma meantone]]||690.485||Close to [[33edo]]
|-
|-
|[[9/17-comma meantone]]
|[[1/2-comma meantone]]||691.202||Close to [[92edo]], [[59edo]]. Historically significant (see [[historical temperaments]]). Everything up to this point does not meet the original historical definition of meantone (the whole tone being between 10/9 and 9/8).
|690.569
|Close to [[33edo]]
|-
|[[1/2-comma meantone]]||691.202||Close to [[92edo]]. Historically significant (see [[historical temperaments]]). Everything up to this point does not meet the original historical definition of meantone (the whole tone being between 10/9 and 9/8).
|-
|[[8/17-comma meantone]]
|691.834
|Close to [[59edo]].
|-
|-
|[[7/15-comma meantone]]||691.919||Close to [[85edo]]
|[[7/15-comma meantone]]||691.919||Close to [[85edo]]
|-
|-
|[[6/13-comma meantone]]||692.029||
|[[6/13-comma meantone]]||692.029||  
|-
|[[5/11-comma meantone]]||692.179||
|-
|-
|[[26edo]]
|[[5/11-comma meantone]]||692.179||  
|692.308
|The largest MOS scale this can generate is 26 tone.
|-
|-
|[[4/9-comma meantone]]||692.397||
|[[4/9-comma meantone]]||692.397||Close to [[26edo]]
|-
|-
|[[7/16-comma meantone]]||692.546||
|[[7/16-comma meantone]]||692.546||  
|-
|-
|[[3/7-comma meantone]]||692.738||
|[[3/7-comma meantone]]||692.738||  
|-
|-
|[[5/12-comma meantone]]||692.994||Close to [[71edo]]
|[[5/12-comma meantone]]||692.994||Close to [[71edo]]
Line 188: Line 114:
|[[2/5-comma meantone]]||693.352||Close to [[45edo]]
|[[2/5-comma meantone]]||693.352||Close to [[45edo]]
|-
|-
|[[7/18-comma meantone]]||693.591||
|[[7/18-comma meantone]]||693.591||  
|-
|[[5/13-comma meantone]]||693.683||
|-
|-
|[[Split Golden ratio comma meantone]]
|[[5/13-comma meantone]]||693.683||Close to [[64edo]]
|693.740
|Close to [[64edo]]
|-
|-
|[[3/8-comma meantone]]||693.890||Close to [[83edo]]
|[[3/8-comma meantone]]||693.890||Close to [[83edo]]
|-
|-
|[[4/11-comma meantone]]|| 694.134||Almost exactly 1/3-''Pythagorean'' comma meantone
|[[4/11-comma meantone]]||694.134||Almost exactly 1/3-''Pythagorean'' comma meantone
|-
|[[5/14-comma meantone]]||694.274 ||
|-
|[[6/17-comma meantone]]||694.365||Everything up to this point generates 12 and 19 and 26 tone MOS scales.
|-
|-
|[[19edo]]
|[[5/14-comma meantone]]||694.274||  
|694.737
|The largest MOS scale this can generate is 19 tone. Historically significant (see [[historical temperaments]]).
|-
|-
|[[1/3-comma meantone]]||694.786||Everything from this point onwards generates 12 and 19 and 31 tone MOS scales. Historically significant (see [[historical temperaments]]).
|[[6/17-comma meantone]]||694.365||
|-
|-
|
|[[1/3-comma meantone]]||694.786||Close to [[19edo]]. Historically significant (see [[historical temperaments]]).
|694.819
|As flat of [[Pythagorean tuning]] as [[22edo]] is sharp.
|-
|-
|[[5/16-comma meantone]]||695.234 ||
|[[5/16-comma meantone]]||695.234||  
|-
|-
|[[4/13-comma meantone]]||695.338||  
|[[4/13-comma meantone]]||695.338||  
|-
|-
|[[Lucy tuning]]
|[[3/10-comma meantone]]||695.503||Close to [[88edo]]
|695.493
|Historically significant (see [[historical temperaments]]). Close to [[88edo]].
|-
|[[3/10-comma meantone]]||695.503||
|-
|-
|[[5/17-comma meantone]]||695.630||
|[[5/17-comma meantone]]||695.630||
|-
|-
|[[7/24-comma meantone]]
|[[2/7-comma meantone]]||695.810||Close to [[69edo]]. Historically significant (see [[historical temperaments]]).
|695.682
|Close to [[69edo]].
|-
|-
|[[2/7-comma meantone]]||695.810||Historically significant (see [[historical temperaments]]).
