6656/6655: Difference between revisions

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| Comma = yes

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'''6656/6655''', the '''jacobin comma''', [https://en.xen.wiki/index.php?title=List_of_superparticular_intervals&diff=next&oldid=18864 apparently named] by [[Gene Ward Smith]] in 2014, 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''', [https://en.xen.wiki/index.php?title=List_of_superparticular_intervals&diff=next&oldid=18864 apparently named] by [[Gene Ward Smith]] in 2014, 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]])([[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. 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]].

Subgroup: 2.3.5.7.11.13

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]]:

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

{| class="right-all"

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

|-

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

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

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

|-

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

| ⟨ || 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

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

{{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 }}

{~~{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~~ }}

==== Septendecimal jacobin ====

Subgroup: 2.3.5.7.11.13.17

Comma list: 6656/6655, 12376/12375

Mapping:

{| 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 ET sequence: {{Optimal ET sequence| 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 }}

== See also ==

* [[13-limit]]

* [[Unnoticeable comma]]

* [[List of superparticular intervals]]

[[Category:Jacobin]]

[[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 4: / Line 4: @@
 | Comma = yes
 }}
-'''6656/6655''', the '''jacobin comma''', [https://en.xen.wiki/index.php?title=List_of_superparticular_intervals&diff=next&oldid=18864 apparently named] by [[Gene Ward Smith]] in 2014, 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''', [https://en.xen.wiki/index.php?title=List_of_superparticular_intervals&diff=next&oldid=18864 apparently named] by [[Gene Ward Smith]] in 2014, 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]])([[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. 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]].
-Subgroup: 2.3.5.7.11.13
+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>
-[{{val| 1 0 0 0 0 -9 }}, <br>
+{| class="right-all"
-{{val| 0 1 0 0 0 0 }}, <br>
+|-
-{{val| 0 0 1 0 0 1 }}, <br>
+| [⟨ || 1 || 0 || 0 || 0 || 0 || -9 || ],
-{{val| 0 0 0 1 0 0 }}, <br>
+|-
-{{val| 0 0 0 0 1 3 }}]
+| ⟨ || 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
-Mapping generators: ~2, ~3, ~5, ~7, ~11
+{{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 }}
-{{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 }}
+==== 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 ET sequence: {{Optimal ET sequence| 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 }}
 == See also ==
-* [[13-limit]]
 * [[Unnoticeable comma]]
 * [[List of superparticular intervals]]
 [[Category:Jacobin]]
+[[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?}}