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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 ===
Line 9: Line 9:
* 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 optimized 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:  
* {{EDOs|7, 12, 17 and 5}} edos (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 ===
=== Preservation of meantone behavior ===
Temperaments that fall outside of the "[[Historical temperaments|historically considered 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 "[[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.
Temperaments that fall outside of the "diamond monotone" range preserve even fewer meantone properties, but they are also included for completeness.


== Flatter than flattest allowed historical meantone ==
== The table ==
 
=== Flatter than flattest historically-defined meantone ===
{| class="wikitable mw-collapsible"
{| class="wikitable mw-collapsible"
|+Spectrum of meantone tunings 1/1-comma to 1/2-comma
|+Spectrum of meantone tunings 1/1-comma to 1/2-comma
Line 63: 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 73: 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 85: 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 91: 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 99: 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 105: 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 115: 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 205: 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 225: 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 231: 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 237: Line 239:
|
|
|-
|-
|[[Golden ratio-comma meantone|Golden ratio-comma]]  
|[[1/φ-comma meantone|1/ϕ-comma]]
|688.663
|688.663
|
|
Line 245: 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 261: 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 267: 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 279: 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 298: Line 300:
|}
|}


== Historically considered meantone ==
=== Historically-defined meantone ===
{| class="wikitable mw-collapsible"
{| class="wikitable mw-collapsible"
|+Spectrum of meantone tunings 10/21-comma to 1/22-comma
|+Spectrum of meantone tunings 10/21-comma to 1/22-comma
Line 309: 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 323: 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 335: 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 343: 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 353: 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 361: 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 367: 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 423: 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 429: 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 437: 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 467: 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 540: Line 542:
|}
|}


== Negative-harmonic-defined meantone (most often approached as [[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 666: 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 708: 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 718: 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 730: 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 778: 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 790: 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 802: 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 812: 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 830: 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 889: 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 970: Line 972:
|
|
|-
|-
|[[Negative Golden ratio-comma meantone|Negative Golden ratio-comma]]  
|[[-1/φ-comma meantone|-1/ϕ-comma]]
|715.247
|715.247
|
|
Line 980: 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 1,222: 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]].
|-
|-
|}
|}
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