|[[5/18-comma meantone]]||695.981||Close to [[50edo]]. Historically significant (see [[historical temperaments]]).
|-
|-
|[[5/18-comma meantone]]||695.981||Historically significant (see [[historical temperaments]]).
|[[3/11-comma meantone]]||696.090||Close to [[50edo]]
|-
|[[50edo]]
|696.000
|The largest MOS scale this can generate is 50 tone. Historically significant (see [[historical temperaments]]).
|-
|[[3/11-comma meantone]]||696.090||Almost exactly 1/4-''Pythagorean'' comma meantone
|-
|[[7/26-comma meantone]]||696.165||Historically significant (see [[historical temperaments]]).
|-
|[[Golden meantone]]
|696.214
|Historically significant (see [[historical temperaments]]).
|-
|
|696.218
|As flat of [[Pythagorean tuning]] as [[39edo]] is sharp. Negative harmonically significant tuning with flat fifth (see [[negative harmony temperaments]]).
|-
|-
|[[4/15-comma meantone]]||696.220||Close to [[81edo]].
|[[7/26-comma meantone]]||696.165||Close to [[golden meantone]]. Historically significant (see [[historical temperaments]]).
|-
|-
|[[Quarter-comma meantone|1/4-comma meantone]]||696.578||Historically significant (see [[historical temperaments]]).
|[[4/15-comma meantone]]||696.220||Close to [[81edo]], close to [[golden meantone]]
|-
|-
|[[31edo]]
|[[Quarter-comma meantone|1/4-comma meantone]]||696.578||Close to [[31edo]]. Historically significant (see [[historical temperaments]]).
|696.774
|The largest MOS scale this can generate is 31 tone. Historically significant (see [[historical temperaments]]).
|-
|-
|[[4/17-comma meantone]]||696.895||
|[[4/17-comma meantone]]||696.895||
Line 264: Line 154:
|[[2/9-comma meantone]]||697.176||Close to 5-limit and undecimal CTE tunings. Historically significant (see [[historical temperaments]]).
|[[2/9-comma meantone]]||697.176||Close to 5-limit and undecimal CTE tunings. Historically significant (see [[historical temperaments]]).
|-
|-
|[[74edo]]
|[[3/14-comma meantone]]||697.346||Close to [[74edo]]. Historically significant (see [[historical temperaments]]).
|697.297
|The largest MOS scale this can generate is 74 tone. Historically significant (see [[historical temperaments]]).
|-
|[[3/14-comma meantone]]||697.346||Historically significant (see [[historical temperaments]]).
|-
|[[1/5-comma meantone]]||697.654||Historically significant (see [[historical temperaments]]).
|-
|[[43edo]]
|697.654
|The largest MOS scale this can generate is 43 tone. Historically significant (see [[historical temperaments]]).
|-
|[[3/16-comma meantone]]||697.923||
|-
|-
|
|[[1/5-comma meantone]]||697.654||Close to [[43edo]]. Historically significant (see [[historical temperaments]]).
|698.023
|As flat of [[Pythagorean tuning]] as [[17edo]] is sharp.
|-
|-
|[[2/11-comma meantone]]||698.045||
|[[3/16-comma meantone]]||697.923||  
|-
|-
|[[55edo]]
|[[2/11-comma meantone]]||698.045||Close to [[55edo]]
|698.182
|The largest MOS scale this can generate is 55 tone. Historically significant (see [[historical temperaments]]).
|-
|-
|[[3/17-comma meantone]]||698.245||Negative harmonically significant tuning with flat fifth (see [[negative harmony temperaments]]).
|[[3/17-comma meantone]]||699.425||
|-
|-
|[[1/6-comma meantone]]||698.371||Historically significant (see [[historical temperaments]]).
|[[1/6-comma meantone]]||698.371||Close to [[67edo]]. Historically significant (see [[historical temperaments]]).
|-
|-
|[[4/25-comma meantone]]||698.514||Close to [[67edo]]. Historically significant (see [[historical temperaments]]).
|[[4/25-comma meantone]]||698.514||Close to [[67edo]]. Historically significant (see [[historical temperaments]]).
|-
|-
|[[3/19-comma meantone]]
|[[2/13-comma meantone]]||698.646||  
|698.559
|
|-
|[[2/13-comma meantone]]||698.646||Close to [[79edo]].
|-
|-
|[[1/7-comma meantone]]||698.883||Close to [[91edo]]. Historically significant (see [[historical temperaments]]).
