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 = 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]]. 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]]), or can be expressed in the 2.3.7.13 subgroup as ([[169/168]])/([[729/728]]), i.e. S13/S27.
+[[Category:Commas with unknown etymology]]