34edo: Difference between revisions

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== 34edo and logarithmic phi ==

As a Fibonacci number, 34edo contains a fraction of an octave which is a close approximation to the logarithmic phi – 21 degrees of 34edo, approximately 741.2 cents. Repeated iterations of this interval generates [[MOSScales|Moment of Symmetry]] scales with near-phi relationships between the step sizes. As a 2.3.5.13 temperament, the 21\34 generator is an approximate 20/13, and the temperament tempers out 512/507 and {{monzo|-6 2 6 0 0 -13}}. From the tempering of 512/507, two 16/13 neutral thirds are an approximate 3/2, defining an essentially tempered neutral triad with a sharp rather than a flat fifth. ~~But~~, ~~to be clear~~ the ~~harmonic~~ ratio of phi is ~ 833 cents, and the equal divisions of octave approximating this interval closely are 13edo and [[36edo]].

As a Fibonacci number, 34edo contains a fraction of an octave which is a close approximation to the logarithmic phi – 21 degrees of 34edo, approximately 741.2 cents. Repeated iterations of this interval generates [[MOSScales|Moment of Symmetry]] scales with near-phi relationships between the step sizes. As a 2.3.5.13 temperament, the 21\34 generator is an approximate 20/13, and the temperament tempers out 512/507 and {{monzo|-6 2 6 0 0 -13}}. From the tempering of 512/507, two 16/13 neutral thirds are an approximate 3/2, defining an essentially tempered neutral triad with a sharp rather than a flat fifth. (On the other hand, the frequency ratio phi is ~ 833 cents, and the equal divisions of octave approximating this interval closely are 13edo and [[36edo]].ll

== 34edo as a regular temperament ==

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 == 34edo and logarithmic phi ==
-As a Fibonacci number, 34edo contains a fraction of an octave which is a close approximation to the logarithmic phi – 21 degrees of 34edo, approximately 741.2 cents. Repeated iterations of this interval generates [[MOSScales|Moment of Symmetry]] scales with near-phi relationships between the step sizes. As a 2.3.5.13 temperament, the 21\34 generator is an approximate 20/13, and the temperament tempers out 512/507 and {{monzo|-6 2 6 0 0 -13}}. From the tempering of 512/507, two 16/13 neutral thirds are an approximate 3/2, defining an essentially tempered neutral triad with a sharp rather than a flat fifth. But, to be clear the harmonic ratio of phi is ~ 833 cents, and the equal divisions of octave approximating this interval closely are 13edo and [[36edo]].
+As a Fibonacci number, 34edo contains a fraction of an octave which is a close approximation to the logarithmic phi – 21 degrees of 34edo, approximately 741.2 cents. Repeated iterations of this interval generates [[MOSScales|Moment of Symmetry]] scales with near-phi relationships between the step sizes. As a 2.3.5.13 temperament, the 21\34 generator is an approximate 20/13, and the temperament tempers out 512/507 and {{monzo|-6 2 6 0 0 -13}}. From the tempering of 512/507, two 16/13 neutral thirds are an approximate 3/2, defining an essentially tempered neutral triad with a sharp rather than a flat fifth. (On the other hand, the frequency ratio phi is ~ 833 cents, and the equal divisions of octave approximating this interval closely are 13edo and [[36edo]].ll
 == 34edo as a regular temperament ==