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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.  
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.  


Temperaments that fall outside of the "historically considered meantone", or even the diamond monotone, range will not provide most of the properties that meantone usually provides, but they are included for completeness.
== Scope of table ==


== Scope ==
=== 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:  
A small number of additional temperaments are included. Not too many, to avoid clutter, just the bare minimum:  
* 7, 12, 17 and 5 [[edo]]s (to delineate [[MOS]] shapes)
* {{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"''.  
* any tunings listed under "[[historical temperaments]]" (e.g. 4/25-comma), ''but only the ones of the form "n-comma"''.


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.  
== 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.


All tunings on this table are close (within 1/2 degree of closing) to some arbitrarily large [[EDO]], but labeling them beyond [[100edo]] would clutter the table.
== The table ==


== Flatter than flattest allowed historical meantone ==
=== Flatter than flattest historically-defined meantone ===
{| class="wikitable mw-collapsible"
{| class="wikitable mw-collapsible"
|+Spectrum of meantone tunings 1/1-comma to 11/21-comma
|+Spectrum of meantone tunings 1/1-comma to 1/2-comma
!Meantone Temperament!!Generator (cents)!!Comments
!Meantone Temperament!!Generator (cents)!!Comments
|-
|-
Line 45: Line 65:
|
|
|-
|-
|[[15/16-comma meantone|15/16-comma]] ||681.793|| Close to [[44edo]]
|[[15/16-comma meantone|15/16-comma]] ||681.793|| Close to [[44edo]].
|-
|-
|[[14/15-comma meantone|14/15-comma]] ||681.883||
|[[14/15-comma meantone|14/15-comma]] ||681.883||
Line 55: Line 75:
|[[11/12-comma meantone|11/12-comma]] ||682.241||  
|[[11/12-comma meantone|11/12-comma]] ||682.241||  
|-
|-
|[[10/11-comma meantone|10/11-comma]] ||682.404||Close to [[51edo]]
|[[10/11-comma meantone|10/11-comma]] ||682.404||Close to [[51edo]].
|-
|-
|[[19/21-comma meantone|19/21-comma]]  
|[[19/21-comma meantone|19/21-comma]]  
Line 67: Line 87:
|
|
|-
|-
|[[8/9-comma meantone|8/9-comma]] ||682.838||Close to [[58edo]]
|[[8/9-comma meantone|8/9-comma]] ||682.838||Close to [[58edo]].
|-
|-
|[[15/17-comma meantone|15/17-comma]]  
|[[15/17-comma meantone|15/17-comma]]  
Line 73: Line 93:
|
|
|-
|-
|[[7/8-comma meantone|7/8-comma]] ||683.137||Close to [[65edo]]
|[[7/8-comma meantone|7/8-comma]] ||683.137||Close to [[65edo]].
|-
|-
|[[13/15-comma meantone|13/15-comma]] ||683.316||Close to [[72edo]]
|[[13/15-comma meantone|13/15-comma]] ||683.316||Close to [[72edo]].
|-
|-
|[[19/22-comma meantone|19/22-comma]]  
|[[19/22-comma meantone|19/22-comma]]  
Line 81: Line 101:
|
|
|-
|-
|[[6/7-comma meantone|6/7-comma]] ||683.521||Close to [[79edo]]
|[[6/7-comma meantone|6/7-comma]] ||683.521||Close to [[79edo]].
|-
|-
|[[17/20-comma meantone|17/20-comma]]  
|[[17/20-comma meantone|17/20-comma]]  
Line 87: Line 107:
|
|
|-
|-
|[[11/13-comma meantone|11/13-comma]] ||683.757||Close to [[86edo]]
|[[11/13-comma meantone|11/13-comma]] ||683.757||Close to [[86edo]].
|-
|-
|[[16/19-comma meantone|16/19-comma]]  
|[[16/19-comma meantone|16/19-comma]]  
Line 97: Line 117:
|Close to [[93edo]]
|Close to [[93edo]]
|-
|-
|[[5/6-comma meantone|5/6-comma]] ||684.033|| Close to [[100edo]]
|[[5/6-comma meantone|5/6-comma]] ||684.033|| Close to [[100edo]].
|-
|-
|[[14/17-comma meantone|14/17-comma]]  
|[[14/17-comma meantone|14/17-comma]]  
Line 187: Line 207:
|
|
|-
|-
|[[2/3-comma meantone|2/3-comma]] ||687.617||Close to [[89edo]]
|[[2/3-comma meantone|2/3-comma]] ||687.617||Close to [[89edo]].
|-
|-
|[[17/26-comma meantone|17/26-comma]]
|[[17/26-comma meantone|17/26-comma]]
Line 207: Line 227:
|[[9/14-comma meantone|9/14-comma]] ||688.129||
|[[9/14-comma meantone|9/14-comma]] ||688.129||
|-
|-
|[[7/11-comma meantone|7/11-comma]] ||688.269||Close to [[68edo]]
|[[7/11-comma meantone|7/11-comma]] ||688.269||Close to [[68edo]].
|-
|-
|[[12/19-comma meantone|12/19-comma]]  
|[[12/19-comma meantone|12/19-comma]]  
Line 213: Line 233:
|
|
|-
|-
|[[5/8-comma meantone|5/8-comma]] ||688.514||Close to [[61edo]]
|[[5/8-comma meantone|5/8-comma]] ||688.514||Close to [[61edo]] and [[43/32]].
|-
|-
|[[13/21-comma meantone|13/21-comma]]  
|[[13/21-comma meantone|13/21-comma]]  
Line 219: Line 239:
|
|
|-
|-
|[[Golden ratio-comma meantone|Golden ratio-comma]]  
|[[1/φ-comma meantone|1/ϕ-comma]]
|688.663
|688.663
|
|
Line 227: Line 247:
|[[11/18-comma meantone|11/18-comma]]  
|[[11/18-comma meantone|11/18-comma]]  
|688.812
|688.812
|Close to [[54edo]]
|Close to [[54edo]].
|-
|-
|[[14/23-comma meantone|14/23-comma]]   
|[[14/23-comma meantone|14/23-comma]]   
