6656/6655: Difference between revisions

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{{Infobox Interval

~~| Ratio = 6656/6655~~

~~| Monzo = 9 0 -1 0 -3 1~~

~~| Cents = 0.26012~~

| Name = jacobin comma

| Color name = ~~thotrilu~~-agu comma

| Color name = Thotrilu-agu comma

| Comma = yes

}}

'''6656/6655''', the '''jacobin comma''', is a [[13-limit]] (also 2.5.11.13 [[subgroup]]) [[superparticular]] ~~interval~~ of about 0.26{{cent}}. It is the difference between a stack of three [[11/8]] superfourths and one [[13/10]] [[naiadic]] plus an [[octave]]. In terms of commas, it is the difference between [[364/363]] and [[385/384]], between [[2080/2079]] and [[3025/3024]] as well as between [[4096/4095]] and [[10648/10647]]. In the 17-limit, it factors neatly into 12376/12375 × [[~~sparkisma|~~14400/14399]].

'''6656/6655''', the '''jacobin comma''', is an [[unnoticeable comma|unnoticeable]] [[13-limit]] (also 2.5.11.13 [[subgroup]]) [[superparticular]] [[comma]] of about 0.26{{cent}}. It is the difference between a stack of three [[11/8]] superfourths and one [[13/10]] [[naiadic]] plus an [[octave]] ([[13/5]]).

In terms of commas, it is the difference between [[364/363]] and [[385/384]], between [[2080/2079]] and [[3025/3024]] as well as between [[4096/4095]] and [[10648/10647]]. In the [[17-limit]], it factors neatly into ([[12376/12375]])⋅([[14400/14399]]).

== Temperaments ==

By tempering it out, the '''jacobin temperament''' is defined. ~~Perhaps most remarkably~~, ~~'''~~[[1789edo]]~~'''~~ is an edo that [[support]]s the jacobin temperament. ~~You may find a list~~ of ~~good JI-approximating edos that support this temperament below~~. Although ~~it is more rational to use such edos for this temperament,~~ 1789edo has a unique position due to its number of steps being a hallmark year of the French Revolution.

By tempering it out, the '''jacobin temperament''' is defined. Interestingly, [[1789edo]] is an edo that [[support]]s the jacobin temperament - and is consistent to distance 2 on the subgroup 2.5.11.13, the subgroup of the comma. Although 1789edo has a unique position due to its number of steps being a hallmark year of the {{W|French Revolution}}, it is more rational to use the other edos for this temperament. Miscellaneous temperaments tempering out this comma are collected in [[The Jacobins]].

~~Subgroup:~~ 2.~~3.5.7~~.11.13

The 17-limit factorization shows us a natural path of extension by adding 12376/12375. Equally, it can be conceptualized as a rank-2 temperament in 2.13/10.11 subgroup, in which case it is called ''jacobin-naiadic'', after [[13/10]] being called a naiadic by interval class.

[[~~Mapping~~]]~~: ~~

See [[The Jacobins#Jacobin]] for technical data.

~~[{{val| 1 0 0 0 0 -9 }}, ~~

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

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

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

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

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

== Etymology ==

This comma was apparently named by [[Gene Ward Smith]] in 2014 for reasons unknown<ref>[https://en.xen.wiki/index.php?title=List_of_superparticular_intervals&diff=next&oldid=18864 Xenharmonic Wiki | ''List of superparticular intervals (Revision as of 23:46, 13 June 2014 by Wikispaces>genewardsmith)'']</ref>. An hypothesis is that "jacobin" refers to the {{w|Jacobins}}, a political group founded in 1789 during the French Revolution, since this comma is tempered out by [[1789edo]], a tuning which, furthermore, is [[consistent]] to distance 2 in the 2.5.11.13-subgroup-limited [[13-odd-limit]], reinforcing the importance of this comma in this tuning.

{{~~Val list~~|~~legend=1| 9~~, 15, 22, 26, ~~31f~~, 37, ~~39df, 41, 46, 63, 72, 87, 111, 152f, 183, 198, 224, 270, 494, 764, 1012, 1084, 1236, 1506, 2814, 2901, 3125, 3395, 8026e, 8296e, 11421e, 11691e, 12927e, 13421e, 16322ee, 16816ee }}~~

== See also ==

* [[13-limit]]

* [[Unnoticeable comma]]

* [[List of superparticular intervals]]

~~[[Category:13-limit]]~~

== References ==

~~[[Category:Unnoticeable commas]]~~

~~[[Category:Superparticular]]~~

[[Category:Jacobin]]

[[Category:Commas with unknown etymology]]