|[[1/7-comma meantone]]||698.883||Close to [[79edo]], [[91edo]]. Historically significant (see [[historical temperaments]]).
|-
|-
|[[2/15-comma meantone]]||699.088||
|[[2/15-comma meantone]]||699.088||  
|-
|-
|[[1/8-comma meantone]]||699.267||
|[[1/8-comma meantone]]||699.267||  
|-
|-
|[[2/17-comma meantone]]||699.425||
|[[2/17-comma meantone]]||699.425||  
|-
|-
|[[1/9-comma meantone]]||699.565||Negative harmonically significant tuning with flat fifth (see [[negative harmony temperaments]]).
|[[1/9-comma meantone]]||699.565||  
|-
|-
|[[1/10-comma meantone]]||699.804||
|[[1/10-comma meantone]]||699.804||  
|-
|-
|[[1/11-comma meantone]]||700.000||Everything up to this point generates 12 and 19 and 31 tone MOS scales.
|[[1/11-comma meantone]]||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]].)
|[[12edo]]||700.000||The largest MOS scale this can generate is 12 tone. Historically significant (see [[historical temperaments]].)
|-
|-
|[[1/12-comma meantone]]||700.163||Everything from this point onwards generates 12 and 17 and 29 tone MOS scales.
|[[1/12-comma meantone]]||700.163||Everything from this point onwards generates 12 and 17 tone MOS scales.
|-
|-
|[[1/13-comma meantone]]||700.301||
|[[1/13-comma meantone]]||700.301||  
|-
|-
|[[1/14-comma meantone]]||700.419||Negative harmonically significant tuning with flat fifth (see [[negative harmony temperaments]]).
|[[1/14-comma meantone]]||700.419||  
|-
|-
|[[1/15-comma meantone]]||700.521||Negative harmonically significant tuning with flat fifth (see [[negative harmony temperaments]]).
|[[1/15-comma meantone]]||700.521||  
|-
|-
|[[1/16-comma meantone]]||700.611||
|[[1/16-comma meantone]]||700.611||  
|-
|-
|[[1/17-comma meantone]]||700.690||
|[[1/17-comma meantone]]||700.690||  
|-
|-
|[[1/18-comma meantone]]||700.760||
|[[1/18-comma meantone]]||700.760||  
|-
|-
|0/1-comma meantone||701.955||[[Pythagorean tuning]]. Historically significant (see [[historical temperaments]].) Negative harmonically significant (see [[negative harmony temperaments]]). Everything from this point onwards does not meet the original historical definition of meantone (the whole tone being between 10/9 and 9/8).
|0/1-comma meantone||701.955||[[Pythagorean tuning]]. Historically significant (see [[historical temperaments]].) Everything from this point onwards does not meet the original historical definition of meantone (the whole tone being between 10/9 and 9/8).
|-
|-
|[[-1/18-comma meantone]]
|[[-1/18-comma meantone]]
Line 336: Line 206:
|
|
|-
|-
| -[[-1/17-comma meantone|1/17-comma meantone]]
|[[-1/17-comma meantone]]
|703.220
|703.220
|
|
Line 346: Line 216:
|[[-1/15-comma meantone]]
|[[-1/15-comma meantone]]
|703.389
|703.389
|Close to 11/13 third-kleisma temperament. Historically significant tuning with sharp fifth (see [[historical temperaments]]).
|Close to 11/13 third-kleisma temperament.
|-
|-
|[[-1/14-comma meantone]]
|[[-1/14-comma meantone]]
|703.491
|703.491
|Close to [[29edo]]. Historically significant tuning with sharp fifth (see [[historical temperaments]]).
|Close to [[29edo]].
|-
|-
|[[-1/13-comma meantone]]
|[[-1/13-comma meantone]]
Line 358: Line 228:
|[[-1/12-comma meantone]]
|[[-1/12-comma meantone]]
|703.747
|703.747
|
|  
|-
|
|703.910
|As sharp of [[Pythagorean tuning]] as [[55edo|12edo]] is flat. Negative harmonically significant (see [[negative harmony temperaments]]).
|-
|-
|[[-1/11-comma meantone]]
|[[-1/11-comma meantone]]
|703.910
|703.910
|
|About as sharp of [[Pythagorean tuning]] as [[12edo]] is flat.
|-
|-
|[[-1/10-comma meantone]]
|[[-1/10-comma meantone]]
Line 374: Line 240:
|[[-1/9-comma meantone]]
|[[-1/9-comma meantone]]
|704.344
|704.344
|Close to [[46edo]], 11/7 quarter-kleisma temperament. Historically significant tuning with sharp fifth (see [[historical temperaments]]).
