Twelfth root of two

<h2 id="numerical-value">Numerical value</h2>
<p>The <a href="/facts/Nth_root/XrRjtJGe">twelfth root</a> of <a href="/facts/Two/u5WC44t9">two</a> to 20 significant figures is 1.0594630943592952646.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> Fraction approximations in increasing order of accuracy include ⁠18/17⁠, ⁠89/84⁠, ⁠196/185⁠, ⁠1657/1564⁠, and ⁠18904/17843⁠.
</p>
<h2 id="the-equal-tempered-chromatic-scale">The equal-tempered chromatic scale</h2>
<p>A <a href="/facts/Interval_(music)/aFwkkgpY">musical interval</a> is a ratio of frequencies and the <a href="/facts/Equal_temperament/8AwSblW3">equal-tempered</a> chromatic scale divides the <a href="/facts/Octave/RA4gWvkV">octave</a> (which has a ratio of 2:1) into twelve equal parts. Each note has a frequency that is 21⁄12 times that of the one below it.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
</p><p>Applying this value successively to the tones of a chromatic scale, starting from A above <a href="/facts/Middle_C/FdvRz0zL">middle C</a> (known as <a href="/facts/A440_(pitch_standard)/WIUBW1x2">A4</a>) with a frequency of 440 Hz, produces the following sequence of <a href="/facts/Pitch_(music)/fQAK0JsC">pitches</a>:
</p>
<table><tbody><tr><th>Note</th><th>Standard interval name(s)relating to A 440</th><th>Frequency (Hz)</th><th>Multiplier</th><th>Coefficient(to six decimal places)</th><th>Just intonation ratio</th><th>Difference(± <a href="/facts/Cent_(music)/lsm8TJgX">cents</a>)</th></tr><tr><td>A</td><td><a href="/facts/Unison/XnNeISEv">Unison</a></td><td>440.00</td><td>20⁄12</td><td>1.000000</td><td>1</td><td>0</td></tr><tr><td>A♯/B♭</td><td><a href="/facts/Minor_second/lVBsLYSA">Minor second/Half step/Semitone</a></td><td>466.16</td><td>21⁄12</td><td>1.059463</td><td>≈ 16⁄15</td><td>+11.73</td></tr><tr><td>B</td><td><a href="/facts/Major_second/x94d53Fo">Major second/Full step/Whole tone</a></td><td>493.88</td><td>22⁄12</td><td>1.122462</td><td>≈ 9⁄8</td><td>−3.91</td></tr><tr><td>C</td><td><a href="/facts/Minor_third/4y8zBxwY">Minor third</a></td><td>523.25</td><td>23⁄12</td><td>1.189207</td><td>≈ 6⁄5</td><td>+15.64</td></tr><tr><td>C♯/D♭</td><td><a href="/facts/Major_third/7CPRURg4">Major third</a></td><td>554.37</td><td>24⁄12</td><td><a href="/facts/Cube_root_of_two/eIHS1ybp">1.259921</a></td><td>≈ 5⁄4</td><td>−13.69</td></tr><tr><td>D</td><td><a href="/facts/Perfect_fourth/IA1GPHBw">Perfect fourth</a></td><td>587.33</td><td>25⁄12</td><td>1.334839</td><td>≈ 4⁄3</td><td>−1.96</td></tr><tr><td>D♯/E♭</td><td><a href="/facts/Tritone/fH2JqArb">Augmented fourth/Diminished fifth/Tritone</a></td><td>622.25</td><td>26⁄12</td><td><a href="/facts/Square_root_of_two/SD0sag94">1.414213</a></td><td>≈ 7⁄5</td><td>+17.49</td></tr><tr><td>E</td><td><a href="/facts/Perfect_fifth/PTIaHPuI">Perfect fifth</a></td><td>659.26</td><td>27⁄12</td><td>1.498307</td><td>≈ 3⁄2</td><td>+1.96</td></tr><tr><td>F</td><td><a href="/facts/Minor_sixth/lyMBj5By">Minor sixth</a></td><td>698.46</td><td>28⁄12</td><td>1.587401</td><td>≈ 8⁄5</td><td>+13.69</td></tr><tr><td>F♯/G♭</td><td><a href="/facts/Major_sixth/noVTP950">Major sixth</a></td><td>739.99</td><td>29⁄12</td><td>1.681792</td><td>≈ 5⁄3</td><td>−15.64</td></tr><tr><td>G</td><td><a href="/facts/Minor_seventh/Aa7YIt0e">Minor seventh</a></td><td>783.99</td><td>210⁄12</td><td>1.781797</td><td>≈ 16⁄9</td><td>+3.91</td></tr><tr><td>G♯/A♭</td><td><a href="/facts/Major_seventh/tWPXmCNB">Major seventh</a></td><td>830.61</td><td>211⁄12</td><td>1.887748</td><td>≈ 15⁄8</td><td>−11.73</td></tr><tr><td>A</td><td><a href="/facts/Octave/RA4gWvkV">Octave</a></td><td>880.00</td><td>212⁄12</td><td>2.000000</td><td>2</td><td>0</td></tr></tbody></table>
<p>The final A (A5: 880 Hz) is exactly twice the frequency of the lower A (A4: 440 Hz), that is, one octave higher.
</p>
<h3>Other tuning scales</h3>
<p>Other tuning scales use slightly different interval ratios:
</p>
<ul><li>The <a href="/facts/Just_intonation/TM91PueU">just</a> or <a href="/facts/Pythagorean_tuning/VjGlelBG">Pythagorean</a> perfect fifth is 3/2, and the difference between the equal tempered perfect fifth and the just is a <a href="/facts/Grad_(musical_interval)/9qECwYym">grad</a>, the twelfth root of the <a href="/facts/Pythagorean_comma/xsj2MvI5">Pythagorean comma</a> (
  
