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{{c}}.

Despite its small size, 3edt has an excellent approximation to the 13th harmonic: 7 steps of 3edt is only 2.63{{c}} 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 {{nowrap|(13/9)3 / (3/1) {{=}} [[2197/2187]]}}, the threedie, and {{nowrap|(75/52)3 / (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]])2.

=== 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]]

* [[Triton]]

* [[~~Tricot~~]]

* [[Alphatricot]]

=== Fractional-octave temperaments ===

* [[Augene]], [[augmented]], [[august]] - can be seen as a superset of [[3edo]] and 3edt

* [[Augene]], [[augmented (temperament)|augmented]], [[august]] – can be seen as a superset of [[3edo]] and 3edt

* [[Soviet ferris wheel]] - [[20edo]] and 3edt

* [[Soviet ferris wheel]] – [[20edo]] and 3edt

* [[Akjayland]] - [[21edo]] and 3edt

* [[Akjayland]] – [[21edo]] and 3edt

* [[Oganesson]] - [[118edo]] and 3edt

* [[Oganesson]] – [[118edo]] and 3edt

~~[[Category:Edt]]~~

== See also ==

[[~~Category:Edonoi~~]]

* [[Alpha, beta, and gamma family of equal divisions]]

@@ Line 1: / Line 1: @@
 {{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]].
+{{ED 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{{c}}.
+Despite its small size, 3edt has an excellent approximation to the 13th harmonic: 7 steps of 3edt is only 2.63{{c}} 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 {{nowrap|(13/9)<sup>3</sup> / (3/1) {{=}} [[2197/2187]]}}, the threedie, and  {{nowrap|(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 in equal|3|3|1|columns=15}}
 == 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]]
 * [[Triton]]
-* [[Tricot]]
+* [[Alphatricot]]
 === Fractional-octave temperaments ===
-* [[Augene]], [[augmented]], [[august]] - can be seen as a superset of [[3edo]] and 3edt
+* [[Augene]], [[augmented (temperament)|augmented]], [[august]] – can be seen as a superset of [[3edo]] and 3edt
-* [[Soviet ferris wheel]] - [[20edo]] and 3edt
+* [[Soviet ferris wheel]] – [[20edo]] and 3edt
-* [[Akjayland]] - [[21edo]] and 3edt
+* [[Akjayland]] – [[21edo]] and 3edt
-* [[Oganesson]] - [[118edo]] and 3edt
+* [[Oganesson]] – [[118edo]] and 3edt
-[[Category:Edt]]
+== See also ==
-[[Category:Edonoi]]
+* [[Alpha, beta, and gamma family of equal divisions]]