|Close to [[46edo]], 11/7 quarter-kleisma temperament.
|-
|-
|[[2/17-comma meantone|-2/17-comma meantone]]
|[[-2/17-comma meantone]]
|704.483
|704.483
|
|
Line 390: Line 256:
|[[-1/7-comma meantone]]
|[[-1/7-comma meantone]]
|705.027
|705.027
|Close to [[80edo]]. Negative harmonically significant (see [[negative harmony temperaments]]).
|Close to [[80edo]].
|-
|-
|[[-2/13-comma meantone]]
|[[-2/13-comma meantone]]
Line 400: Line 266:
|
|
|-
|-
| -[[-4/25-comma meantone|4/25-comma meantone]]
|[[-4/25-comma meantone]]
|705.396
|705.396
|Negative harmonically significant (see [[negative harmony temperaments]]).
|
|-
|-
|[[-1/6-comma meantone]]
|[[-1/6-comma meantone]]
|705.538
|705.538
|Negative harmonically significant (see [[negative harmony temperaments]]).
|  
|-
|
|705.728
|As sharp of [[Pythagorean tuning]] as [[55edo]] is flat. Negative harmonically significant (see [[negative harmony temperaments]]).
|-
|-
|[[-3/17-comma meantone]]
|[[-3/17-comma meantone]]
|705.750
|705.750
|Historically significant tuning with sharp fifth (see [[historical temperaments]].
|About as sharp of [[Pythagorean tuning]] as [[55edo]] is flat.
|-
|-
|[[-2/11-comma meantone]]
|[[-2/11-comma meantone]]
|705.865
|705.865
|Everything up to this point generates 12 and 17 and 29 tone MOS scales.
|Everything up to this point generates 17 and 29 tone MOS scales.
|-
|-
|[[17edo]]
|[[17edo]]
|705.882
|705.882
|Simplest tuning for Middle Eastern neutral third scales. The largest MOS scale this can generate is 17 tone.
|Vaguely resembles Middle Eastern neutral third scales.
|-
|-
|[[-3/16-comma meantone]]
|[[-3/16-comma meantone]]
|705.987
|705.987
|Everything from this point onwards generates 12 and 17 and 22 tone MOS scales.
|Everything from this point onwards generates 17 and 22 tone MOS scales.
|-
|
|706.236
|As sharp of [[Pythagorean tuning]] as [[43edo]] is flat. Negative harmonically significant (see [[negative harmony temperaments]]).
|-
|-
|[[-1/5-comma meantone]]
|[[-1/5-comma meantone]]
|706.256
|706.256
|Negative harmonically significant (see [[negative harmony temperaments]]).
|About as sharp of [[Pythagorean tuning]] as [[43edo]] is flat.
|-
|-
|[[-3/14-comma meantone]]
|[[-3/14-comma meantone]]
|706.563
|706.563
|Negative harmonically significant (see [[negative harmony temperaments]]).
|  
|-
|
|706.613
|As sharp of [[Pythagorean tuning]] as [[74edo]] is flat. Negative harmonically significant (see [[negative harmony temperaments]]).
|-
|-
|[[-2/9-comma meantone]]
|[[-2/9-comma meantone]]
|706.734
|706.734
|Negative harmonically significant (see [[negative harmony temperaments]]).
|
|-
|-
|[[-3/13-comma meantone]]
|[[-3/13-comma meantone]]
Line 454: Line 308:
|[[-4/17-comma meantone]]
|[[-4/17-comma meantone]]
|707.015
|707.015
|
|About as sharp of [[Pythagorean tuning]] as [[74edo]] is flat.
|-
|-
|[[Negative Quarter-comma meantone]]
|707.332
|
|
|707.136
|As sharp of [[Pythagorean tuning]] as [[74edo]] is flat. Negative harmonically significant (see [[negative harmony temperaments]]).
|-
|-
|[[Negative Quarter-comma meantone|-1/4-comma meantone]]
|[[-4/15-comma meantone]]
|707.332
|Negative harmonically significant (see [[negative harmony temperaments]]).
|-
| -[[4/15-comma meantone]]
|707.690
|707.690
|
|About as sharp of [[Pythagorean tuning]] as [[Golden meantone]] is flat.
|-
|[[39edo]]
|707.692
|Historically significant tuning with sharp fifth (see [[historical temperaments]].) The largest MOS scale this can generate is 39 tone.
