Editing Twelfth root of two (section)

{{Short description|Algebraic irrational number}}
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 | image1 = 4Octaves.and.Frequencies.svg
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 | caption1 = Octaves (12 semitones) increase exponentially when measured on a linear frequency scale (Hz).
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 | caption2  = Octaves are equally spaced when measured on a logarithmic scale (cents).
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The '''twelfth root of two''' or <math>\sqrt[12]{2}</math> (or [[Nth root#Identities and properties|equivalently]] <math>2^{1/12}</math>) is an [[algebraic number|algebraic]] [[irrational number]], approximately equal to 1.0594631. It is most important in Western [[music theory]], where it represents the [[frequency]] [[ratio]] ([[musical interval]]) of a [[semitone]] ({{audio|Minor second on C.mid|Play}}) in [[twelve-tone equal temperament]]. This number was proposed for the first time in relationship to [[musical tuning]] in the sixteenth and seventeenth centuries. It allows measurement and comparison of different intervals (frequency ratios) as consisting of different numbers of a single interval, the equal tempered semitone (for example, a minor third is 3 semitones, a major third is 4 semitones, and perfect fifth is 7 semitones).{{efn|"The smallest interval in an equal-tempered scale is the ratio <math>r^n=p</math>, so <math>r=\sqrt[n]p</math>, where the ratio ''r'' divides the ratio ''p'' ({{=}} 2/1 in an octave) into ''n'' equal parts."<ref name="Crest">Joseph, George Gheverghese (2010). ''[[The Crest of the Peacock]]: Non-European Roots of Mathematics'', p.294-5. Third edition. Princeton. {{ISBN|9781400836369}}.</ref>}} A semitone itself is divided into 100 [[cent (music)|cents]] (1 cent = <math>\sqrt[1200]{2}=2^{1/1200}</math>).