    
      
        
          
            
              531441
              
                /
              
              524288
            
            
              12
            
          
        
      
    
    {\textstyle {\sqrt[{12}]{531441/524288}}}
  
).</li>
<li>The equal tempered <a href="/facts/Bohlen%25E2%2580%2593Pierce_scale/D8dw0g3R">Bohlen–Pierce scale</a> uses the interval of the thirteenth root of three (
  
    
      
        
          
            3
            
              13
            
          
        
      
    
    {\textstyle {\sqrt[{13}]{3}}}
  
).</li>
<li>Stockhausen's <i><a href="/facts/Studie_II/ccbYIH58">Studie II</a></i> (1954) makes use of the twenty-fifth root of five (
  
    
      
        
          
            5
            
              25
            
          
        
      
    
    {\textstyle {\sqrt[{25}]{5}}}
  
), a compound major third divided into 5×5 parts.</li>
<li>The <a href="/facts/Delta_scale/aKqWjUTE">delta scale</a> is based on ≈
  
    
      
        
          
            
              3
              
                /
              
              2
            
            
              50
            
          
        
      
    
    {\textstyle {\sqrt[{50}]{3/2}}}
  
.</li>
<li>The <a href="/facts/Gamma_scale/D2T1cRtz">gamma scale</a> is based on ≈
  
    
      
        
          
            
              3
              
                /
              
              2
            
            
              20
            
          
        
      
    
    {\textstyle {\sqrt[{20}]{3/2}}}
  
.</li>
<li>The <a href="/facts/Beta_scale/F69V6Ujg">beta scale</a> is based on ≈
  
    
      
        
          
            
              3
              
                /
              
              2
            
            
              11
            
          
        
      
    
    {\textstyle {\sqrt[{11}]{3/2}}}
  
.</li>
<li>The <a href="/facts/Alpha_scale/4oNIeXHf">alpha scale</a> is based on ≈
  
    
      
        
          
            
              3
              
                /
              
              2
            
            
              9
            
          
        
      
    
    {\textstyle {\sqrt[{9}]{3/2}}}
  
.</li></ul>
<h2 id="pitch-adjustment">Pitch adjustment</h2>
<p class="note">See also: <a href="/facts/Audio_time_stretching_and_pitch_scaling/647HEoTb">Audio time stretching and pitch scaling</a></p>

<p>Since the frequency ratio of a semitone is close to 106% (
  
    
      
        100
        
          
            2
            
              12
            
          
        
        ≈
        105.946
      
    
    {\textstyle 100{\sqrt[{12}]{2}}\approx 105.946}
  
), increasing or decreasing the playback speed of a recording by 6% will shift the pitch up or down by about one semitone, or "half-step". Upscale <a href="/facts/Reel-to-reel_audio_tape_recording/x4MLKeVB">reel-to-reel magnetic tape recorders</a> 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 <a href="/facts/Pitch_shift/D59blj0P">pitch shifting</a> to achieve similar results, ranging from <a href="/facts/Cent_(music)/lsm8TJgX">cents</a> up to several half-steps. Reel-to-reel adjustments also affect the tempo of the recorded sound, while digital shifting does not.
</p>
<h2 id="history">History</h2>
<p>Historically this number was proposed for the first time in relationship to musical tuning in 1580 (drafted, rewritten 1610) by <a href="/facts/Simon_Stevin/Nb7y9QvD">Simon Stevin</a>.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> In 1581 Italian musician <a href="/facts/Vincenzo_Galilei/x4phV3dd">Vincenzo Galilei</a> may be the first European to suggest twelve-tone equal temperament.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> The twelfth root of two was first calculated in 1584 by the Chinese mathematician and musician <a href="/facts/Zhu_Zaiyu/rDNRDAJW">Zhu Zaiyu</a> using an abacus to reach twenty four decimal places accurately,<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> calculated circa 1605 by Flemish mathematician <a href="/facts/Simon_Stevin/Nb7y9QvD">Simon Stevin</a>,<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> in 1636 by the French mathematician <a href="/facts/Marin_Mersenne/Yklw7IGN">Marin Mersenne</a> and in 1691 by German musician <a href="/facts/Andreas_Werckmeister/Wv4J5cQr">Andreas Werckmeister</a>.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a>
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Fret/5vzMbyET">Fret</a></li>
<li><a href="/facts/Just_intonation/TM91PueU">Just intonation § Practical difficulties</a></li>
<li><a href="/facts/Music_and_mathematics/MncsslXb">Music and mathematics</a></li>
<li><a href="/facts/Piano_key_frequencies/lRaf1MAS">Piano key frequencies</a></li>
<li><a href="/facts/Scientific_pitch_notation/rq2QnnyE">Scientific pitch notation</a></li>
<li><a href="/facts/Twelve-tone_technique/hj3FwBNk">Twelve-tone technique</a></li>
<li><i><a href="/facts/The_Well-Tempered_Clavier/nJWK5Rzb">The Well-Tempered Clavier</a></i></li></ul>
<h2 id="notes">Notes</h2>