Line 243: Line 263:
|
|
|-
|-
|[[7/12-comma meantone|7/12-comma]] ||689.410||Close to [[47edo]]
|[[7/12-comma meantone|7/12-comma]] ||689.410||Close to [[47edo]].
|-
|-
|[[11/19-comma meantone|11/19-comma]]  
|[[11/19-comma meantone|11/19-comma]]  
Line 249: Line 269:
|
|
|-
|-
|[[4/7-comma meantone|4/7-comma]] ||689.666||Close to [[87edo]]
|[[4/7-comma meantone|4/7-comma]] ||689.666||Close to [[87edo]].
|-
|-
|[[9/16-comma meantone|9/16-comma]] ||689.858||  
|[[9/16-comma meantone|9/16-comma]] ||689.858||  
|-
|-
|[[5/9-comma meantone|5/9-comma]] ||690.007||Close to [[40edo]]
|[[5/9-comma meantone|5/9-comma]] ||690.007||Close to [[40edo]].
|-
|-
|[[11/20-comma meantone|11/20-comma]]  
|[[11/20-comma meantone|11/20-comma]]  
Line 261: Line 281:
|[[6/11-comma meantone|6/11-comma]] ||690.224||  
|[[6/11-comma meantone|6/11-comma]] ||690.224||  
|-
|-
|[[7/13-comma meantone|7/13-comma]] ||690.375||Close to [[73edo]]
|[[7/13-comma meantone|7/13-comma]] ||690.375||Close to [[73edo]].
|-
|-
|[[8/15-comma meantone|8/15-comma]] ||690.485||
|[[8/15-comma meantone|8/15-comma]] ||690.485||
Line 276: Line 296:
|690.690
|690.690
|Close to [[33edo]]
|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 considered meantone ==
=== Historically-defined meantone ===
{| class="wikitable mw-collapsible"
{| class="wikitable mw-collapsible"
|+Spectrum of meantone tunings 1/2-comma to 1/22-comma
|+Spectrum of meantone tunings 10/21-comma to 1/22-comma
!Temperament!!Generator (cents)!!Comments
!Temperament!!Generator (cents)!!Comments
|-
|[[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.
|-
|-
|[[10/21-comma meantone|10/21-comma]]
|[[10/21-comma meantone|10/21-comma]]
Line 291: Line 311:
|[[9/19-comma meantone|9/19-comma]]  
|[[9/19-comma meantone|9/19-comma]]  
|691.768
|691.768
|Close to [[85edo]]
|Close to [[85edo]].
|-
|-
|[[8/17-comma meantone|8/17-comma]]  
|[[8/17-comma meantone|8/17-comma]]  
Line 305: Line 325:
|[[9/20-comma meantone|9/20-comma]]  
|[[9/20-comma meantone|9/20-comma]]  
|692.277
|692.277
|Close to [[26edo]]
|Close to [[26edo]].
|-
|-
|[[4/9-comma meantone|4/9-comma]] ||692.397||
|[[4/9-comma meantone|4/9-comma]] ||692.397||
Line 317: Line 337:
|
|
|-
|-
|[[5/12-comma meantone|5/12-comma]] ||692.994||Close to [[71edo]]
|[[5/12-comma meantone|5/12-comma]] ||692.994||Close to [[71edo]].
|-
|-
|[[7/17-comma meantone|7/17-comma]] ||693.099||
|[[7/17-comma meantone|7/17-comma]] ||693.099||
Line 325: Line 345:
|
|
|-
|-
|[[2/5-comma meantone|2/5-comma]] ||693.352||Close to [[45edo]]
|[[2/5-comma meantone|2/5-comma]] ||693.352||Close to [[45edo]].
|-
|-
|[[9/23-comma meantone|9/23-comma]]  
|[[9/23-comma meantone|9/23-comma]]  
Line 335: Line 355:
|[[5/13-comma meantone|5/13-comma]] ||693.683||
|[[5/13-comma meantone|5/13-comma]] ||693.683||
|-
|-
|[[Split Golden ratio-comma meantone|Split Golden ratio-comma]]  
|[[1/(φ+1)-comma meantone|1/(ϕ+1)-comma]]
|693.740
|693.740
|Close to [[64edo]]
|Close to [[64edo]].
|-
|-
|[[8/21-comma meantone|8/21-comma]]  
|[[8/21-comma meantone|8/21-comma]]  
Line 343: Line 363:
|
|
|-
|-
|[[3/8-comma meantone|3/8-comma]] ||693.890||Close to [[83edo]]
|[[3/8-comma meantone|3/8-comma]] ||693.890||Close to [[83edo]].
|-
|-
|[[7/19-comma meantone|7/19-comma]]  
|[[7/19-comma meantone|7/19-comma]]  
Line 349: Line 369:
|
|
|-
|-
|[[4/11-comma meantone|4/11-comma]] ||694.134||Almost exactly 1/3-''Pythagorean'' comma meantone
|[[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||  
|[[5/14-comma meantone|5/14-comma]] ||694.274||  
Line 405: Line 425:
|[[7/26-comma meantone|7/26-comma]] ||696.165||Close to [[golden meantone]]. Historically significant (see [[historical temperaments]]).
|[[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||
|[[4/15-comma meantone|4/15-comma]] ||696.220||Close to [[5-limit]] meantone [[POTE]] tuning.
|-
|-
|[[5/19-comma meantone|5/19-comma]]  
|[[5/19-comma meantone|5/19-comma]]  
Line 411: Line 431:
|Close to [[81edo]].
|Close to [[81edo]].
|-
|-
|[[Quarter-comma meantone|1/4-comma]] ||696.578||Historically significant (see [[historical temperaments]]).
|[[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]]  
|[[5/21-comma meantone|5/21-comma]]  
Line 419: Line 439:
|[[4/17-comma meantone|4/17-comma]] ||696.895||
|[[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.
|[[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]]   
|[[5/22-comma meantone|5/22-comma]]   
Line 449: Line 469:
|
|
|-
|-
|[[1/6-comma meantone|1/6-comma]] ||698.371||Historically significant (see [[historical temperaments]]). Everything up to this point has a fifth which is noticeably flat of [[Pythagorean tuning]].
|[[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]].
|[[4/25-comma meantone|4/25-comma]] ||698.514||Close to [[67edo]].
Line 522: Line 542:
|}
|}


