13edo: Difference between revisions

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== Approximation to irrational intervals ==

=== ~~Acoustic phi~~ ===

=== Golden ratio ===

13edo has a very ~~close~~ approximation of [[acoustic phi]] (9\13), with only -2.3 cents of error. [[23edo]] and [[36edo]] ~~are even closer, but unlike all closer edos~~, 13edo has ~~no other interval that represents any ratio from the Fibonacci sequence~~ (~~3/2, 5/3, 8/5, 13/8,~~ 21/13, ~~etc~~.~~) except~~ of ~~course for 1/1 and 2/1~~. In a ~~way, one could say that 13edo is the only~~ edo ~~that tempers the ratios~~ of ~~the Fibonacci sequence into a single interval~~.

13edo has a very good approximation of [[acoustic phi]] (9\13), with only -2.3 cents of error. The next better approximations are in [[23edo]] and [[36edo]]. As a coincidence, 13edo also has a very close appoximation of [[logarithmic phi]] (21\13), with only -3.2 cents of error. Logarithmic phi has some interesting applications in [[Metallic MOS]].

Not until [[144edo|144]] do we find a better edo in terms of relative error on both of these two intervals.

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 == Approximation to irrational intervals ==
-=== Acoustic phi ===
+=== Golden ratio ===
-edo has a very close approximation of [[acoustic phi]] (9\13), with only -2.3 cents of error. [[23edo]] and [[36edo]] are even closer, but unlike all closer edos, 13edo has no other interval that represents any ratio from the Fibonacci sequence (3/2, 5/3, 8/5, 13/8, 21/13, etc.) except of course for 1/1 and 2/1. In a way, one could say that 13edo is the only edo that tempers the ratios of the Fibonacci sequence into a single interval.
+edo has a very good approximation of [[acoustic phi]] (9\13), with only -2.3 cents of error. The next better approximations are in [[23edo]] and [[36edo]]. As a coincidence, 13edo also has a very close appoximation of [[logarithmic phi]] (21\13), with only -3.2 cents of error. Logarithmic phi has some interesting applications in [[Metallic MOS]].
+Not until [[144edo|144]] do we find a better edo in terms of relative error on both of these two intervals.
 See also: [[9edϕ]]
-=== Logarithmic phi ===
-As a coïncidence, 13edo also has a very close appoximation of [[logarithmic phi]] (21\13), with only -3.2 cents of error.
-Not until [[144edo|144]] do we find a better edo in terms of relative error on these two intervals.
-However, it should be noted that when we are hearing logarithmic phi, we are in fact hearing 2<sup>ϕ</sup> ≃ 3.070. While this interval can still be used in a way or another as a useful tone in a piece of music, it doesn't correspond to anything. When it comes to acoustic phi, we are truly hearing the mathematical constant ϕ ≃ 1.6180.
-That being said, logarithmic phi has interesting applications as [[Metallic MOS]], and in particular the fractal-like possibilities of self-similar subdivision of musical scales.
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