3edt: Difference between revisions
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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. | 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=== | === Odd harmonics === | ||
{{Harmonics in equal|3|3|1 | {{Harmonics in equal|3|3|1}} | ||
== Relationship to octave temperaments == | == Relationship to octave temperaments == | ||
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: | 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 === | === Simple octave temperaments === | ||
* [[Liese]] | * [[Liese]] | ||
* [[Triton]] | * [[Triton]] | ||
* [[Tricot]] | * [[Tricot]] | ||
=== Fractional-octave temperaments === | === Fractional-octave temperaments === | ||
* [[Augene]], [[augmented]], [[august]] - can be seen as a superset of [[3edo]] and 3edt | * [[Augene]], [[augmented]], [[august]] - can be seen as a superset of [[3edo]] and 3edt | ||
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* [[Akjayland]] - [[21edo]] and 3edt | * [[Akjayland]] - [[21edo]] and 3edt | ||
* [[Oganesson]] - [[118edo]] and 3edt | * [[Oganesson]] - [[118edo]] and 3edt | ||