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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
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| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| | 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''. |
| : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-11-19 17:25:15 UTC</tt>.<br>
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| : The original revision id was <tt>181280597</tt>.<br>
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| : The revision comment was: <tt></tt><br>
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| The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
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| <h4>Original Wikitext content:</h4> | |
| <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">A [[http://mathworld.wolfram.com/UnivariatePolynomial.html|univariate polynomial]] a0x^n + a1x^(n-1) + ... + an whose coefficients ai are integers (or equivalently, rational numbers) has //roots// which are known as [[http://en.wikipedia.org/wiki/Algebraic_number|algebraic numbers]]. A root is a value r for which the [[http://en.wikipedia.org/wiki/Polynomial_function#Polynomial_functions|polynomial function]] f(x) = a0x^n + a1x^(n-1) + ... + a0 satisfies f(r) = 0. If r is a [[http://Real%20number|real number]], it is a //real algebraic number//. | |
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| 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^(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 [[http://en.wikipedia.org/wiki/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 {{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. |
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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 [[http://en.wikipedia.org/wiki/Newton%27s_method|Newton's method]] can be used. A refinement of Newton's method is the [[http://en.wikipedia.org/wiki/Durand%E2%80%93Kerner_method|Durand–Kerner method]].</pre></div> | | 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}}. |
| <h4>Original HTML content:</h4>
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| <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Algebraic number</title></head><body>A <a class="wiki_link_ext" href="http://mathworld.wolfram.com/UnivariatePolynomial.html" rel="nofollow">univariate polynomial</a> a0x^n + a1x^(n-1) + ... + an whose coefficients ai are integers (or equivalently, rational numbers) has <em>roots</em> which are known as <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Algebraic_number" rel="nofollow">algebraic numbers</a>. A root is a value r for which the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Polynomial_function#Polynomial_functions" rel="nofollow">polynomial function</a> f(x) = a0x^n + a1x^(n-1) + ... + a0 satisfies f(r) = 0. If r is a <a class="wiki_link_ext" href="http://Real%20number" rel="nofollow">real number</a>, it is a <em>real algebraic number</em>. <br />
| | [[Category:Math]] |
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| | [[Category:Number theory]] |
| Real algebraic numbers which are not also rational numbers turn up in various places in musical tuning theory. For instance, the intervals in the <a class="wiki_link" href="/Target%20tunings">target tunings</a> 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. <a class="wiki_link" href="/Generators">Generators</a> for <a class="wiki_link" href="/linear%20temperaments">linear temperaments</a> which are real algebraic numbers can have interesting properties in terms of the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Combination_tone" rel="nofollow">combination tones</a> they produce. <br />
| | [[Category:todo:increase applicability]] |
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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 <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Newton%27s_method" rel="nofollow">Newton's method</a> can be used. A refinement of Newton's method is the <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Durand%E2%80%93Kerner_method" rel="nofollow">Durand–Kerner method</a>.</body></html></pre></div>
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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.
