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.~~26012 [[~~cent~~]]s~~. It is the difference between a stack of three [[11/8]] superfourths and one [[13/10]] [[naiadic]] plus an [[octave]].

'''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 ~~supports~~ the jacobin temperament. ~~Simpler~~ edos that support this temperament ~~include~~ {{~~EDOs~~| 37, 72, ~~135~~, ~~159~~, 183, ~~194~~, ~~320~~ }}. ~~On the larger end~~, ~~the list includes edos like~~ {{~~EDOs~~| 3395 ~~and 3125~~ }}. ~~Although it is more rational to use JI-approximating edos like the above mentioned~~ 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. You may find a list of good JI-approximating edos that support this temperament below. Although 1789edo has a unique position due to its number of steps being a hallmark year of the 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]].

The 17-limit factorization shows us a natural path of extension, also given below.

=== Jacobin ===

[[Subgroup]]: 2.3.5.7.11.13

[[Comma list]]: 6656/6655

[[Mapping]]: <br>

{| class="right-all"

|-

| [⟨ || 1 || 0 || 0 || 0 || 0 || -9 || ],

|-

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

|-

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

|-

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

|-

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

|}

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

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1199.9887{{c}}, ~3/2 = 701.9663{{c}}, ~5/4 = 386.3430{{c}}, ~7/4 = 968.8484{{c}}, ~11/8 = 551.3967{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9661{{c}}, ~5/4 = 386.3383{{c}}, ~7/4 = 968.8455{{c}}, ~11/8 = 551.3980{{c}}

{{Optimal ET sequence|legend=1| 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 }}

[[Badness]] (Sintel): 0.416

==== Septendecimal jacobin ====

Subgroup: 2.3.5.7.11.13.17

Comma list: 6656/6655, 12376/12375

Mapping: <br>

{| class="right-all"

|-

| [⟨ || 1 || 0 || 0 || 0 || 0 || -9 || 6 || ],

|-

| ⟨ || 0 || 1 || 0 || 0 || 0 || 0 || 2 || ],

|-

| ⟨ || 0 || 0 || 1 || 0 || 0 || 1 || 2 || ],

|-

| ⟨ || 0 || 0 || 0 || 1 || 0 || 0 || -1 || ],

|-

| ⟨ || 0 || 0 || 0 || 0 || 1 || 3 || -2 || ]]

|}

Optimal tunings:

* WE: ~2 = 1199.9890{{c}}, ~3/2 = 701.9675{{c}}, ~5/4 = 386.3466{{c}}, ~7/4 = 968.8454{{c}}, ~11/8 = 551.3945{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.9670{{c}}, ~5/4 = 386.3410{{c}}, ~7/4 = 968.8434{{c}}, ~11/8 = 551.3964{{c}}

{{Optimal ET sequence|legend=0| 15g, 22, 37g, 39dfg, 41g, 50, 63g, 72, 111, 152f, 159, 183, 239f, 248, 270, 311, 422, 494, 581, 742, 764, 814, 1075, 1236, 1395, 1506, 2000, 2581, 2814, 2901, 3323, 3395, 8296e, 11691e, 16322ee, 17086cdeeg, 21223cdeefg }}

Badness (Sintel): 0.239

== 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>.

== See also ==

* [[13-limit]]

* [[Unnoticeable comma]]

* [[List of superparticular intervals]]

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

== References ==

[[Category:~~Ratio~~]]

~~[[Category:Unnoticeable~~ comma]]

[[Category:Jacobin]]

~~[[Category:Superparticular]]~~

[[Category:Commas with unknown etymology]]

{{Todo|add etymology|comment=I know it's something to do with the French Revolution, but what, exactly? Spell it all out please, why exactly does the name "jacobin" uniquely fit this particular comma?}}

