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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. It is one of the simplest commas [[tempering out|tempered out]] in [[311edo|311et]], a highly notable general-purpose equal temperament.  


== Commatic relations ==
== Commatic relations ==
This comma identifies itself as the difference between the following superparticular pairs:  
This comma identifies itself as the difference between the following superparticular pairs:  
* [[34/33]] and [[36/35]]
* [[34/33]] and [[36/35]]
* [[85/84]] and [[99/98]]
* [[85/84]] and [[99/98]]
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== Temperaments ==
== Temperaments ==
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.  
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.  
 
=== Dakotic ===
[[Subgroup]]: 2.3.5.7.11.17
 
[[Subgroup val|Subgroup-val]] [[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 || ]]
|}
: mapping generators: ~2, ~3, ~5, ~7, ~11
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0443{{c}}, ~3/2 = 702.2453{{c}}, ~5/4 = 385.9853{{c}}, ~7/4 = 968.3869{{c}}, ~11/8 = 551.7162{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2568{{c}}, ~5/4 = 386.0203{{c}}, ~7/4 = 968.4102{{c}}, ~11/8 = 551.7760{{c}}
 
{{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 }}
 
[[Badness]] (Sintel): 0.236


=== Dakotismic ===
=== Dakotismic ===
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[[Mapping]]: <br>
[[Mapping]]: <br>
[{{val| 1 0 0 0 0 0 1 }}<br>
{| class="right-all"
{{val| 0 1 0 0 0 0 3 }}<br>
|-
{{val| 0 0 1 0 0 0 -1 }}<br>
| [⟨ || 1 || 0 || 0 || 0 || 0 || 0 || 1 || ],
{{val| 0 0 0 1 0 0 -1 }}<br>
|-
{{val| 0 0 0 0 1 0 1 }}<br>
| ⟨ || 0 || 1 || 0 || 0 || 0 || 0 || 3 || ],
{{val| 0 0 0 0 0 1 0 }}]
|-
 
| ⟨ || 0 || 0 || 1 || 0 || 0 || 0 || -1 || ],
Mapping generators: ~2, ~3, ~5, ~7, ~11, ~13  
|-
| ⟨ || 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  


[[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 tuning]]s:
* [[WE]]: ~2 = 1200.0443{{c}}, ~3/2 = 702.2453{{c}}, ~5/4 = 385.9853{{c}}, ~7/4 = 968.3869{{c}}, ~11/8 = 551.7162{{c}}, ~13/8 = 840.3943{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.2568{{c}}, ~5/4 = 386.0203{{c}}, ~7/4 = 968.4102{{c}}, ~11/8 = 551.7760{{c}}, ~13/8 = 840.4446{{c}}


{{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 }}*
{{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 }}*
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<nowiki>*</nowiki> [[optimal patent val]]: [[861edo|861]]
<nowiki>*</nowiki> [[optimal patent val]]: [[861edo|861]]


=== Dakotic ===
[[Badness]] (Sintel): 0.478
[[Subgroup]]: 2.3.5.7.11.17
 
[[Sval]] [[mapping]]: <br>
[{{val| 1 0 0 0 0 1 }}<br>
{{val| 0 1 0 0 0 3 }}<br>
{{val| 0 0 1 0 0 -1 }}<br>
{{val| 0 0 0 1 0 -1 }}<br>
{{val| 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 ==
== 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.  
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.  
== See also ==
* [[List of superparticular intervals]]


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