== Negative-harmonic-defined meantone (synonymous with [[Superpyth]]) ==
=== Negative harmony theory-defined meantone (most often approached as [[superpyth]]) ===
{| class="wikitable mw-collapsible"
{| class="wikitable mw-collapsible"
|+Spectrum of meantone tunings 0/1-comma to -10/21-comma
|+Spectrum of meantone tunings 0/1-comma to -10/21-comma
Line 632: Line 652:
|[[-1/6-comma meantone|-1/6-comma]]  
|[[-1/6-comma meantone|-1/6-comma]]  
|705.538
|705.538
| Everything from this point onwards has a fifth which is noticeably sharp of [[Pythagorean tuning]].
| 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]]   
|[[-4/23-comma meantone|-4/23-comma]]   
Line 648: Line 668:
|[[17edo]]
|[[17edo]]
|705.882
|705.882
|Vaguely resembles Middle Eastern neutral third scales.
|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]]  
|[[-3/16-comma meantone|-3/16-comma]]  
Line 690: Line 710:
|About as sharp of [[Pythagorean tuning]] as [[31edo]] is flat.
|About as sharp of [[Pythagorean tuning]] as [[31edo]] is flat.
|-
|-
|[[Negative Quarter-comma meantone|Negative Quarter-comma]]  
|[[-1/4-comma meantone|-1/4-comma]]
|707.332
|707.332
|
|
Line 700: Line 720:
|[[-4/15-comma meantone|-4/15-comma]]  
|[[-4/15-comma meantone|-4/15-comma]]  
|707.690
|707.690
|About as sharp of [[Pythagorean tuning]] as [[Golden meantone]] is flat.
|About as sharp of [[Pythagorean tuning]] as [[golden meantone]] is flat.
|-
|-
|[[-7/26-comma meantone|-7/26-comma]]  
|[[-7/26-comma meantone|-7/26-comma]]  
Line 712: Line 732:
|[[-5/18-comma meantone|-5/18-comma]]  
|[[-5/18-comma meantone|-5/18-comma]]  
|707.930
|707.930
|About as sharp of [[Pythagorean tuning]] as [[50edo]] is flat. Close to [[86edo|100edo]]
|About as sharp of [[Pythagorean tuning]] as [[50edo]] is flat. Close to [[100edo]].
|-
|-
|[[-2/7-comma meantone|-2/7-comma]]  
|[[-2/7-comma meantone|-2/7-comma]]  
Line 760: Line 780:
|[[-9/26-comma meantone|-9/26-comma]]  
|[[-9/26-comma meantone|-9/26-comma]]  
|709.399
|709.399
|
|Close to [[2.3.7-limit]] superpyth [[POTE]] tuning.
|-
|-
|[[-8/23-comma meantone|-8/23-comma]]  
|[[-8/23-comma meantone|-8/23-comma]]  
Line 772: Line 792:
|[[-6/17-comma meantone|-6/17-comma]]  
|[[-6/17-comma meantone|-6/17-comma]]  
|709.545
|709.545
|
|Close to [[11-limit]] superpyth [[CTE]] tuning.
|-
|-
|[[-5/14-comma meantone|-5/14-comma]]  
|[[-5/14-comma meantone|-5/14-comma]]  
|709.636
|709.636
|Close to [[93edo]]
|Close to [[93edo]]. Close to [[2.3.7-limit]] and [[7-limit]] superpyth CTE tunings.
|-
|-
|[[-4/11-comma meantone|-4/11-comma]]  
|[[-4/11-comma meantone|-4/11-comma]]  
Line 784: Line 804:
|[[-7/19-comma meantone|-7/19-comma]]  
|[[-7/19-comma meantone|-7/19-comma]]  
|709.878
|709.878
|
|Close to [[13-limit]] superpyth CTE tuning.
|-
|-
|[[-3/8-comma meantone|-3/8-comma]]  
|[[-3/8-comma meantone|-3/8-comma]]  
Line 794: Line 814:
|
|
|-
|-
|[[Negative Split Golden ratio-comma meantone|Negative Split Golden ratio-comma]]  
|[[-1/(φ+1)-comma meantone|-1/(ϕ+1)-comma]]
|710.170
|710.170
|
|Close to [[11-limit]] superpyth POTE tuning.
|-
|-
|[[-5/13-comma meantone|-5/13-comma]]  
|[[-5/13-comma meantone|-5/13-comma]]  
|710.227
|710.227
|Close to [[49edo]].
|Close to [[49edo]]. Close to [[7-limit]] superpyth POTE tuning.
|-
|-
|[[-7/18-comma meantone|-7/18-comma]]  
|[[-7/18-comma meantone|-7/18-comma]]  
Line 812: Line 832:
|[[-2/5-comma meantone|-2/5-comma]]  
|[[-2/5-comma meantone|-2/5-comma]]  
|710.558
|710.558
|
|Close to [[13-limit]] superpyth POTE tuning.
|-
|-
|[[-9/22-comma meantone|-9/22-comma]]  
|[[-9/22-comma meantone|-9/22-comma]]  
Line 871: Line 891:
|}
|}