|-
|
|707.696
|As sharp of [[Pythagorean tuning]] as [[50edo|Golden meantone]] is flat. Negative harmonically significant (see [[negative harmony temperaments]]).
|-
|-
|[[-7/26-comma meantone]]
|[[-7/26-comma meantone]]
|707.745
|707.745
|Negative harmonically significant (see [[negative harmony temperaments]]).
|
|-
|-
|[[-3/11-comma meantone]]
|[[-3/11-comma meantone]]
|707.820
|707.820
|Almost exactly -1/4-''Pythagorean'' comma meantone
|Almost exactly -1/4-''Pythagorean'' comma meantone
|-
|
|707.910
|As sharp of [[Pythagorean tuning]] as [[50edo]] is flat. Negative harmonically significant (see [[negative harmony temperaments]]).
|-
|-
|[[-5/18-comma meantone]]
|[[-5/18-comma meantone]]
|707.930
|707.930
|Negative harmonically significant (see [[negative harmony temperaments]]).
|About as sharp of [[Pythagorean tuning]] as [[50edo]] is flat.
|-
|-
|[[-2/7-comma meantone]]
|[[-2/7-comma meantone]]
|708.100
|708.100
|Negative harmonically significant (see [[negative harmony temperaments]]).
|
|-
|-
|[[-7/24-comma meantone]]
|[[-7/24-comma meantone]]
Line 506: Line 344:
|[[-3/10-comma meantone]]
|[[-3/10-comma meantone]]
|708.407
|708.407
|
|About sharp of [[Pythagorean tuning]] as [[19edo|Lucy tuning]] is flat. Nearly as sharp of [[Pythagorean tuning]] as [[88edo]] is flat.
|-
|
|708.417
|As sharp of [[Pythagorean tuning]] as [[19edo|Lucy tuning]] is flat. Nearly as sharp of [[Pythagorean tuning]] as [[88edo]] is flat. Negative harmonically significant (see [[negative harmony temperaments]]).
|-
|-
|[[-4/13-comma meantone]]
|[[-4/13-comma meantone]]
Line 518: Line 352:
|[[-5/16-comma meantone]]
|[[-5/16-comma meantone]]
|708.675
|708.675
|
|  
|-
|[[22edo]]
|709.091
|Isomorphic to [[Indian music|Indian]] shrutis. The largest MOS scale this can generate is 22 tone.
|-
|-
|[[-1/3-comma meantone]]
|[[-1/3-comma meantone]]
|709.124
|709.124
|Negative harmonically significant (see [[negative harmony temperaments]]).
|Close to [[22edo]]. About as sharp of [[Pythagorean tuning]] as [[19edo]] is flat.
|-
|
|709.173
|As sharp of [[Pythagorean tuning]] as [[19edo]] is flat. Negative harmonically significant (see [[negative harmony temperaments]]).
|-
|-
|[[-6/17-comma meantone]]
|[[-6/17-comma meantone]]
Line 542: Line 368:
|[[-4/11-comma meantone]]
|[[-4/11-comma meantone]]
|709.775
|709.775
|Almost exactly -1/3-''Pythagorean'' comma meantone
|Almost exactly -1/3-''Pythagorean'' comma meantone.
|-
|-
|[[-3/8-comma meantone]]
|[[-3/8-comma meantone]]
Line 562: Line 388:
|[[-2/5-comma meantone]]
|[[-2/5-comma meantone]]
|710.558
|710.558
|Negative harmonically significant (see [[negative harmony temperaments]]).
|
|-
|-
|[[-7/17-comma meantone]]
|[[-7/17-comma meantone]]
Line 584: Line 410:
|
|
|-
|-
|
|[[-5/11-comma meantone]]
|711.602
|As sharp of [[Pythagorean tuning]] as [[26edo|19edo]] is flat.
|-
|[[5/11-comma meantone|-5/11-comma meantone]]
|711.731
|711.731
|
|
Line 606: Line 428:
|[[-1/2-comma meantone]]
|[[-1/2-comma meantone]]
|712.708
|712.708
|Close to [[32edo]]. Negative harmonically significant (see [[negative harmony temperaments]]). Everything from this point onwards does not meet the negative harmonic definition of meantone (the whole tone being between 9/8 and 729/640).
|Close to [[32edo]]. Everything from this point onwards does not meet the negative harmonic definition of meantone (the whole tone being between 9/8 and 729/640).
