Algebraic number - Revision history

FloraC: + link

2026-02-08T08:37:01Z

+ link

← Older revision		Revision as of 08:37, 8 February 2026
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	Finding numerical values for real algebraic numbers is probably best left to a computer algebra package, but iterative methods such as {{w\|Newton's method}} can be used. A refinement of Newton's method is the {{w\|Durand–Kerner method}}.		Finding numerical values for real algebraic numbers is probably best left to a computer algebra package, but iterative methods such as {{w\|Newton's method}} can be used. A refinement of Newton's method is the {{w\|Durand–Kerner method}}.

			== External links ==
			* [https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_19436.html Yahoo! Tuning Group \| ''Algebraic generators'']

	[[Category:Math]]		[[Category:Math]]
	[[Category:Number theory]]		[[Category:Number theory]]
	~~[[Category:todo:~~increase applicability]]		{{Todo\|increase applicability}}

ArrowHead294 at 13:01, 14 March 2025

2025-03-14T13:01:45Z

← Older revision		Revision as of 13:01, 14 March 2025
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	{{Wikipedia}}		{{Wikipedia}}
	A [http://mathworld.wolfram.com/UnivariatePolynomial.html univariate polynomial] ''a''<sub>0</sub>''x''<sup>''n''</sup> ~~{{nowrap\|~~+ ''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''.		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.

ArrowHead294 at 18:40, 9 January 2025

2025-01-09T18:40:50Z

← Older revision		Revision as of 18:40, 9 January 2025
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	{{Wikipedia}}		{{Wikipedia}}
	A [http://mathworld.wolfram.com/UnivariatePolynomial.html univariate polynomial] ''a''<sub>0</sub>''x''<sup>''n''</sup> + ''a''<sub>1</sub>''x''<sup>(''n'' - 1)</sup> + … + ''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]] ''f'' (''x'') = ''a''<sub>0</sub>''x''<sup>''n''</sup> + ''a''<sub>1</sub>''x''<sup>(''n'' - 1)</sup> + … + ''a''<sub>''n''</sub> satisfies ''f'' (''r'') = 0. If ''r'' is a ~~[[Wikipedia: Real number~~\|real number]], it is a ''real algebraic number''.		A [http://mathworld.wolfram.com/UnivariatePolynomial.html univariate polynomial] ''a''<sub>0</sub>''x''<sup>''n''</sup> {{nowrap\|+ ''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 ''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.

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

ArrowHead294: {{Wikipedia}} template is fixed now

2024-10-29T02:25:03Z

{{Wikipedia}} template is fixed now

← Older revision		Revision as of 02:25, 29 October 2024
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	{{Wikipedia~~\|Algebraic number~~}}		{{Wikipedia}}
	A [http://mathworld.wolfram.com/UnivariatePolynomial.html univariate polynomial] ''a''<sub>0</sub>''x''<sup>''n''</sup> + ''a''<sub>1</sub>''x''<sup>(''n'' - 1)</sup> + … + ''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]] ''f'' (''x'') = ''a''<sub>0</sub>''x''<sup>''n''</sup> + ''a''<sub>1</sub>''x''<sup>(''n'' - 1)</sup> + … + ''a''<sub>''n''</sub> satisfies ''f'' (''r'') = 0. If ''r'' is a [[Wikipedia: Real number\|real number]], it is a ''real algebraic number''.		A [http://mathworld.wolfram.com/UnivariatePolynomial.html univariate polynomial] ''a''<sub>0</sub>''x''<sup>''n''</sup> + ''a''<sub>1</sub>''x''<sup>(''n'' - 1)</sup> + … + ''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]] ''f'' (''x'') = ''a''<sub>0</sub>''x''<sup>''n''</sup> + ''a''<sub>1</sub>''x''<sup>(''n'' - 1)</sup> + … + ''a''<sub>''n''</sub> satisfies ''f'' (''r'') = 0. If ''r'' is a [[Wikipedia: Real number\|real number]], it is a ''real algebraic number''.

