Radian: Difference between revisions

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{{Infobox Interval

| Ratio = 2^{\frac{1}{2\pi} }

| Cents = 190.98593171027437

| Name = radian, radial major second, radial whole tone

}}

'''Radian''', '''radial major second''', or a '''radial whole tone''' is an interval of <math>\frac{1200}{2\pi}</math>, or 190.98593 cents.

The interval has an interpretation that relates to all [[EDO|EDOs]]. Since pitch classes in all equal divisions of the octave form in the shape of a circle, radian interval therefore occurs as the radius of this perceptional circle.

~~Radian is 13 cents below just [[9/8]] and 9 cents below [[12edo]] major second of exactly 1/6 of a circle, or 200 cents.~~

== Approximations ==

~~== Approximations ==~~

Closest equal temperament approximations of the radian can be derived from the continued fraction of 1/2pi: 4\[[25edo|25]], 7\[[44edo|44]], 53\333, and 113\710. 7\44 and 113\710 are complementary to the historically notable 22/7 and 355/113 approximations of pi.

~~Closest equal temperament approximations~~ of the ~~radian can be derived from the continued fraction~~ of ~~1/2pi: 4\~~[[~~25edo|25~~]], 7\[[~~44edo|44~~]]~~, 53\333, and 113\710. 7\44 and 113\710 are complementary to the historically notable 22/7 and 355/113 approximations of pi~~.

Radian is 13 cents below just [[9/8]] and 9 cents below [[12edo]] major second of exactly 1/6 of a circle, or 200 cents. It is definitionally the [[meantone]] whole tone of [[Lucy tuning]], which is well approximated by [[88edo]].

Other approximations include 5\[[31edo|31]], which is also for the meantone, 11\69, the local meantone as well, and 13\82. Starting with 88edo, difference between the radian (14 steps out of 88) and 9/8 (15 steps out of 88) is visible.

== ~~Relative~~ Radian ==

== Radian in other intervals of equivalence ==

The '''Relative Radian''' is ~~an interval based on~~ a ~~given interval~~. ~~In being~~ the ~~interval of~~ <math>\frac{1}{2\pi}</math> ~~of this interval, pitches, it is the generalization~~ of the ~~common radian.~~

The '''Relative Radian''' is a generalization of the common radian to nonoctave intervals of equivalence. Just as the octave radian, it is defined as <math>\frac{1}{2\pi}</math> of the original interval on the logarithmic scale.

~~The~~ interval ~~has an interpretation that relates to equal divisions of all intervals. Since pitch classes in equal divisions of all intervals form in~~ the ~~shape of a circle, radian interval therefore occurs as the radius of this perceptional circle~~.

Every noticeable interval is the relative radian of some [[EDO]] of size <math>\frac{4800\pi}{7}</math> or smaller.

Every noticeable interval is the relative radian of some [[EDO]] of size <math>\frac{4800\pi}{7}</math> or smaller{{Clarify}}.

[[Category:Transcendental]]

@@ Line 1: / Line 1: @@
+{{Novelty}}
+{{Infobox Interval
+| Ratio = 2^{\frac{1}{2\pi} }
+| Cents = 190.98593171027437
+| Name = radian, radial major second, radial whole tone
+}}
 '''Radian''', '''radial major second''', or a '''radial whole tone''' is an interval of <math>\frac{1200}{2\pi}</math>, or 190.98593 cents.
 The interval has an interpretation that relates to all [[EDO|EDOs]]. Since pitch classes in all equal divisions of the octave form in the shape of a circle, radian interval therefore occurs as the radius of this perceptional circle.
-Radian is 13 cents below just [[9/8]] and 9 cents below [[12edo]] major second of exactly 1/6 of a circle, or 200 cents.
+== Approximations ==
-== Approximations ==
+Closest equal temperament approximations of the radian can be derived from the continued fraction of 1/2pi: 4\[[25edo|25]], 7\[[44edo|44]], 53\333, and 113\710. 7\44 and 113\710 are complementary to the historically notable 22/7 and 355/113 approximations of pi.
-Closest equal temperament approximations of the radian can be derived from the continued fraction of 1/2pi: 4\[[25edo|25]], 7\[[44edo|44]], 53\333, and 113\710. 7\44 and 113\710 are complementary to the historically notable 22/7 and 355/113 approximations of pi.
+Radian is 13 cents below just [[9/8]] and 9 cents below [[12edo]] major second of exactly 1/6 of a circle, or 200 cents. It is definitionally the [[meantone]] whole tone of [[Lucy tuning]], which is well approximated by [[88edo]].
 Other approximations include 5\[[31edo|31]], which is also for the meantone, 11\69, the local meantone as well, and 13\82. Starting with 88edo, difference between the radian (14 steps out of 88) and 9/8 (15 steps out of 88) is visible.
-== Relative Radian ==
+== Radian in other intervals of equivalence ==
-The '''Relative Radian''' is an interval based on a given interval. In being the interval of <math>\frac{1}{2\pi}</math> of this interval,  pitches, it is the generalization of the common radian.
+The '''Relative Radian''' is a generalization of the common radian to nonoctave intervals of equivalence. Just as the octave radian, it is defined as <math>\frac{1}{2\pi}</math> of the original interval on the logarithmic scale.
-The interval has an interpretation that relates to equal divisions of all intervals. Since pitch classes in equal divisions of all intervals form in the shape of a circle, radian interval therefore occurs as the radius of this perceptional circle.
-Every noticeable interval is the relative radian of some [[EDO]] of size <math>\frac{4800\pi}{7}</math> or smaller.
+Every noticeable interval is the relative radian of some [[EDO]] of size <math>\frac{4800\pi}{7}</math> or smaller{{Clarify}}.
+[[Category:Transcendental]]