2edt: Difference between revisions

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Created page with "'''2EDT''', if the attempt is made to use it as an actual scale, would divide the tritave into two equal parts, each of size 950.9775 cents, which is to say sqrt(3) as..."
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→Harmonics: added more columns so that harmonic 13 is shown, since it is mentioned right above
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'''2EDT''', if the attempt is made to use it as an actual scale, would divide the [[3/1|tritave]] into two equal parts, each of size 950.9775 cents, which is to say sqrt(3) as a frequency ratio. If we want to consider it to be a temperament, it tempers out [[4/3]] as well as [[1edo]].
{{Infobox ET}}
{{ED intro}}


==Factoids about 2EDT==
== Theory ==
[[26/15]] and 45/26 are [[Nearest just interval|good rational representations]] of the square root of 3.
As a temperament in the 3.5 [[subgroup]], it tempers out [[27/25]], equating 5/3 with 9/5.


2EDT is closely related to the [[Very high accuracy temperaments|monzismic temperament]], which tempers out |54 -37 2>, the monzisma.
Since [[26/15]] is a [[convergent]] of sqrt(3), 26/15 (and its [[tritave complement]] 45/26) are good rational representations of the square root of 3. 2edt thus tempers out (26/15)<sup>2</sup> / (3/1) = [[676/675]], the island comma.


[[Category:Edt]]
=== Harmonics ===
[[Category:Edonoi]]
{{Harmonics in equal|2|3|1|columns = 14}}
 
== Relationship to octave temperaments ==
One step of 2edt can represent the generator for any rank-2 octavated temperament which takes 2 generators to reach the 3rd harmonic, such as [[monzismic]].

Latest revision as of 04:34, 6 March 2026

← 1edt 2edt 3edt →
Prime factorization 2 (prime) (highly composite)
Step size 950.978 ¢ 
Octave 1\2edt (950.978 ¢)
(convergent)
Consistency limit 3
Distinct consistency limit 2

2 equal divisions of the tritave, perfect twelfth, or 3rd harmonic (abbreviated 2edt or 2ed3), is a nonoctave tuning system that divides the interval of 3/1 into 2 equal parts of about 951 ¢ each. Each step represents a frequency ratio of 31/2, or the square root of 3.

Theory

As a temperament in the 3.5 subgroup, it tempers out 27/25, equating 5/3 with 9/5.

Since 26/15 is a convergent of sqrt(3), 26/15 (and its tritave complement 45/26) are good rational representations of the square root of 3. 2edt thus tempers out (26/15)2 / (3/1) = 676/675, the island comma.

Harmonics

Approximation of harmonics in 2edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Error Absolute (¢) -249 +0 +453 +67 -249 +435 +204 +0 -182 -347 +453 +314 +186 +67
Relative (%) -26.2 +0.0 +47.6 +7.0 -26.2 +45.8 +21.4 +0.0 -19.2 -36.5 +47.6 +33.1 +19.6 +7.0
Steps
(reduced) 		1
(1) 		2
(0) 		3
(1) 		3
(1) 		3
(1) 		4
(0) 		4
(0) 		4
(0) 		4
(0) 		4
(0) 		5
(1) 		5
(1) 		5
(1) 		5
(1)

Relationship to octave temperaments

One step of 2edt can represent the generator for any rank-2 octavated temperament which takes 2 generators to reach the 3rd harmonic, such as monzismic.

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