3edt: Difference between revisions

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Eliora (talk | contribs)
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a lot of temperaments have 3/1 reached in 3 generators, elaborate
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clean up
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{{Infobox ET}}
{{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 ==
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===
{{Harmonics in equal|3|3|1|intervals=odd}}


== Relationship to octave temperaments ==
== 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 ===
=== Simple octave temperaments ===
* [[Liese]]
* [[Liese]]

Revision as of 12:20, 28 April 2024

← 2edt 3edt 4edt →
Prime factorization 3 (prime)
Step size 633.985 ¢ 
Octave 2\3edt (1267.97 ¢)
(convergent)
Consistency limit 4
Distinct consistency limit 3

Template:EDO intro

Theory

3edt can be thought of as 2edo with the 3/1 made just, by 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 rational approximations, 13/9 and 75/52, both which are convergents. 3edt thus tempers out (13/9)3 / (3/1) = 2197/2187, the threedie, and (75/52)3 / (3/1) = 140625/140608, the catasma.

Odd harmonics

Approximation of odd harmonics in 3edt
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +0 -250 -199 +0 +287 -3 -250 +167 -26 -199 +278
Relative (%) +0.0 -39.5 -31.4 +0.0 +45.2 -0.4 -39.5 +26.3 -4.0 -31.4 +43.8
Steps
(reduced) 		3
(0) 		4
(1) 		5
(2) 		6
(0) 		7
(1) 		7
(1) 		7
(1) 		8
(2) 		8
(2) 		8
(2) 		9
(0)

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:

Simple octave temperaments

Fractional-octave temperaments

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