1716/1715: Difference between revisions

(One intermediate revision by the same user not shown)

Line 4:

| Comma = yes

}}

The '''lummic comma''', '''1716/1715''', is a [[13-limit]] [[superparticular]] [[comma]] measuring about 1.0{{cent}}. It often arises as the difference between simpler commas: ([[176/175]])/([[196/195]]), ([[351/350]])/([[441/440]]), ([[1573/1568]])/([[385/384]]), ([[540/539]])/([[1575/1573]]), and ([[1001/1000]])/([[2401/2400]]). It factors into ([[2080/2079]])([[9801/9800]]).

The '''lummic comma''', '''1716/1715''', is an [[unnoticeable comma|unnoticeable]] [[13-limit]] [[superparticular]] [[comma]] measuring about 1.0{{cent}}. It often arises as the difference between simpler commas: ([[176/175]])/([[196/195]]), ([[351/350]])/([[441/440]]), ([[1573/1568]])/([[385/384]]), ([[540/539]])/([[1575/1573]]), and ([[1001/1000]])/([[2401/2400]]). It factors into ([[2080/2079]])⋅([[9801/9800]]).

Tempering it out leads to the rank-5 '''lummic ~~temperament~~'''. It is tempered out by several notable edos including [[72edo]], [[270edo]] and [[400edo]] among others.

== Temperaments ==

[[Tempering out]] this comma leads to the rank-5 '''lummic''' temperament. It is tempered out by several notable edos including [[72edo]], [[270edo]] and [[400edo]] among others.

[[Subgroup]]: 2.3.5.7.11.13

[[Mapping]]: <br>

{| class="right-all"

|-

| [⟨ || 1 || 0 || 0 || 0 || 0 || -2 || ],

|-

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

|-

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

|-

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

|-

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

|}

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

[[Optimal tuning]]s:

: [[WE]]: ~2 = 1199.9814{{c}}, ~3/2 = 701.9502{{c}}, ~5/4 = 386.4011{{c}}, ~7/4 = 969.0830{{c}}, ~11/8 = 551.2624{{c}}

: [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9462{{c}}, ~5/4 = 386.3868{{c}}, ~7/4 = 969.0766{{c}}, ~11/8 = 551.2373{{c}}

{{Optimal ET sequence|legend=1| 22f, 26, 27e, 31, 45ef, 46, 58, 72, 103, 121, 130, 193, 198, 224, 270, 494, 764, 1137, 1258, 1289, 1419, 1559, 1689, 2183d }}

[[Badness]] (Sintel): 0.318

== Etymology ==

This comma was named after [[Carl Lumma]]. Perhaps interestingly, a ''lummisma'' was first proposed by [[Gene Ward Smith]] in 2011 as a name for 1001/1000, but Carl Lumma pointed out that 1716/1715 was "better"<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_97947.html Yahoo! Tuning Group | ''Short commas'']</ref>, which might ultimately led to this comma being named so.

== References ==

[[Category:Lummic]]

[[Category:Commas named after composers]]

[[Category:Commas named after music theorists]]

@@ Line 4: / Line 4: @@
 | Comma = yes
 }}
-The '''lummic comma''', '''1716/1715''', is a [[13-limit]] [[superparticular]] [[comma]] measuring about 1.0{{cent}}. It often arises as the difference between simpler commas: ([[176/175]])/([[196/195]]), ([[351/350]])/([[441/440]]), ([[1573/1568]])/([[385/384]]), ([[540/539]])/([[1575/1573]]), and ([[1001/1000]])/([[2401/2400]]). It factors into ([[2080/2079]])([[9801/9800]]).
+The '''lummic comma''', '''1716/1715''', is an [[unnoticeable comma|unnoticeable]] [[13-limit]] [[superparticular]] [[comma]] measuring about 1.0{{cent}}. It often arises as the difference between simpler commas: ([[176/175]])/([[196/195]]), ([[351/350]])/([[441/440]]), ([[1573/1568]])/([[385/384]]), ([[540/539]])/([[1575/1573]]), and ([[1001/1000]])/([[2401/2400]]). It factors into ([[2080/2079]])⋅([[9801/9800]]).
-Tempering it out leads to the rank-5 '''lummic temperament'''. It is tempered out by several notable edos including [[72edo]], [[270edo]] and [[400edo]] among others.
+== Temperaments ==
+[[Tempering out]] this comma leads to the rank-5 '''lummic''' temperament. It is tempered out by several notable edos including [[72edo]], [[270edo]] and [[400edo]] among others.
+[[Subgroup]]: 2.3.5.7.11.13
+[[Mapping]]: <br>
+{| class="right-all"
+|-
+| [⟨ || 1 || 0 || 0 || 0 || 0 || -2 || ],
+|-
+| ⟨ || 0 || 1 || 0 || 0 || 0 || -1 || ],
+|-
+| ⟨ || 0 || 0 || 1 || 0 || 0 || 1 || ],
+|-
+| ⟨ || 0 || 0 || 0 || 1 || 0 || 3 || ],
+|-
+| ⟨ || 0 || 0 || 0 || 0 || 1 || -1 || ]]
+|}
+: mapping generators: ~2, ~3, ~5, ~7, ~11
+[[Optimal tuning]]s:
+: [[WE]]: ~2 = 1199.9814{{c}}, ~3/2 = 701.9502{{c}}, ~5/4 = 386.4011{{c}}, ~7/4 = 969.0830{{c}}, ~11/8 = 551.2624{{c}}
+: [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9462{{c}}, ~5/4 = 386.3868{{c}}, ~7/4 = 969.0766{{c}}, ~11/8 = 551.2373{{c}}
+{{Optimal ET sequence|legend=1| 22f, 26, 27e, 31, 45ef, 46, 58, 72, 103, 121, 130, 193, 198, 224, 270, 494, 764, 1137, 1258, 1289, 1419, 1559, 1689, 2183d }}
+[[Badness]] (Sintel): 0.318
+== Etymology ==
+This comma was named after [[Carl Lumma]]. Perhaps interestingly, a ''lummisma'' was first proposed by [[Gene Ward Smith]] in 2011 as a name for 1001/1000, but Carl Lumma pointed out that 1716/1715 was "better"<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_97947.html Yahoo! Tuning Group | ''Short commas'']</ref>, which might ultimately led to this comma being named so.
+== References ==
 [[Category:Lummic]]
 [[Category:Commas named after composers]]
 [[Category:Commas named after music theorists]]