595/594: Difference between revisions

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| Comma = yes
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'''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]]
Line 37: Line 36:


== 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.  


=== Dakotismic ===
=== Dakotismic ===
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| ⟨ || 0 || 0 || 0 || 0 || 0 || 1 || 0 || ]]
| ⟨ || 0 || 0 || 0 || 0 || 0 || 1 || 0 || ]]
|}
|}
: mapping generators: ~2, ~3, ~5, ~7, ~11, ~13  
: 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]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 702.2950{{c}}, ~5/4 = 386.0705{{c}}, ~7/4 = 968.4704{{c}}, ~11/8 = 551.8578{{c}}, ~13/8 = 840.5277{{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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[[Subgroup]]: 2.3.5.7.11.17
[[Subgroup]]: 2.3.5.7.11.17


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


: sval mapping generators: ~2, ~3, ~5, ~7, ~11
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~3/2 = 702.2950{{c}}, ~5/4 = 386.0705{{c}}, ~7/4 = 968.4704{{c}}, ~11/8 = 551.8578{{c}}
 
[[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 }}
{{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 }}

Revision as of 10:29, 22 March 2026

Interval information
Ratio 595/594
Factorization 2-1 × 3-3 × 5 × 7 × 11-1 × 17
Monzo [-1 -3 1 1 -1 0 1⟩
Size in cents 2.912085¢
Name dakotisma
Color name 17o1uzy2, soluzoyo 2nd
FJS name [math]\displaystyle{ \text{m2}^{5,7,17}_{11} }[/math]
Special properties superparticular,
reduced
Tenney norm (log2 nd) 18.4311
Weil norm (log2 max(n, d)) 18.4335
Wilson norm (sopfr(nd)) 51
Comma size unnoticeable
S-expression S34⋅S35
Open this interval in xen-calc

595/594, the dakotisma, is a small 17-limit comma measuring about 2.91 cents. It is one of the simplest commas tempered out in 311et, a highly notable general-purpose equal temperament.

Commatic relations

This comma identifies itself as the difference between the following superparticular pairs:

not to mention some nonsuperparticular but useful relations:

It factors into the following superparticular pairs:

not to mention some nonsuperparticular but useful relations:

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.

Dakotismic

Subgroup: 2.3.5.7.11.13.17

Mapping:

[⟨ 1 0 0 0 0 0 1 ],
⟨ 0 1 0 0 0 0 3 ],
⟨ 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

Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~3/2 = 702.2950 ¢, ~5/4 = 386.0705 ¢, ~7/4 = 968.4704 ¢, ~11/8 = 551.8578 ¢, ~13/8 = 840.5277 ¢

Optimal ET sequence: 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 patent val: 861

Dakotic

Subgroup: 2.3.5.7.11.17

Subgroup val mapping:

[⟨ 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 (CTE): ~2 = 1200.0000 ¢, ~3/2 = 702.2950 ¢, ~5/4 = 386.0705 ¢, ~7/4 = 968.4704 ¢, ~11/8 = 551.8578 ¢

Optimal ET sequence: 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.

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