Radian: Difference between revisions
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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. | 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. | ||
== 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. | |||
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. | |||
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. | 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. | ||
== | == Radian in other intervals of equivalence == | ||
The '''Relative Radian''' is | 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. | ||
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}}. | ||
Revision as of 10:45, 17 June 2022
Radian, radial major second, or a radial whole tone is an interval of [math]\displaystyle{ \frac{1200}{2\pi} }[/math], or 190.98593 cents.
The interval has an interpretation that relates to all 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.
Approximations
Closest equal temperament approximations of the radian can be derived from the continued fraction of 1/2pi: 4\25, 7\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.
Other approximations include 5\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.
Radian in other intervals of equivalence
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]\displaystyle{ \frac{1}{2\pi} }[/math] of the original interval on the logarithmic scale.
Every noticeable interval is the relative radian of some EDO of size [math]\displaystyle{ \frac{4800\pi}{7} }[/math] or smaller[clarification needed].