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{{Infobox Interval
{{Infobox Interval
| Name = dakotisma
| Name = dakotisma
| Color name = 17o1uzy2, soluzoyo 2nd
| Comma = yes
| Comma = yes
}}
}}
'''595/594''', the '''dakotisma''', is a [[small comma|small]] [[17-limit]] [[comma]] measuring about 2.91 [[cent]]s. Named after [[Scott Dakota]], it is one of the simplest commas tempered out in [[311edo|311et]], a highly notable general-purpose equal temperament.  
'''595/594''', the '''dakotisma''', is a [[small comma|small]] [[17-limit]] [[comma]] measuring about 2.91 [[cent]]s. Named after [[Scott Dakota]], it is one of the simplest commas tempered out in [[311edo|311et]], a highly notable general-purpose equal temperament.  


It identifies itself as the difference between the following superparticular pairs:  
== Commatic relations ==
This comma identifies itself as the difference between the following superparticular pairs:  


* ([[34/33]])/([[36/35]])
* [[34/33]] and [[36/35]]
* ([[85/84]])/([[99/98]])
* [[85/84]] and [[99/98]]
* ([[100/99]])/([[120/119]])
* [[100/99]] and [[120/119]]
* ([[136/135]])/([[176/175]])
* [[136/135]] and [[176/175]]
* ([[221/220]])/([[351/350]])
* [[221/220]] and [[351/350]]
* ([[289/288]])/([[561/560]])
* [[289/288]] and [[561/560]]
* ([[325/324]])/([[715/714]])
* [[325/324]] and [[715/714]]
* ([[364/363]])/([[936/935]])
* [[364/363]] and [[936/935]]
* ([[385/384]])/([[1089/1088]])
* [[385/384]] and [[1089/1088]]
* ([[441/440]])/([[1701/1700]])
* [[441/440]] and [[1701/1700]]
* ([[442/441]])/([[1716/1715]])
* [[442/441]] and [[1716/1715]]
* ([[540/539]])/([[5832/5831]])
* [[540/539]] and [[5832/5831]]
* (561/560)/([[9801/9800]])
* 561/560 and [[9801/9800]]
 
not to mention some nonsuperparticular but useful relations:
* [[154/153]] and [[245/243]]


It factors into the following superparticular pairs:  
It factors into the following superparticular pairs:  
* ([[1156/1155]])([[1225/1224]])
* [[1156/1155]] and [[1225/1224]]
* ([[833/832]])([[2080/2079]])
* [[833/832]] and [[2080/2079]]
* ([[625/624]])([[12376/12375]])
* [[625/624]] and [[12376/12375]]
 
not to mention some nonsuperparticular but useful relations:
* [[1575/1573]] and [[2431/2430]]
* [[1377/1375]] and [[4375/4374]]
* [[1275/1274]] and [[4459/4455]]


== Temperaments ==
== Temperaments ==
Tempering out this comma in the full 17-limit gives the rank-6 '''dakotismic temperament''', and enables the related essentially tempered chords. You may find a list of good equal temperaments that support this temperament below.  
Tempering out this comma in the full 17-limit gives the rank-6 '''dakotismic temperament''', or in the 2.3.5.7.11.17 subgroup, the rank-5 '''dakotic temperament''', enabling [[dakotismic chords]]. You may find a list of good equal temperaments that support these temperaments below.  


=== Dakotismic ===
[[Subgroup]]: 2.3.5.7.11.13.17
[[Subgroup]]: 2.3.5.7.11.13.17


[[Mapping]]:  
[[Mapping]]: <br>
<br>[{{val| 1 0 0 0 0 0 1 }},  
{| class="right-all"
<br>{{val| 0 1 0 0 0 0 3 }},  
|-
<br>{{val| 0 0 1 0 0 0 -1 }},  
| [⟨ || 1 || 0 || 0 || 0 || 0 || 0 || 1 || ],
<br>{{val| 0 0 0 1 0 0 -1 }},  
|-
<br>{{val| 0 0 0 0 1 0 1 }},  
| ⟨ || 0 || 1 || 0 || 0 || 0 || 0 || 3 || ],
<br>{{val| 0 0 0 0 0 1 0 }}]
|-
| ⟨ || 0 || 0 || 1 || 0 || 0 || 0 || -1 || ],
|-
| ⟨ || 0 || 0 || 0 || 1 || 0 || 0 || -1 || ],
|-
| ⟨ || 0 || 0 || 0 || 0 || 1 || 0 || 1 || ],
|-
| ⟨ || 0 || 0 || 0 || 0 || 0 || 1 || 0 || ]]
|}


Mapping generators: ~2, ~3, ~5, ~7, ~11, ~13  
: mapping generators: ~2, ~3, ~5, ~7, ~11, ~13  


{{Val list|legend=1| 12f, 14cf, 15g, 19eg, 22, 26, 27eg, 34d, 38df, 41, 46, 58, 72, 121, 140, 171, 190g, 212g, 217, 289, 311, 668, 694g, 740g, 1051dg }}*
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 702.2950, ~5/4 = 386.0705, ~7/4 = 968.4704, ~11/8 = 551.8578, ~13/8 = 840.5277
 
{{Optimal ET sequence|legend=1| 12f, 14cf, 15g, 19eg, 22, 26, 27eg, 34d, 38df, 41, 46, 58, 72, 121, 140, 171, 190g, 212g, 217, 289, 311, 668, 694g, 740g, 1051dg }}*


<nowiki>*</nowiki> [[optimal patent val]]: [[861edo|861]]
<nowiki>*</nowiki> [[optimal patent val]]: [[861edo|861]]
=== Dakotic ===
[[Subgroup]]: 2.3.5.7.11.17
[[Sval]] [[mapping]]: <br>
{| class="right-all"
|-
| [⟨ || 1 || 0 || 0 || 0 || 0 || 1 || ],
|-
| ⟨ || 0 || 1 || 0 || 0 || 0 || 3 || ],
|-
| ⟨ || 0 || 0 || 1 || 0 || 0 || -1 || ],
|-
| ⟨ || 0 || 0 || 0 || 1 || 0 || -1 || ],
|-
| ⟨ || 0 || 0 || 0 || 0 || 1 || 1 || ]]
|}
: sval mapping generators: ~2, ~3, ~5, ~7, ~11
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~3/2 = 702.2950, ~5/4 = 386.0705, ~7/4 = 968.4704, ~11/8 = 551.8578
{{Optimal ET sequence|legend=1| 12, 14c, 19eg, 22, 27eg, 31g, 34d, 38d, 41, 46, 58, 68, 72, 118, 171, 190g, 193, 212g, 217, 239, 311, 1051dg }}
== Etymology ==
The dakotisma was named by [[Praveen Venkataramana]] in 2022 in honor of [[Scott Dakota]]'s pioneering study of this comma and of its role in 311et.


[[Category:Dakotismic]]
[[Category:Dakotismic]]
[[Category:Commas named after composers]]
[[Category:Commas named after music theorists]]
Retrieved from "https://en.xen.wiki/w/595/594"