<h2 id="further-reading">Further reading</h2>
<ul><li><a href="/facts/James_Murray_Barbour/mMxKhNez">Barbour, J. M.</a> (1933). "A Sixteenth Century Chinese Approximation for π". <i><a href="/facts/American_Mathematical_Monthly/Te6CHvkg">American Mathematical Monthly</a></i>. 40 (2): 69–73. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F2300937">10.2307/2300937</a>. <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/2300937">2300937</a>.</li>
<li><a href="/facts/Alexander_John_Ellis/8U03rgzm">Ellis, Alexander</a>; <a href="/facts/Hermann_von_Helmholtz/oFISedae">Helmholtz, Hermann</a> (1954). <i><a href="/facts/On_the_Sensations_of_Tone/AWRR5wdP">On the Sensations of Tone</a></i>. Dover Publications. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-486-60753-4. {{cite book}}: ISBN / Date incompatibility (help)</li>
<li><a href="/facts/Harry_Partch/YP5kVpDh">Partch, Harry</a> (1974). <i><a href="/facts/Genesis_of_a_Music/BEcmkhsy">Genesis of a Music</a></i>. Da Capo Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-306-80106-X.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>"The smallest interval in an equal-tempered scale is the ratio 
  
    
      
        
          r
          
            n
          
        
        =
        p
      
    
    {\displaystyle r^{n}=p}
  
, so 
  
    
      
        r
        =
        
          
            p
            
              n
            
          
        
      
    
    {\displaystyle r={\sqrt[{n}]{p}}}
  
, where the ratio r divides the ratio p (= 2/1 in an octave) into n equal parts."[1] <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Sloane, N. J. A. (ed.). "Sequence A010774 (Decimal expansion of 12th root of 2)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. <a href="/wiki/Neil_Sloane" target="_blank">/wiki/Neil_Sloane</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>"Equal temperament | Definition & Facts | Britannica". www.britannica.com. Retrieved 2024-06-03. <a href="https://www.britannica.com/art/equal-temperament" target="_blank">https://www.britannica.com/art/equal-temperament</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Christensen, Thomas (2002), The Cambridge History of Western Music Theory, Cambridge University Press, p. 205, ISBN 978-0521686983 <a href="978-0521686983" target="_blank">978-0521686983</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Joseph, George Gheverghese (2010). The Crest of the Peacock: Non-European Roots of Mathematics, p.294-5. Third edition. Princeton. ISBN 9781400836369. <a href="/wiki/The_Crest_of_the_Peacock" target="_blank">/wiki/The_Crest_of_the_Peacock</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Joseph, George Gheverghese (2010). The Crest of the Peacock: Non-European Roots of Mathematics, p.294-5. Third edition. Princeton. ISBN 9781400836369. <a href="/wiki/The_Crest_of_the_Peacock" target="_blank">/wiki/The_Crest_of_the_Peacock</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Joseph, George Gheverghese (2010). The Crest of the Peacock: Non-European Roots of Mathematics, p.294-5. Third edition. Princeton. ISBN 9781400836369. <a href="/wiki/The_Crest_of_the_Peacock" target="_blank">/wiki/The_Crest_of_the_Peacock</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
<li id="fn:8"><p>Goodrich, L. Carrington (2013). A Short History of the Chinese People, [unpaginated]. Courier. ISBN 9780486169231. Cites: Chu Tsai-yü (1584). New Remarks on the Study of Resonant Tubes. <a href="https://books.google.com/books?id=ofVAAQAAQBAJ&q=%22twelfth+root+of+two%22&pg=PT182" target="_blank">https://books.google.com/books?id=ofVAAQAAQBAJ&q=%22twelfth+root+of+two%22&pg=PT182</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li>
</ol>

Twelfth root of two open-in-new

Twelfth root of two