Radian

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This page presents a novelty topic. It features ideas which are less likely to find practical applications in xenharmonic music. It may contain numbers that are impractically large, exceedingly complex or chosen arbitrarily.

Novelty topics are often developed by a single person or a small group. As such, this page may also feature idiosyncratic terms, notations or conceptual frameworks.

Interval information
Expression [math]2^{\frac{1}{2\pi} }[/math]
Size in cents 190.98593¢
Names radian,
radial major second,
radial whole tone
Special properties reduced
Harmonic entropy
(Shannon, [math]\sqrt{n\cdot d}[/math])
~4.56002 bits

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 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]\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[clarification needed].