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{{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.  
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.  


== Overview of table ==
== Scope of table ==


=== Characteristics included ===
=== Characteristics included ===
Some of the characteristics this table mentions for each temperament include:
Some of the characteristics this table mentions for each temperament include:
* Whether it saw historical (pre-1950) use
* 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 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 an optimised tuning of meantone (e.g. 7-limit [[CTE]])
* 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 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 about equally sharp of [[3/2]] as some other listed temperament is flat
Line 23: Line 23:
=== Special cases included ===
=== 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 small [[MOS]] shapes and boundaries of diamond monotone)
* {{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"''.


== Cautions ==
== 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.


=== Preservation of meantone behavior ===
Temperaments that fall outside of the "diamond monotone" range preserve even fewer meantone properties, but they are also included for completeness.
Temperaments that fall outside of the "[[Historical temperaments|historically considered meantone]]" range will not possess most of the 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 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 64: 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 74: 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 86: 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 92: 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 100: 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 106: 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 116: 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 206: 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 226: 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 232: 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 238: Line 239:
|
|
|-
|-
|[[Golden ratio-comma meantone|Golden ratio-comma]]  
|[[1/φ-comma meantone|1/ϕ-comma]]
|688.663
|688.663
|
|
Line 246: 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 262: 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 268: 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 280: 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 295: 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 310: 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 324: 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 336: 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 344: 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 354: 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 362: 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 368: 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 424: 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 430: 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 438: 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 468: 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 flat of [[Pythagorean tuning]] by at least the [[just-noticeable distance]].
|[[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 541: 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 667: 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 709: 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 719: 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 731: 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 779: 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 791: 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 803: 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 813: 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 831: 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 890: 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 971: Line 972:
|
|
|-
|-
|[[Negative Golden ratio-comma meantone|Negative Golden ratio-comma]]  
|[[-1/φ-comma meantone|-1/ϕ-comma]]
|715.247
|715.247
|
|
Line 981: 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 1,223: 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]].
|-
|-
|}
|}
Retrieved from "https://en.xen.wiki/w/User:BudjarnLambeth/Table_of_n-comma_meantone_generators"