2197/2187: Difference between revisions

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~~Threedie~~, ~~the interval 13^3~~/~~3^7~~, is a 3.13 analog of the 2.5 [[128/125|diesis]]~~, 2^~~7~~/5^3~~. ~~Tempering by~~ it, ~~instead of chopping~~ the octave into three sharp 5/4s, ~~chops~~ the tritave ~~(3)~~ into three slightly flat 13/9s, since (13/9)^3/3 = 2197/2187. ~~596~~ provides the optimal patent val for the corresponding rank ~~five~~ temperament.

{{Infobox Interval

| Name = threedie

| Color name = s3o32, satritho 2nd, Satritho comma

| Comma = yes

}}

The '''threedie''', '''2197/2187''', is a 3.13-[[subgroup]] analog of the 2.5-subgroup [[128/125|diesis]]. Measuring about 7.9{{cent}}, it is a [[small comma]].

== Temperaments ==

Analogous to [[tempering out]] the diesis, which divides the octave into three sharp [[5/4]]'s, tempering out the threedie divides the [[tritave]] into three slightly flat [[13/9]]'s, since (13/9)3/3 = 2197/2187. Furthermore, tempering out the threedie divides [[9/8]] into three instances of [[27/26]]. [[596edo]] provides the optimal patent val for the corresponding rank-5 temperament.

It factors into simpler commas in the [[2.3.5.13 subgroup]] as ([[325/324]])([[676/675]]), and tempering out both of them leads to the 2.3.5.13 [[kleismic]] temperament (also known as ''cata''). It can also be expressed in the [[2.3.7.13 subgroup]] as ([[169/168]])/([[729/728]]), i.e. {{S|13}}/S27.

[[Category:Commas with unknown etymology]]

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-Threedie, the interval 13^3/3^7, is a 3.13 analog of the 2.5 [[128/125|diesis]], 2^7/5^3. Tempering by it, instead of chopping the octave into three sharp 5/4s, chops the tritave (3) into three slightly flat 13/9s, since (13/9)^3/3 = 2197/2187. 596 provides the optimal patent val for the corresponding rank five temperament.
+{{Infobox Interval
+| Name = threedie
+| Color name = s3o<sup>3</sup>2, satritho 2nd,<br>Satritho comma
+| Comma = yes
+}}
+The '''threedie''', '''2197/2187''', is a 3.13-[[subgroup]] analog of the 2.5-subgroup [[128/125|diesis]]. Measuring about 7.9{{cent}}, it is a [[small comma]].
+== Temperaments ==
+Analogous to [[tempering out]] the diesis, which divides the octave into three sharp [[5/4]]'s, tempering out the threedie divides the [[tritave]] into three slightly flat [[13/9]]'s, since (13/9)<sup>3</sup>/3 = 2197/2187. Furthermore, tempering out the threedie divides [[9/8]] into three instances of [[27/26]]. [[596edo]] provides the optimal patent val for the corresponding rank-5 temperament.
+It factors into simpler commas in the [[2.3.5.13 subgroup]] as ([[325/324]])([[676/675]]), and tempering out both of them leads to the 2.3.5.13 [[kleismic]] temperament (also known as ''cata''). It can also be expressed in the [[2.3.7.13 subgroup]] as ([[169/168]])/([[729/728]]), i.e. {{S|13}}/S27.
+[[Category:Commas with unknown etymology]]