|-
|-
| -[[9/17-comma meantone]]
|[[-9/17-comma meantone]]
|713.340
|713.340
|
|
|-
|-
| -[[-8/15-comma meantone|8/15-comma meantone]]
|[[-8/15-comma meantone]]
|713.425
|713.425
|
|
Line 664: Line 486:
|Close to [[52edo]].
|Close to [[52edo]].
|-
|-
| -[[-7/11-comma meantone|7/11-comma meantone]]
|[[-7/11-comma meantone]]
|715.641
|715.641
|
|
Line 700: Line 522:
|
|
|-
|-
|[[12/17-comma meantone|-12/17-comma meantone]]
|[[-12/17-comma meantone]]
|717.136
|717.136
|Close to [[82edo]].
|Close to [[82edo]].
Line 720: Line 542:
|
|
|-
|-
|[[19/26-comma meantone|-19/26-comma meantone]]
|[[-19/26-comma meantone]]
|717.671
|717.671
|
|
|-
|-
|[[11/15-comma meantone|-11/15-comma meantone]]
|[[-11/15-comma meantone]]
|717.726
|717.726
|
|
Line 730: Line 552:
|[[-3/4-comma meantone]]
|[[-3/4-comma meantone]]
|718.085
|718.085
|
|About as sharp of [[Pythagorean tuning]] as [[7edo]] is flat.
|-
|
|718.196
|As sharp of [[Pythagorean tuning]] as [[7edo]] is flat.
|-
|-
|[[-13/17-comma meantone]]
|[[-13/17-comma meantone]]
Line 770: Line 588:
|[[-5/6-comma meantone]]
|[[-5/6-comma meantone]]
|719.877
|719.877
|Everything up to this point generates 12 and 17 and 22 tone MOS scales.
|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.'''
|[[5edo]]||720.000||The largest MOS scale this can generate is 5 tone. '''Upper boundary of 5-limit diamond monotone.'''

Revision as of 01:56, 26 December 2024

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Here are all meantone tunings that can be written in the form "n-comma meantone", where n is a fraction between 0 and 1 with a denominator 18 or smaller. Also included are 5, 7 and 12edo (to delineate MOS shapes), as well as a few other notable meantone tunings (e.g. 4/25-comma).

The comma being divided here is the syntonic comma (81/80).

Temperaments that fall outside of the diamond monotone range will not provide most of the advantages that meantone usually provides, but they are included for completeness.

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

Temperament Generator (cents) Comments
1/1-comma meantone 680.449 Close to 30edo
15/16-comma meantone 681.793
14/15-comma meantone 681.883 Close to 44edo
13/14-comma meantone 681.985
12/13-comma meantone 682.103
11/12-comma meantone 682.241
12/11-comma meantone 682.404 Close to 51edo
9/10-comma meantone 682.599
8/9-comma meantone 682.838 Close to 58edo
7/8-comma meantone 683.137 Close to 65edo
13/15-comma meantone 683.316 Close to 72edo
6/7-comma meantone 683.521 Close to 79edo
11/13-comma meantone 683.757 Close to 86edo
5/6-comma meantone 684.033
9/11-comma meantone 684.359
13/16-comma meantone 684.481
4/5-comma meantone 684.75
11/14-comma meantone 685.057
7/9-comma meantone 685.228
10/13-comma meantone 685.412 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 685.825 Everything from this point onwards generates 12 and 19 tone MOS scales.
11/15-comma meantone 686.184
8/11-comma meantone 686.314
5/7-comma meantone 686.593
7/10-comma meantone 686.901
9/13-comma meantone 687.066
11/16-comma meantone 687.169
2/3-comma meantone 687.617 Close to 89edo
9/14-comma meantone 688.129 Close to 75edo
7/11-comma meantone 688.269 Close to 68edo
5/8-comma meantone 688.514 Close to 61edo
8/13-comma meantone 688.720 Close to 54edo
3/5-comma meantone 689.051
7/12-comma meantone 689.410 Close to 47edo
4/7-comma meantone 689.666 Close to 87edo
9/16-comma meantone 689.858
5/9-comma meantone 690.007 Close to 40edo
6/11-comma meantone 690.224
7/13-comma meantone 690.375 Close to 73edo
8/15-comma meantone 690.485 Close to 33edo
1/2-comma meantone 691.202 Close to 92edo, 59edo. Historically significant (see historical temperaments). Everything up to this point does not meet the original historical definition of meantone (the whole tone being between 10/9 and 9/8).
