2197/2187: Difference between revisions
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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. | 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 as ([[325/324]])([[676/675]]), or can be expressed as ([[169/168]])/([[729/728]]), i.e. S13/S27. | 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. | ||