Algebraic number

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A univariate polynomial a0x^n + a1x^(n-1) + ... + an whose coefficients ai are integers (or equivalently, rational numbers) has roots which are known as algebraic numbers. A root is a value r for which the polynomial function f(x) = a0x^n + a1x^(n-1) + ... + an satisfies f(r) = 0. If r is a real number, it is a real algebraic number.

Real algebraic numbers which are not also rational numbers turn up in various places in musical tuning theory. For instance, the intervals in the target tunings minimax or least squares are real algebraic numbers. An example of this is 1/4-comma meantone, where 2 and 5 are not tempered but 3 is tempered to 2 5^(1/4), a root of x^4-80. Generators for linear temperaments which are real algebraic numbers can have interesting properties in terms of the combination tones they produce.

Finding numerical values for real algebraic numbers is probably best left to a computer algebra package, but iterative methods such as Newton's method can be used. A refinement of Newton's method is the Durand–Kerner method.