3edt: Difference between revisions
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3edt can be thought of as [[2edo]] with the 3/1 made just, by [[Stretched tuning|stretching]] the octave by 67.97 cents. | 3edt can be thought of as [[2edo]] with the 3/1 made just, by [[Stretched tuning|stretching]] the octave by 67.97 cents. | ||
Despite its small size, 3edt has an excellent approximation to the 13th harmonic: 7 steps of 3edt is only 2.63 cents flat of 13/1. | Despite its small size, 3edt has an excellent approximation to the 13th harmonic: 7 steps of 3edt is only 2.63 cents flat of 13/1. One step of 3edt has two good 13-limit [[Nearest just interval|rational approximations]], [[13/9]] and 75/52, both which are [[convergent]]s. 3edt thus tempers out (13/9)<sup>3</sup> / (3/1) = [[2197/2187]], the threedie, and (75/52)<sup>3</sup> / (3/1) = [[140625/140608]], the catasma. The good approximation for 13/9 and 75/52 also implies a good approximation for 25/4, or ([[5/2]])<sup>2</sup>. | ||
=== Harmonics === | === Harmonics === | ||