3edt: Difference between revisions

(10 intermediate revisions by 5 users not shown)

Line 1:

== Theory ==

3edt can be thought of as [[2edo]] with the 3/1 made just, by [[Stretched tuning|stretching]] the octave by 67.97{{c}}.

~~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{{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.

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)3 / (3/1) = [[2197/2187]], the threedie, and (75/52)3 / (3/1) = [[140625/140608]], the catasma.

=== Harmonics ===

== Relationship to octave temperaments ==

Line 17:

Line 16:

* [[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

== See also ==

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

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro}}
+{{ED intro}}
 == Theory ==
+edt can be thought of as [[2edo]] with the 3/1 made just, by [[Stretched tuning|stretching]] the octave by 67.97{{c}}.
-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{{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>.
-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.
 === Harmonics ===
-{{Harmonics in equal|3|3|1}}
+{{Harmonics in equal|3|3|1|columns=15}}
 == Relationship to octave temperaments ==
@@ Line 17: / Line 16: @@
 * [[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
+== See also ==
+* [[Alpha, beta, and gamma family of equal divisions]]