595/594: Difference between revisions

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{{Infobox Interval

| Name = dakotisma

| Color name = 17o1uzy2, soluzoyo 2nd

| 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.

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:

* ([[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:

not to mention some nonsuperparticular but useful relations:

* ([[1575/1573]])([[2431/2430]])

* [[1575/1573]] and [[2431/2430]]

* ([[1377/1375]])([[4375/4374]])

* [[1377/1375]] and [[4375/4374]]

* ([[1275/1274]])([[4459/4455]])

* [[1275/1274]] and [[4459/4455]]

== 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

[[Mapping]]:

[[Mapping]]:

[~~{{val~~| 1 0 0 0 0 0 1 }},

{| class="right-all"

~~ {{val~~| 0 1 0 0 0 0 3 }},

|-

~~ {{val~~| 0 0 1 0 0 0 -1 }},

| [⟨ || 1 || 0 || 0 || 0 || 0 || 0 || 1 || ],

~~ {{val~~| 0 0 0 1 0 0 -1 }},

|-

~~ {{val~~| 0 0 0 0 1 0 1 }},

| ⟨ || 0 || 1 || 0 || 0 || 0 || 0 || 3 || ],

~~ {{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

[[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

{{~~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 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]]

=== Dakotic ===

[[Subgroup]]: 2.3.5.7.11.17

[[Sval]] [[mapping]]:

{| 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:Commas named after composers]]

[[Category:Commas named after music theorists]]

@@ Line 1: / Line 1: @@
 {{Infobox Interval
 | Name = dakotisma
+| Color name = 17o1uzy2, soluzoyo 2nd
 | 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.
-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:
-* ([[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:
+not to mention some nonsuperparticular but useful relations:
-* ([[1575/1573]])([[2431/2430]])
+* [[1575/1573]] and [[2431/2430]]
-* ([[1377/1375]])([[4375/4374]])
+* [[1377/1375]] and [[4375/4374]]
-* ([[1275/1274]])([[4459/4455]])
+* [[1275/1274]] and [[4459/4455]]
 == 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
-[[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
+[[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
-{{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 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]]
+=== 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:Commas named after composers]]
+[[Category:Commas named after music theorists]]