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.

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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. ~~EDOs which temper out~~ the ~~difference between the radian and exactly 1/6~~ of ~~a circle are said to temper out the~~ [[~~engineer's comma~~]], ~~as it equates pi with 3~~.

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.

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

[[Category:Transcendental]]

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+{{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.
@@ Line 7: / Line 13: @@
 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. EDOs which temper out the difference between the radian and exactly 1/6 of a circle are said to temper out the [[engineer's comma]], as it equates pi with 3.
+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.
@@ Line 15: / Line 21: @@
 Every noticeable interval is the relative radian of some [[EDO]] of size <math>\frac{4800\pi}{7}</math> or smaller{{Clarify}}.
+[[Category:Transcendental]]