Algebraic number

A univariate polynomial a₀xⁿ + a₁x^{(n - 1)} + … + a_n whose coefficients a_i 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) = a₀xⁿ + a₁x^{(n - 1)} + … + a_n 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 x⁴ - 80. Generators for linear temperaments which are real algebraic numbers can have interesting properties in terms of the combination tones they produce. Algebraic numbers are also relevant to JI-agnostic delta-rational harmony, as tunings of MOS scales with exact delta-rational values for a certain chord have generators that are algebraic numbers in the linear frequency domain.

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.