@@ 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 0.26012 [[cent]]s. It is the difference between a stack of three [[11/8]] superfourths and one [[13/10]] [[naiadic]] plus an [[octave]].
+'''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 supports the jacobin temperament. Simpler edos that support this temperament include {{EDOs| 37, 72, 135, 159, 183, 194, 320 }}. On the larger end, the list includes edos like {{EDOs| 3395 and 3125 }}. Although it is more rational to use JI-approximating edos like the above mentioned 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. You may find a list of good JI-approximating edos that support this temperament below. Although 1789edo has a unique position due to its number of steps being a hallmark year of the 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]].
+The 17-limit factorization shows us a natural path of extension, also given below.
+=== Jacobin ===
+[[Subgroup]]: 2.3.5.7.11.13
+[[Comma list]]: 6656/6655
+[[Mapping]]: <br>
+{| class="right-all"
+|-
+| [⟨ || 1 || 0 || 0 || 0 || 0 || -9 || ],
+|-
+| ⟨ || 0 || 1 || 0 || 0 || 0 || 0 || ],
+|-
+| ⟨ || 0 || 0 || 1 || 0 || 0 || 1 || ],
+|-
+| ⟨ || 0 || 0 || 0 || 1 || 0 || 0 || ],
+|-
+| ⟨ || 0 || 0 || 0 || 0 || 1 || 3 || ]]
+|}
+: mapping generators: ~2, ~3, ~5, ~7, ~11
+[[Optimal tuning]]s:
+* [[WE]]: ~2 = 1199.9887{{c}}, ~3/2 = 701.9663{{c}}, ~5/4 = 386.3430{{c}}, ~7/4 = 968.8484{{c}}, ~11/8 = 551.3967{{c}}
+* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9661{{c}}, ~5/4 = 386.3383{{c}}, ~7/4 = 968.8455{{c}}, ~11/8 = 551.3980{{c}}
+{{Optimal ET sequence|legend=1| 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 }}
+[[Badness]] (Sintel): 0.416
+==== Septendecimal jacobin ====
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 6656/6655, 12376/12375
+Mapping: <br>
+{| class="right-all"
+|-
+| [⟨ || 1 || 0 || 0 || 0 || 0 || -9 || 6 || ],
+|-
+| ⟨ || 0 || 1 || 0 || 0 || 0 || 0 || 2 || ],
+|-
+| ⟨ || 0 || 0 || 1 || 0 || 0 || 1 || 2 || ],
+|-
+| ⟨ || 0 || 0 || 0 || 1 || 0 || 0 || -1 || ],
+|-
+| ⟨ || 0 || 0 || 0 || 0 || 1 || 3 || -2 || ]]
+|}
+Optimal tunings:
+* WE: ~2 = 1199.9890{{c}}, ~3/2 = 701.9675{{c}}, ~5/4 = 386.3466{{c}}, ~7/4 = 968.8454{{c}}, ~11/8 = 551.3945{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.9670{{c}}, ~5/4 = 386.3410{{c}}, ~7/4 = 968.8434{{c}}, ~11/8 = 551.3964{{c}}
+{{Optimal ET sequence|legend=0| 15g, 22, 37g, 39dfg, 41g, 50, 63g, 72, 111, 152f, 159, 183, 239f, 248, 270, 311, 422, 494, 581, 742, 764, 814, 1075, 1236, 1395, 1506, 2000, 2581, 2814, 2901, 3323, 3395, 8296e, 11691e, 16322ee, 17086cdeeg, 21223cdeefg }}
+Badness (Sintel): 0.239
+== 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>.
 == See also ==
-* [[13-limit]]
-* [[Unnoticeable comma]]
 * [[List of superparticular intervals]]
-[[Category:13-limit]]
+== References ==
-[[Category:Ratio]]
-[[Category:Unnoticeable comma]]
+[[Category:Jacobin]]
-[[Category:Superparticular]]
+[[Category:Commas with unknown etymology]]
+{{Todo|add etymology|comment=I know it's something to do with the French Revolution, but what, exactly? Spell it all out please, why exactly does the name "jacobin" uniquely fit this particular comma?}}