595/594: Difference between revisions
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'''595/594''', the '''dakotisma''', is a [[small comma|small]] [[17-limit]] [[comma]] measuring about 2.91 [[cent]]s. | '''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 | 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 = | [[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 | ||
[[ | [[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 | |||
[[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 = | |||
{{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 }} | ||