transcendental number

The set of real numbers can be divided into two subsets, the algebraic numbers and the transcendental numbers. Algebraic numbers are the roots of polynomial equations with integer coefficients. Transcendentals are not. π (pi) and e have been proved transcendental.

According to set theory, the number of algebraic numbers is countably infinite (i.e., they can be put in a one-to-one correspondence with the integers) while the number of transendental numbers is uncountably infinite. So the possibilities for transcendental scales are hopelessly unlimited; one could not even begin to classify them.

[from Paul Erlich]

(Erlich made the following observation about transcendental numbers as a basis for tuning:)

Acoustically, the distinction between algebraic and transcendental numbers is meaningless. Just, equal-tempered, meantone-type, and golden scales account for only an infinitesimal minority of scales defined by algebraic numbers. As rational numbers have a distinct acoustical quality, one might ask for numbers that are "most irrational". It turns out that these numbers, the noble numbers, are algebraic as well. So while there are tons of interesting nj-net scales, I can see no possible impetus for transcendental scales.

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