Fitzgerald Lee: Fixed Wikipedia thingy

2024-10-28T11:10:31Z

Fixed Wikipedia thingy

← Older revision		Revision as of 11:10, 28 October 2024
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	{{Wikipedia}}		{{Wikipedia\|Algebraic number}}
	A [http://mathworld.wolfram.com/UnivariatePolynomial.html univariate polynomial] ''a''<sub>0</sub>''x''<sup>''n''</sup> + ''a''<sub>1</sub>''x''<sup>(''n'' - 1)</sup> + … + ''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]] ''f'' (''x'') = ''a''<sub>0</sub>''x''<sup>''n''</sup> + ''a''<sub>1</sub>''x''<sup>(''n'' - 1)</sup> + … + ''a''<sub>''n''</sub> satisfies ''f'' (''r'') = 0. If ''r'' is a [[Wikipedia: Real number\|real number]], it is a ''real algebraic number''.		A [http://mathworld.wolfram.com/UnivariatePolynomial.html univariate polynomial] ''a''<sub>0</sub>''x''<sup>''n''</sup> + ''a''<sub>1</sub>''x''<sup>(''n'' - 1)</sup> + … + ''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]] ''f'' (''x'') = ''a''<sub>0</sub>''x''<sup>''n''</sup> + ''a''<sub>1</sub>''x''<sup>(''n'' - 1)</sup> + … + ''a''<sub>''n''</sub> satisfies ''f'' (''r'') = 0. If ''r'' is a [[Wikipedia: Real number\|real number]], it is a ''real algebraic number''.

FloraC: Cleanup

2024-09-07T13:47:18Z

Cleanup

← Older revision		Revision as of 13:47, 7 September 2024
Line 1:		Line 1:
	A [http://mathworld.wolfram.com/UnivariatePolynomial.html univariate polynomial] ''a''<sub>0</sub>''x''<sup>''n''</sup> + ''a''<sub>1</sub>''x''<sup>(''n'' - 1)</sup> + … + ''a''<sub>''n''</sub> whose coefficients ''a''<sub>''i''</sub> are integers (or equivalently, rational numbers) has roots which are known as ~~[[Wikipedia: Algebraic number\|~~algebraic numbers]]. A root is a value ''r'' for which the [[Wikipedia: Polynomial #Polynomial functions\|polynomial function]] ''f'' (''x'') = ''a''<sub>0</sub>''x''<sup>''n''</sup> + ''a''<sub>1</sub>''x''<sup>(''n'' - 1)</sup> + … + ''a''<sub>''n''</sub> satisfies ''f'' (''r'') = 0. If ''r'' is a [[Wikipedia: Real number\|real number]], it is a ''real algebraic number''.		{{Wikipedia}}
			A [http://mathworld.wolfram.com/UnivariatePolynomial.html univariate polynomial] ''a''<sub>0</sub>''x''<sup>''n''</sup> + ''a''<sub>1</sub>''x''<sup>(''n'' - 1)</sup> + … + ''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]] ''f'' (''x'') = ''a''<sub>0</sub>''x''<sup>''n''</sup> + ''a''<sub>1</sub>''x''<sup>(''n'' - 1)</sup> + … + ''a''<sub>''n''</sub> satisfies ''f'' (''r'') = 0. If ''r'' is a [[Wikipedia: Real number\|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 ''x''<sup>4</sup> - 80. [[Generators]] for [[linear temperament]]s which are real algebraic numbers can have interesting properties in terms of the ~~[[Wikipedia: Combination~~ tone\|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.		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 ''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.

	Finding numerical values for real algebraic numbers is probably best left to a computer algebra package, but iterative methods such as ~~[[Wikipedia: Newton's method~~\|Newton's method]] can be used. A refinement of Newton's method is the ~~[[Wikipedia: Durand–Kerner method~~\|Durand–Kerner method]].		Finding numerical values for real algebraic numbers is probably best left to a computer algebra package, but iterative methods such as {{w\|Newton's method}} can be used. A refinement of Newton's method is the {{w\|Durand–Kerner method}}.

