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| {{no footnotes|date=June 2011}}
| | Painting Positions Worker Nigel Golick from Barrhead, likes puppetry, new launch property singapore and train spotting. Gains encouragement through travel and just spent 9 months at Swiss Alps Jungfrau-Aletsch.<br><br>My page [http://www.quipux.org.ec/?q=node/32426 singapore new launch property] |
| The '''twelfth root of two''' or <math>\sqrt[12]{2}</math> is an [[algebraic number|algebraic]] [[irrational number]]. It is most important in [[music theory]], where it represents the [[frequency]] [[ratio]] of a [[semitone]] in [[Twelve-tone equal temperament]].
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| ==Numerical value==
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| Its value is 1.05946309435929..., which is slightly more than {{frac|18|17}} ≈ 1.0588. Better approximations are {{frac|196|185}} ≈ 1.059459 or {{frac|18904|17843}} ≈ 1.0594630948.
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| ==The equal-tempered chromatic scale==
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| Since a [[interval (music)|musical interval]] is a ratio of frequencies, the equal-tempered chromatic scale divides the [[octave]] (which has a ratio of 2:1) into [[12 (number)|twelve]] equal parts.
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| Applying this value successively to the tones of a chromatic scale, starting from '''A''' above [[middle C|middle '''C''']] with a frequency of 440 Hz, produces the following sequence of [[pitch (music)|pitch]]es:
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| {| class="wikitable" style="text-align: center"
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| ! Note<br/>
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| ! Frequency<br/> Hz
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| ! Multiplier<br/>
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| ! Coefficient<br/>(to six places)
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| |-
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| | A || 440.00 || 2{{sup|0/12}} || 1.000000
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| |-
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| | A{{music|#}}/B{{music|b}} || 466.16 || 2{{sup|1/12}} || 1.059463
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| |-
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| | B || 493.88 || 2{{sup|2/12}} || 1.122462
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| |-
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| | C || 523.25 || 2{{sup|3/12}} || 1.189207
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| |-
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| | C{{music|#}}/D{{music|b}} || 554.37 || 2{{sup|4/12}} || 1.259921
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| |-
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| | D || 587.33 || 2{{sup|5/12}} || 1.334839
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| |-
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| | D{{music|#}}/E{{music|b}} || 622.25 || 2{{sup|6/12}} || 1.414213
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| |-
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| | E || 659.26 || 2{{sup|7/12}} || 1.498307
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| |-
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| | F || 698.46 || 2{{sup|8/12}} || 1.587401
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| |-
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| | F{{music|#}}/G{{music|b}} || 739.99 || 2{{sup|9/12}} || 1.681792
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| |-
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| | G || 783.99 || 2{{sup|10/12}} || 1.781797
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| |-
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| | G{{music|#}}/A{{music|b}} || 830.61 || 2{{sup|11/12}} || 1.887748
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| |-
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| | A || 880.00 || 2{{sup|12/12}} || 2.000000
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| |}
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| The final '''A''' (880 Hz) is twice the frequency of the lower '''A''' (440 Hz), that is, one octave higher.
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| ==Pitch adjustment==
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| Since the frequency ratio of a semitone is close to 106%, increasing or decreasing the playback speed of a recording by 6% will shift the pitch up or down one semitone, or "half-step". Upscale [[reel-to-reel audio tape recording|reel-to-reel magnetic tape recorders]] typically have pitch adjustments of up to ±6%, generally used to match the playback or recording pitch to other music sources having slightly different tunings (or possibly recorded on equipment that was not running at quite the right speed). Modern recording studios utilize digital [[pitch shift]]ing to achieve the same results, ranging from [[cent (music)|cents]] up to several half-steps.
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| ==History==
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| The twelfth root of two was calculated accurately by the Chinese court [[astronomer]], [[historian]], [[physicist]] and [[mathematician]] [[Zhu Zaiyu, Prince of Zheng]] of the [[Ming Dynasty]]. In 1584, Zhu published a work 律呂精義 ''A clear explanation of that which concerns the ''律'' [equal temperament]''. Prince Zhu made note of the difference between his ideal mathematically-tuned 呂 (ancient music instrument), which gave the theoretical music instrument lengths for 12-tone equal temperament correct to 25 places, implemented with an 81-column abacus and calculated the cubic root of the square root of the square root of 2, obtaining <math>\sqrt[3]{\sqrt{\sqrt{2}}} = \sqrt [12] {2} \approx 1.059463094359295264561825,</math> which coincidentally applied a form of [[Pythagorean tuning]].
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| Calculated again in 1636 by the French mathematician [[Marin Mersenne]], and as the techniques for calculating [[logarithm]]s develop, the original approach for calculation would eventually become trivial.
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| ==See also==
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| * [[Just intonation#Historical_reasons_for_disuse|Just Intonation]]'s history of temperaments.
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| * [[Music and mathematics]]
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| * [[Piano key frequencies]]
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| * [[Scientific pitch notation]]
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| * [[Well-Tempered Clavier]]
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| * [[Musical tuning]]
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| * [[Nth root]]
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| ==Further reading==
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| * Barbour, J.M.. ''A Sixteenth Century Approximation for Pi,'' The American Mathematical Monthly, Vol. 40, no. 2, 1933. Pp. 69–73.
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| * Ellis, Alexander and Hermann Helmholtz. ''On the Sensations of Tone''. Dover Publications, 1954. ISBN 0-486-60753-4
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| * Partch, Harry. ''[[Genesis of a Music]]''. Da Capo Press, 1974. ISBN 0-306-80106-X
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| [[Category:Mathematical constants]]
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| [[Category:Algebraic numbers]]
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| [[Category:Irrational numbers]]
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| [[Category:Musical tuning]]
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Painting Positions Worker Nigel Golick from Barrhead, likes puppetry, new launch property singapore and train spotting. Gains encouragement through travel and just spent 9 months at Swiss Alps Jungfrau-Aletsch.
My page singapore new launch property