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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | {{Wikipedia}} |
| 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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