	[[Category:Math]]		[[Category:Math]]
	[[Category:Number theory]]		[[Category:Number theory]]
	[[Category:todo:increase applicability]]		[[Category:todo:increase applicability]]

Inthar at 22:54, 9 February 2024

2024-02-09T22:54:26Z

← Older revision		Revision as of 22:54, 9 February 2024
Line 6:		Line 6:

	[[Category:Math]]		[[Category:Math]]
	[[Category:Number ~~thoery~~]]		[[Category:Number theory]]
	[[Category:todo:increase applicability]]		[[Category:todo:increase applicability]]

Inthar at 22:54, 9 February 2024

2024-02-09T22:54:03Z

← Older revision		Revision as of 22:54, 9 February 2024
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	A [http://mathworld.wolfram.com/UnivariatePolynomial.html univariate polynomial] ''a''<sub>0</sub>''x''<sup>''n''</sup> + ''a''<sub>1</sub>''x''<sup>(''n'' - 1)</sup> + … + ''a''<sub>''n''</sub> whose coefficients ''a''<sub>''i''</sub> are integers (or equivalently, rational numbers) has roots which are known as [[Wikipedia: Algebraic number\|algebraic numbers]]. A root is a value ''r'' for which the [[Wikipedia: Polynomial #Polynomial functions\|polynomial function]] ''f'' (''x'') = ''a''<sub>0</sub>''x''<sup>''n''</sup> + ''a''<sub>1</sub>''x''<sup>(''n'' - 1)</sup> + … + ''a''<sub>''n''</sub> satisfies ''f'' (''r'') = 0. If ''r'' is a [[Wikipedia: Real number\|real number]], it is a ''real algebraic number''.		A [http://mathworld.wolfram.com/UnivariatePolynomial.html univariate polynomial] ''a''<sub>0</sub>''x''<sup>''n''</sup> + ''a''<sub>1</sub>''x''<sup>(''n'' - 1)</sup> + … + ''a''<sub>''n''</sub> whose coefficients ''a''<sub>''i''</sub> are integers (or equivalently, rational numbers) has roots which are known as [[Wikipedia: Algebraic number\|algebraic numbers]]. A root is a value ''r'' for which the [[Wikipedia: Polynomial #Polynomial functions\|polynomial function]] ''f'' (''x'') = ''a''<sub>0</sub>''x''<sup>''n''</sup> + ''a''<sub>1</sub>''x''<sup>(''n'' - 1)</sup> + … + ''a''<sub>''n''</sub> satisfies ''f'' (''r'') = 0. If ''r'' is a [[Wikipedia: Real number\|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 ''x''<sup>4</sup> - 80. [[Generators]] for [[linear temperament]]s which are real algebraic numbers can have interesting properties in terms of the [[Wikipedia: Combination tone\|combination tones]] they produce. Algebraic numbers are also relevant to [[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.		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 ''x''<sup>4</sup> - 80. [[Generators]] for [[linear temperament]]s which are real algebraic numbers can have interesting properties in terms of the [[Wikipedia: Combination tone\|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 [[Wikipedia: Newton's method\|Newton's method]] can be used. A refinement of Newton's method is the [[Wikipedia: Durand–Kerner method\|Durand–Kerner method]].		Finding numerical values for real algebraic numbers is probably best left to a computer algebra package, but iterative methods such as [[Wikipedia: Newton's method\|Newton's method]] can be used. A refinement of Newton's method is the [[Wikipedia: Durand–Kerner method\|Durand–Kerner method]].