7/15-comma meantone 691.919 Close to 85edo
6/13-comma meantone 692.029
5/11-comma meantone 692.179
4/9-comma meantone 692.397 Close to 26edo
7/16-comma meantone 692.546
3/7-comma meantone 692.738
5/12-comma meantone 692.994 Close to 71edo
7/17-comma meantone 693.099
2/5-comma meantone 693.352 Close to 45edo
7/18-comma meantone 693.591
5/13-comma meantone 693.683 Close to 64edo
3/8-comma meantone 693.890 Close to 83edo
4/11-comma meantone 694.134 Almost exactly 1/3-Pythagorean comma meantone
5/14-comma meantone 694.274
6/17-comma meantone 694.365
1/3-comma meantone 694.786 Close to 19edo. Historically significant (see historical temperaments).
5/16-comma meantone 695.234
4/13-comma meantone 695.338
3/10-comma meantone 695.503 Close to 88edo
5/17-comma meantone 695.630
2/7-comma meantone 695.810 Close to 69edo. Historically significant (see historical temperaments).
5/18-comma meantone 695.981 Close to 50edo. Historically significant (see historical temperaments).
3/11-comma meantone 696.090 Close to 50edo
7/26-comma meantone 696.165 Close to golden meantone. Historically significant (see historical temperaments).
4/15-comma meantone 696.220 Close to 81edo, close to golden meantone
1/4-comma meantone 696.578 Close to 31edo. Historically significant (see historical temperaments).
4/17-comma meantone 696.895
3/13-comma meantone 696.992 Close to septimal & tridecimal CTE tunings.
2/9-comma meantone 697.176 Close to 5-limit and undecimal CTE tunings. Historically significant (see historical temperaments).
3/14-comma meantone 697.346 Close to 74edo. Historically significant (see historical temperaments).
1/5-comma meantone 697.654 Close to 43edo. Historically significant (see historical temperaments).
3/16-comma meantone 697.923
2/11-comma meantone 698.045 Close to 55edo
3/17-comma meantone 699.425
1/6-comma meantone 698.371 Close to 67edo. Historically significant (see historical temperaments).
4/25-comma meantone 698.514 Close to 67edo. Historically significant (see historical temperaments).
2/13-comma meantone 698.646
1/7-comma meantone 698.883 Close to 79edo, 91edo. Historically significant (see historical temperaments).
2/15-comma meantone 699.088
1/8-comma meantone 699.267
2/17-comma meantone 699.425
1/9-comma meantone 699.565
1/10-comma meantone 699.804
1/11-comma meantone 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 700.163 Everything from this point onwards generates 12 and 17 tone MOS scales.
1/13-comma meantone 700.301
1/14-comma meantone 700.419
1/15-comma meantone 700.521
1/16-comma meantone 700.611
1/17-comma meantone 700.690
1/18-comma meantone 700.760
0/1-comma meantone 701.955 Pythagorean tuning. Historically significant (see historical temperaments.) Everything from this point onwards does not meet the original historical definition of meantone (the whole tone being between 10/9 and 9/8).
-1/18-comma meantone 703.150
-1/17-comma meantone 703.220
-1/16-comma meantone 703.299
-1/15-comma meantone 703.389 Close to 11/13 third-kleisma temperament.
-1/14-comma meantone 703.491 Close to 29edo.
-1/13-comma meantone 703.609
-1/12-comma meantone 703.747
-1/11-comma meantone 703.910 About as sharp of Pythagorean tuning as 12edo is flat.
-1/10-comma meantone 704.105
-1/9-comma meantone 704.344 Close to 46edo, 11/7 quarter-kleisma temperament.
-2/17-comma meantone 704.483
-1/8-comma meantone 704.643
-2/15-comma meantone 704.823 Close to 63edo.
-1/7-comma meantone 705.027 Close to 80edo.
-2/13-comma meantone 705.350
-3/19-comma meantone 705.350
-4/25-comma meantone 705.396
-1/6-comma meantone 705.538
-3/17-comma meantone 705.750 About as sharp of Pythagorean tuning as 55edo is flat.
-2/11-comma meantone 705.865 Everything up to this point generates 17 and 29 tone MOS scales.
17edo 705.882 Vaguely resembles Middle Eastern neutral third scales.
-3/16-comma meantone 705.987 Everything from this point onwards generates 17 and 22 tone MOS scales.
-1/5-comma meantone 706.256 About as sharp of Pythagorean tuning as 43edo is flat.
-3/14-comma meantone 706.563
-2/9-comma meantone 706.734
-3/13-comma meantone 706.918
-4/17-comma meantone 707.015 About as sharp of Pythagorean tuning as 74edo is flat.