== Sharper than sharpest negative-harmonic-defined meantone ==
=== Sharper than sharpest negative harmonic-defined meantone ===
{| class="wikitable mw-collapsible"
{| class="wikitable mw-collapsible"
|+ Spectrum of meantone tunings -1/2-comma to -1/1-comma
|+ Spectrum of meantone tunings -1/2-comma to -1/1-comma
Line 952: Line 972:
|
|
|-
|-
|[[Negative Golden ratio-comma meantone|Negative Golden ratio-comma]]  
|[[-1/φ-comma meantone|-1/ϕ-comma]]
|715.247
|715.247
|
|
Line 962: Line 982:
|[[-5/8-comma meantone|-5/8-comma]]  
|[[-5/8-comma meantone|-5/8-comma]]  
|715.396
|715.396
|Close to [[52edo]].
|Close to [[52edo]] and 387/256.
|-
|-
|[[-12/19-comma meantone|-12/19-comma]]  
|[[-12/19-comma meantone|-12/19-comma]]  
Line 1,204: Line 1,224:
|[[-1/1-comma meantone|-1/1-comma]]  
|[[-1/1-comma meantone|-1/1-comma]]  
|723.461
|723.461
|Close to [[68edo]]
|Close to [[68edo]].
|-
|-
|}
|}
[[Category:Tables]]
[[Category:Tables]]
[[Category:Meantone]]
[[Category:Meantone]]
Retrieved from "https://en.xen.wiki/w/User:BudjarnLambeth/Table_of_n-comma_meantone_generators"