Inthar at 22:53, 9 February 2024

2024-02-09T22:53:49Z

← Older revision		Revision as of 22:53, 9 February 2024
Line 1:		Line 1:
	A [http://mathworld.wolfram.com/UnivariatePolynomial.html univariate polynomial] ''a''<sub>0</sub>''x''<sup>''n''</sup> + ''a''<sub>1</sub>''x''<sup>(''n'' - 1)</sup> + … + ''a''<sub>''n''</sub> whose coefficients ''a''<sub>''i''</sub> are integers (or equivalently, rational numbers) has roots which are known as [[Wikipedia: Algebraic number\|algebraic numbers]]. A root is a value ''r'' for which the [[Wikipedia: Polynomial #Polynomial functions\|polynomial function]] ''f'' (''x'') = ''a''<sub>0</sub>''x''<sup>''n''</sup> + ''a''<sub>1</sub>''x''<sup>(''n'' - 1)</sup> + … + ''a''<sub>''n''</sub> satisfies ''f'' (''r'') = 0. If ''r'' is a [[Wikipedia: Real number\|real number]], it is a ''real algebraic number''.		A [http://mathworld.wolfram.com/UnivariatePolynomial.html univariate polynomial] ''a''<sub>0</sub>''x''<sup>''n''</sup> + ''a''<sub>1</sub>''x''<sup>(''n'' - 1)</sup> + … + ''a''<sub>''n''</sub> whose coefficients ''a''<sub>''i''</sub> are integers (or equivalently, rational numbers) has roots which are known as [[Wikipedia: Algebraic number\|algebraic numbers]]. A root is a value ''r'' for which the [[Wikipedia: Polynomial #Polynomial functions\|polynomial function]] ''f'' (''x'') = ''a''<sub>0</sub>''x''<sup>''n''</sup> + ''a''<sub>1</sub>''x''<sup>(''n'' - 1)</sup> + … + ''a''<sub>''n''</sub> satisfies ''f'' (''r'') = 0. If ''r'' is a [[Wikipedia: Real number\|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 ''x''<sup>4</sup> - 80. [[Generators]] for [[linear temperament]]s which are real algebraic numbers can have interesting properties in terms of the [[Wikipedia: Combination tone\|combination tones]] they produce. ~~Tunings~~ of MOS scales 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 ''x''<sup>4</sup> - 80. [[Generators]] for [[linear temperament]]s which are real algebraic numbers can have interesting properties in terms of the [[Wikipedia: Combination tone\|combination tones]] they produce. Algebraic numbers are also relevant to [[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 [[Wikipedia: Newton's method\|Newton's method]] can be used. A refinement of Newton's method is the [[Wikipedia: Durand–Kerner method\|Durand–Kerner method]].		Finding numerical values for real algebraic numbers is probably best left to a computer algebra package, but iterative methods such as [[Wikipedia: Newton's method\|Newton's method]] can be used. A refinement of Newton's method is the [[Wikipedia: Durand–Kerner method\|Durand–Kerner method]].