Negative Quarter-comma meantone 707.332
-4/15-comma meantone 707.690 About as sharp of Pythagorean tuning as Golden meantone is flat.
-7/26-comma meantone 707.745
-3/11-comma meantone 707.820 Almost exactly -1/4-Pythagorean comma meantone
-5/18-comma meantone 707.930 About as sharp of Pythagorean tuning as 50edo is flat.
-2/7-comma meantone 708.100
-7/24-comma meantone 708.227
-5/17-comma meantone 708.280
-3/10-comma meantone 708.407 About sharp of Pythagorean tuning as Lucy tuning is flat. Nearly as sharp of Pythagorean tuning as 88edo is flat.
-4/13-comma meantone 708.572
-5/16-comma meantone 708.675
-1/3-comma meantone 709.124 Close to 22edo. About as sharp of Pythagorean tuning as 19edo is flat.
-6/17-comma meantone 709.545
-5/14-comma meantone 709.636
-4/11-comma meantone 709.775 Almost exactly -1/3-Pythagorean comma meantone.
-3/8-comma meantone 710.019
Negative Split Golden ratio comma meantone 710.170
-5/13-comma meantone 710.227 Close to 49edo.
-7/18-comma meantone 710.319
-2/5-comma meantone 710.558
-7/17-comma meantone 710.810
-5/12-comma meantone 710.915
-3/7-comma meantone 711.172 Close to 27edo.
-7/16-comma meantone 711.364
-4/9-comma meantone 711.513
-5/11-comma meantone 711.731
-6/13-comma meantone 711.880 Close to 59edo.
-7/15-comma meantone 711.991
-8/17-comma meantone 712.075
-1/2-comma meantone 712.708 Close to 32edo. Everything from this point onwards does not meet the negative harmonic definition of meantone (the whole tone being between 9/8 and 729/640).
-9/17-comma meantone 713.340
-8/15-comma meantone 713.425
-7/13-comma meantone 713.535 Close to 37edo.
-6/11-comma meantone 713.686
-5/9-comma meantone 713.903
-9/16-comma meantone 714.052
-4/7-comma meantone 714.244 Close to 42edo.
-7/12-comma meantone 714.500
-10/17-comma meantone 714.606
-3/5-comma meantone 714.859 Close to 47edo.
-11/18-comma meantone 715.098
-8/13-comma meantone 715.190
Negative Golden ratio comma meantone
-5/8-comma meantone 715.396 Close to 52edo.
-7/11-comma meantone 715.641
-9/14-comma meantone 715.780 Close to 57edo.
-11/17-comma meantone 715.871
-2/3-comma meantone 716.293 Close to 62edo.
-15/22 comma meantone 716.618 Close to 67edo.
-13/19 comma meantone 716.669 Close to 72edo.
-11/16-comma meantone 716.741
-9/13-comma meantone 716.844 Close to 77edo.
-7/10-comma meantone 717.009
-12/17-comma meantone 717.136 Close to 82edo.
-17/24-comma meantone 717.188 Close to 87edo.
-5/7-comma meantone 717.317 Close to 92edo.
-13/18-comma meantone 717.487 Close to 97edo.
-8/11-comma meantone 717.596
-19/26-comma meantone 717.671
-11/15-comma meantone 717.726
-3/4-comma meantone 718.085 About as sharp of Pythagorean tuning as 7edo is flat.
-13/17-comma meantone 718.401
-10/13-comma meantone 718.498
-7/9-comma meantone 718.682
-11/14-comma meantone 718.853
-4/5-comma meantone 719.160
-13/16-comma meantone 719.429
-9/11-comma meantone 719.551
-14/17-comma meantone 719.666
-5/6-comma meantone 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 meantone 720.020 Everything from this point onwards generates 13 and 18 tone MOS scales.
-16/19-comma meantone 720.066
-11/13-comma meantone 720.153
-6/7-comma meantone 720.399
-13/15-comma meantone 720.594
-7/8-comma meantone 720.773
-15/17-comma meantone 720.931
-8/9-comma meantone 721.017
-9/10-comma meantone 721.311
-10/11-comma meantone 721.506
-11/12-comma meantone 721.669
-12/13-comma meantone 721.807
-13/14-comma meantone 721.925
-14/15-comma meantone 722.028
-15/16-comma meantone 722.117
-16/17-comma meantone 722.196
-17/18-comma meantone 722.266
-1/1-comma meantone 723.461 Close to 68edo
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