Algebraic number: Difference between revisions
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A [http://mathworld.wolfram.com/UnivariatePolynomial.html univariate polynomial] ''a''<sub>0</sub>''x''<sup>''n''</sup> | A [http://mathworld.wolfram.com/UnivariatePolynomial.html univariate polynomial] {{nowrap|''a''<sub>0</sub>''x''<sup>''n''</sup> + ''a''<sub>1</sub>''x''<sup>''n'' − 1</sup>}} + … {{nowrap|+ ''a''<sub>''n''</sub>}} whose coefficients ''a''<sub>''i''</sub> are integers (or equivalently, rational numbers) has roots which are known as '''algebraic numbers'''. A root is a value ''r'' for which the [[Wikipedia: Polynomial #Polynomial functions|polynomial function]] {{nowrap|''f''(''x'') {{=}} ''a''<sub>0</sub>''x''<sup>''n''</sup>}} {{nowrap|+ ''a''<sub>1</sub>''x''<sup>''n'' − 1</sup>}} + … {{nowrap|+ ''a''<sub>''n''</sub>}} satisfies {{nowrap|''f''(''r'') {{=}} 0}}. If ''r'' is a {{w|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|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<sup>1/4</sup>, a root of {{nowrap|''x''<sup>4</sup> − 80}}. [[Generators]] for [[linear temperament]]s which are real algebraic numbers can have interesting properties in terms of the {{w|combination tone|combination tones}} they produce. Algebraic numbers are also relevant to JI-agnostic [[delta-rational]] harmony, as tunings of [[mos scale]]s with exact delta-rational values for a certain chord have generators that are algebraic numbers in the linear frequency domain. | 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|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<sup>1/4</sup>, a root of {{nowrap|''x''<sup>4</sup> − 80}}. [[Generators]] for [[linear temperament]]s which are real algebraic numbers can have interesting properties in terms of the {{w|combination tone|combination tones}} they produce. Algebraic numbers are also relevant to JI-agnostic [[delta-rational]] harmony, as tunings of [[mos scale]]s with exact delta-rational values for a certain chord have generators that are algebraic numbers in the linear frequency domain. | ||
Latest revision as of 13:01, 14 March 2025
A univariate polynomial a0xn + a1xn − 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) = a0xn + a1xn − 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 2×51/4, 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.