Inthar at 22:53, 9 February 2024

2024-02-09T22:53:11Z

← Older revision		Revision as of 22:53, 9 February 2024
Line 1:		Line 1:
	A [http://mathworld.wolfram.com/UnivariatePolynomial.html univariate polynomial] ''a''<sub>0</sub>''x''<sup>''n''</sup> + ''a''<sub>1</sub>''x''<sup>(''n'' - 1)</sup> + … + ''a''<sub>''n''</sub> whose coefficients ''a''<sub>''i''</sub> are integers (or equivalently, rational numbers) has roots which are known as [[Wikipedia: Algebraic number\|algebraic numbers]]. A root is a value ''r'' for which the [[Wikipedia: Polynomial #Polynomial functions\|polynomial function]] ''f'' (''x'') = ''a''<sub>0</sub>''x''<sup>''n''</sup> + ''a''<sub>1</sub>''x''<sup>(''n'' - 1)</sup> + … + ''a''<sub>''n''</sub> satisfies ''f'' (''r'') = 0. If ''r'' is a [[Wikipedia: Real number\|real number]], it is a ''real algebraic number''.		A [http://mathworld.wolfram.com/UnivariatePolynomial.html univariate polynomial] ''a''<sub>0</sub>''x''<sup>''n''</sup> + ''a''<sub>1</sub>''x''<sup>(''n'' - 1)</sup> + … + ''a''<sub>''n''</sub> whose coefficients ''a''<sub>''i''</sub> are integers (or equivalently, rational numbers) has roots which are known as [[Wikipedia: Algebraic number\|algebraic numbers]]. A root is a value ''r'' for which the [[Wikipedia: Polynomial #Polynomial functions\|polynomial function]] ''f'' (''x'') = ''a''<sub>0</sub>''x''<sup>''n''</sup> + ''a''<sub>1</sub>''x''<sup>(''n'' - 1)</sup> + … + ''a''<sub>''n''</sub> satisfies ''f'' (''r'') = 0. If ''r'' is a [[Wikipedia: Real number\|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 ''x''<sup>4</sup> - 80. [[Generators]] for [[linear temperament]]s which are real algebraic numbers can have interesting properties in terms of the [[Wikipedia: Combination tone\|combination tones]] they produce.		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 ''x''<sup>4</sup> - 80. [[Generators]] for [[linear temperament]]s which are real algebraic numbers can have interesting properties in terms of the [[Wikipedia: Combination tone\|combination tones]] they produce. 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 [[Wikipedia: Newton's method\|Newton's method]] can be used. A refinement of Newton's method is the [[Wikipedia: Durand–Kerner method\|Durand–Kerner method]].		Finding numerical values for real algebraic numbers is probably best left to a computer algebra package, but iterative methods such as [[Wikipedia: Newton's method\|Newton's method]] can be used. A refinement of Newton's method is the [[Wikipedia: Durand–Kerner method\|Durand–Kerner method]].

	[[Category:Math]]		[[Category:Math]]
			[[Category:Number thoery]]
	[[Category:todo:increase applicability]]		[[Category:todo:increase applicability]]

← Older revision		Revision as of 22:54, 9 February 2024
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	A [http://mathworld.wolfram.com/UnivariatePolynomial.html univariate polynomial] ''a''<sub>0</sub>''x''<sup>''n''</sup> + ''a''<sub>1</sub>''x''<sup>(''n'' - 1)</sup> + … + ''a''<sub>''n''</sub> whose coefficients ''a''<sub>''i''</sub> are integers (or equivalently, rational numbers) has roots which are known as [[Wikipedia: Algebraic number\|algebraic numbers]]. A root is a value ''r'' for which the [[Wikipedia: Polynomial #Polynomial functions\|polynomial function]] ''f'' (''x'') = ''a''<sub>0</sub>''x''<sup>''n''</sup> + ''a''<sub>1</sub>''x''<sup>(''n'' - 1)</sup> + … + ''a''<sub>''n''</sub> satisfies ''f'' (''r'') = 0. If ''r'' is a [[Wikipedia: Real number\|real number]], it is a ''real algebraic number''.		A [http://mathworld.wolfram.com/UnivariatePolynomial.html univariate polynomial] ''a''<sub>0</sub>''x''<sup>''n''</sup> + ''a''<sub>1</sub>''x''<sup>(''n'' - 1)</sup> + … + ''a''<sub>''n''</sub> whose coefficients ''a''<sub>''i''</sub> are integers (or equivalently, rational numbers) has roots which are known as [[Wikipedia: Algebraic number\|algebraic numbers]]. A root is a value ''r'' for which the [[Wikipedia: Polynomial #Polynomial functions\|polynomial function]] ''f'' (''x'') = ''a''<sub>0</sub>''x''<sup>''n''</sup> + ''a''<sub>1</sub>''x''<sup>(''n'' - 1)</sup> + … + ''a''<sub>''n''</sub> satisfies ''f'' (''r'') = 0. If ''r'' is a [[Wikipedia: Real number\|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 ''x''<sup>4</sup> - 80. [[Generators]] for [[linear temperament]]s which are real algebraic numbers can have interesting properties in terms of the [[Wikipedia: Combination tone\|combination tones]] they produce. Algebraic numbers are also relevant to [[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.		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 ''x''<sup>4</sup> - 80. [[Generators]] for [[linear temperament]]s which are real algebraic numbers can have interesting properties in terms of the [[Wikipedia: Combination tone\|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 [[Wikipedia: Newton's method\|Newton's method]] can be used. A refinement of Newton's method is the [[Wikipedia: Durand–Kerner method\|Durand–Kerner method]].		Finding numerical values for real algebraic numbers is probably best left to a computer algebra package, but iterative methods such as [[Wikipedia: Newton's method\|Newton's method]] can be used. A refinement of Newton's method is the [[Wikipedia: Durand–Kerner method\|Durand–Kerner method]].