@@ Line 1: / Line 1: @@
 {{Infobox Interval
-| Ratio = 6656/6655
-| Monzo = 9 0 -1 0 -3 1
-| Cents = 0.26012
 | Name = jacobin comma
-| Color name = thotrilu-agu comma
+| Color name = Thotrilu-agu comma
+| Comma = yes
 }}
-'''6656/6655''', the '''jacobin comma''', is a [[13-limit]] (also 2.5.11.13 [[subgroup]]) [[superparticular]] interval of about 0.26{{cent}}. It is the difference between a stack of three [[11/8]] superfourths and one [[13/10]] [[naiadic]] plus an [[octave]]. In terms of commas, it is the difference between [[364/363]] and [[385/384]], between [[2080/2079]] and [[3025/3024]] as well as between [[4096/4095]] and [[10648/10647]]. In the 17-limit, it factors neatly into 12376/12375 × [[sparkisma|14400/14399]].
+'''6656/6655''', the '''jacobin comma''', is an [[unnoticeable comma|unnoticeable]] [[13-limit]] (also 2.5.11.13 [[subgroup]]) [[superparticular]] [[comma]] of about 0.26{{cent}}. It is the difference between a stack of three [[11/8]] superfourths and one [[13/10]] [[naiadic]] plus an [[octave]] ([[13/5]]).
+In terms of commas, it is the difference between [[364/363]] and [[385/384]], between [[2080/2079]] and [[3025/3024]] as well as between [[4096/4095]] and [[10648/10647]]. In the [[17-limit]], it factors neatly into ([[12376/12375]])⋅([[14400/14399]]).
 == Temperaments ==
-By tempering it out, the '''jacobin temperament''' is defined. Perhaps most remarkably, '''[[1789edo]]''' is an edo that [[support]]s the jacobin temperament. You may find a list of good JI-approximating edos that support this temperament below. Although it is more rational to use such edos for this temperament, 1789edo has a unique position due to its number of steps being a hallmark year of the French Revolution.
+By tempering it out, the '''jacobin temperament''' is defined. Interestingly, [[1789edo]] is an edo that [[support]]s the jacobin temperament - and is consistent to distance 2 on the subgroup 2.5.11.13, the subgroup of the comma. Although 1789edo has a unique position due to its number of steps being a hallmark year of the {{W|French Revolution}}, it is more rational to use the other edos for this temperament. Miscellaneous temperaments tempering out this comma are collected in [[The Jacobins]].
-Subgroup: 2.3.5.7.11.13
+The 17-limit factorization shows us a natural path of extension by adding 12376/12375. Equally, it can be conceptualized as a rank-2 temperament in 2.13/10.11 subgroup, in which case it is called ''jacobin-naiadic'', after [[13/10]] being called a naiadic by interval class.
-[[Mapping]]: <br>
+See [[The Jacobins#Jacobin]] for technical data.
-[{{val| 1 0 0 0 0 -9 }}, <br>
-{{val| 0 1 0 0 0 0 }}, <br>
-{{val| 0 0 1 0 0 1 }}, <br>
-{{val| 0 0 0 1 0 0 }}, <br>
-{{val| 0 0 0 0 1 3 }}]
-Mapping generators: ~2, ~3, ~5, ~7, ~11
+== Etymology ==
+This comma was apparently named by [[Gene Ward Smith]] in 2014 for reasons unknown<ref>[https://en.xen.wiki/index.php?title=List_of_superparticular_intervals&diff=next&oldid=18864 Xenharmonic Wiki | ''List of superparticular intervals (Revision as of 23:46, 13 June 2014 by Wikispaces>genewardsmith)'']</ref>. An hypothesis is that "jacobin" refers to the {{w|Jacobins}}, a political group founded in 1789 during the French Revolution, since this comma is tempered out by [[1789edo]], a tuning which, furthermore, is [[consistent]] to distance 2 in the 2.5.11.13-subgroup-limited [[13-odd-limit]], reinforcing the importance of this comma in this tuning.
-{{Val list|legend=1| 9, 15, 22, 26, 31f, 37, 39df, 41, 46, 63, 72, 87, 111, 152f, 183, 198, 224, 270, 494, 764, 1012, 1084, 1236, 1506, 2814, 2901, 3125, 3395, 8026e, 8296e, 11421e, 11691e, 12927e, 13421e, 16322ee, 16816ee }}
 == See also ==
-* [[13-limit]]
-* [[Unnoticeable comma]]
 * [[List of superparticular intervals]]
-[[Category:13-limit]]
+== References ==
-[[Category:Unnoticeable commas]]
-[[Category:Superparticular]]
 [[Category:Jacobin]]
+[[Category:Commas with unknown etymology]]