3edt: Difference between revisions

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'''3EDT''', if the attempt is made to use it as an actual scale, would divide the [[3/1|tritave]] into three equal parts, each of size 633.9850 cents, which is to say 3^(1/3) as a frequency ratio. If we want to consider it to be a temperament, it tempers out [[9/8]] as well as [[2edo]].

==Theory==

== Theory ==

75/52 is a [[Nearest just interval|~~good~~ rational ~~representation~~]] of the ~~cube root of~~ 3.

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. This is reinforced by 3edt having 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.

===Odd harmonics===

== Relationship to octave temperaments ==

~~3EDT is closely related~~ to any rank-2 octavated temperament which takes 3 generators to reach the 3rd harmonic~~, of which there's a notable amount of~~.

One step of 3edt can represent the generator to any rank-2 octavated temperament which takes 3 generators to reach the 3rd harmonic. These are:

=== Simple octave temperaments ===

* [[Liese]]

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 {{Infobox ET}}
-'''3EDT''', if the attempt is made to use it as an actual scale, would divide the [[3/1|tritave]] into three equal parts, each of size 633.9850 cents, which is to say 3^(1/3) as a frequency ratio. If we want to consider it to be a temperament, it tempers out [[9/8]] as well as [[2edo]].
+{{EDO intro}}
-==Theory==
+== Theory ==
-/52 is a [[Nearest just interval|good rational representation]] of the cube root of 3.
+edt 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. This is reinforced by 3edt having 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.
+===Odd harmonics===
+{{Harmonics in equal|3|3|1|intervals=odd}}
 == Relationship to octave temperaments ==
-EDT is closely related to any rank-2 octavated temperament which takes 3 generators to reach the 3rd harmonic, of which there's a notable amount of.
+One step of 3edt can represent the generator to any rank-2 octavated temperament which takes 3 generators to reach the 3rd harmonic. These are:
 === Simple octave temperaments ===
 * [[Liese]]