← Older revision		Revision as of 22:53, 9 February 2024
Line 1:		Line 1:
	A [http://mathworld.wolfram.com/UnivariatePolynomial.html univariate polynomial] ''a''<sub>0</sub>''x''<sup>''n''</sup> + ''a''<sub>1</sub>''x''<sup>(''n'' - 1)</sup> + … + ''a''<sub>''n''</sub> whose coefficients ''a''<sub>''i''</sub> are integers (or equivalently, rational numbers) has roots which are known as [[Wikipedia: Algebraic number\|algebraic numbers]]. A root is a value ''r'' for which the [[Wikipedia: Polynomial #Polynomial functions\|polynomial function]] ''f'' (''x'') = ''a''<sub>0</sub>''x''<sup>''n''</sup> + ''a''<sub>1</sub>''x''<sup>(''n'' - 1)</sup> + … + ''a''<sub>''n''</sub> satisfies ''f'' (''r'') = 0. If ''r'' is a [[Wikipedia: Real number\|real number]], it is a ''real algebraic number''.		A [http://mathworld.wolfram.com/UnivariatePolynomial.html univariate polynomial] ''a''<sub>0</sub>''x''<sup>''n''</sup> + ''a''<sub>1</sub>''x''<sup>(''n'' - 1)</sup> + … + ''a''<sub>''n''</sub> whose coefficients ''a''<sub>''i''</sub> are integers (or equivalently, rational numbers) has roots which are known as [[Wikipedia: Algebraic number\|algebraic numbers]]. A root is a value ''r'' for which the [[Wikipedia: Polynomial #Polynomial functions\|polynomial function]] ''f'' (''x'') = ''a''<sub>0</sub>''x''<sup>''n''</sup> + ''a''<sub>1</sub>''x''<sup>(''n'' - 1)</sup> + … + ''a''<sub>''n''</sub> satisfies ''f'' (''r'') = 0. If ''r'' is a [[Wikipedia: Real number\|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 ''x''<sup>4</sup> - 80. [[Generators]] for [[linear temperament]]s which are real algebraic numbers can have interesting properties in terms of the [[Wikipedia: Combination tone\|combination tones]] they produce. ~~Tunings~~ of MOS scales 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 ''x''<sup>4</sup> - 80. [[Generators]] for [[linear temperament]]s which are real algebraic numbers can have interesting properties in terms of the [[Wikipedia: Combination tone\|combination tones]] they produce. Algebraic numbers are also relevant to [[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 [[Wikipedia: Newton's method\|Newton's method]] can be used. A refinement of Newton's method is the [[Wikipedia: Durand–Kerner method\|Durand–Kerner method]].		Finding numerical values for real algebraic numbers is probably best left to a computer algebra package, but iterative methods such as [[Wikipedia: Newton's method\|Newton's method]] can be used. A refinement of Newton's method is the [[Wikipedia: Durand–Kerner method\|Durand–Kerner method]].