Editing Twelfth root of two

{{Short description|Algebraic irrational number}}
{{multiple image|caption_align=center|header_align=center
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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>).

==Numerical value==
The [[nth root|twelfth root]] of [[two]] to 20 significant figures is {{val|1.0594630943592952646}}.<ref>{{Cite OEIS|A010774|Decimal expansion of 12th root of 2}}</ref> Fraction approximations in increasing order of accuracy include {{sfrac|18|17}}, {{sfrac|89|84}}, {{sfrac|196|185}}, {{sfrac|1657|1564}}, and {{sfrac|18904|17843}}.

==The equal-tempered chromatic scale==

A [[interval (music)|musical interval]] is a ratio of frequencies and the [[Equal temperament|equal-tempered]] chromatic scale divides the [[octave]] (which has a ratio of 2:1) into twelve equal parts. Each note has a frequency that is 2{{sup|{{frac|1|12}}}} times that of the one below it.<ref>{{Cite web |title=Equal temperament {{!}} Definition & Facts {{!}} Britannica |url=https://www.britannica.com/art/equal-temperament |access-date=2024-06-03 |website=www.britannica.com |language=en}}</ref>

Applying this value successively to the tones of a chromatic scale, starting from '''A''' above [[middle C|middle '''C''']] (known as [[A440 (pitch standard)|A<sub>4</sub>]]) with a frequency of 440&nbsp;Hz, produces the following sequence of [[pitch (music)|pitch]]es:
{| class="wikitable" style="text-align: center;"
! Note
! Standard interval name(s)<br />relating to A 440
! Frequency<br /> (Hz)
! Multiplier
! Coefficient<br />(to six decimal places)
! {{abbr|Just intonation|for comparison}}<br /> ratio
!{{abbr|Difference|between equal-tempered scale and just intonation}}<br />(± [[Cent_(music)|cents]])
|-
| A     || [[Unison]] || 440.00 || 2{{sup|{{frac|0|12}}}} || {{val|1.000000}}
|1
|align=right|0
|-
| A{{music|#}}/B{{music|b}} || [[Minor second|Minor second/Half step/Semitone]] || 466.16 || 2{{sup|{{frac|1|12}}}} || {{val|1.059463}}
|≈ {{frac|16|15}} <!-- +1.2% -->
|align=right|+11.73
|-
| B     || [[Major second|Major second/Full step/Whole tone]] || 493.88 || 2{{sup|{{frac|2|12}}}} || {{val|1.122462}}
|≈ {{frac|9|8}} <!-- +0.3% -->
|align=right| −3.91
|-
| C     || [[Minor third]] || 523.25 || 2{{sup|{{frac|3|12}}}} || {{val|1.189207}}
|≈ {{frac|6|5}} <!-- +1.1% -->
|align=right| +15.64
|-
| C{{music|#}}/D{{music|b}} || [[Major third]] || 554.37 || 2{{sup|{{frac|4|12}}}} || [[cube root of two#In music theory|{{val|1.259921}}]]
|≈ {{frac|5|4}} <!-- -1.0% -->
|align=right| −13.69
|-
| D     || [[Perfect fourth]] || 587.33 || 2{{sup|{{frac|5|12}}}} || {{val|1.334839}}
|≈ {{frac|4|3}} <!-- -0.2% -->
|align=right| −1.96
|-
| D{{music|#}}/E{{music|b}} || [[Tritone|Augmented fourth/Diminished fifth/Tritone]] || 622.25 || 2{{sup|{{frac|6|12}}}} || [[square root of two|{{val|1.414213}}]]
|≈ {{frac|7|5}} <!-- -1.4% -->
|align=right| +17.49
|-
| E     || [[Perfect fifth]] || 659.26 || 2{{sup|{{frac|7|12}}}} || {{val|1.498307}}
|≈ {{frac|3|2}} <!-- +0.2% -->
|align=right| +1.96
|-
| F     || [[Minor sixth]] || 698.46 || 2{{sup|{{frac|8|12}}}} || {{val|1.587401}}
|≈ {{frac|8|5}} <!-- +1.3% -->
|align=right| +13.69
|-
| F{{music|#}}/G{{music|b}} || [[Major sixth]] || 739.99 || 2{{sup|{{frac|9|12}}}} || {{val|1.681792}}
|≈ {{frac|5|3}} <!-- +1.5% -->
|align=right| −15.64
|-
| G     || [[Minor seventh]] || 783.99 || 2{{sup|{{frac|10|12}}}} || {{val|1.781797}}
|≈ {{frac|16|9}} <!-- +1.8% -->
|align=right| +3.91
|-
| G{{music|#}}/A{{music|b}} || [[Major seventh]] || 830.61 || 2{{sup|{{frac|11|12}}}} || {{val|1.887748}}
|≈ {{frac|15|8}} <!-- -1.3% -->
|align=right| −11.73
|-
| A     || [[Octave]] || 880.00 || 2{{sup|{{frac|12|12}}}} || {{val|2.000000}}
|2
|align=right|0
|}

The final '''A''' (A<sub>5</sub>: 880&nbsp;Hz) is exactly twice the frequency of the lower '''A''' (A<sub>4</sub>: 440&nbsp;Hz), that is, one octave higher.

===Other tuning scales===
Other tuning scales use slightly different interval ratios:
* The [[Just intonation|just]] or [[Pythagorean tuning|Pythagorean]] perfect fifth is 3/2, and the difference between the equal tempered perfect fifth and the just is a [[grad (musical interval)|grad]], the twelfth root of the [[Pythagorean comma]] (<math display=inline>\sqrt[12]{531441/524288}</math>).
* The equal tempered [[Bohlen–Pierce scale]] uses the interval of the thirteenth root of three (<math display=inline>\sqrt[13]{3}</math>).
* Stockhausen's ''[[Studie II]]'' (1954) makes use of the twenty-fifth root of five (<math display=inline>\sqrt[25]{5}</math>), a compound major third divided into 5×5 parts.
* The [[delta scale]] is based on ≈<math display=inline>\sqrt[50]{3/2}</math>.
* The [[gamma scale]] is based on ≈<math display=inline>\sqrt[20]{3/2}</math>.
* The [[beta scale]] is based on ≈<math display=inline>\sqrt[11]{3/2}</math>.
* The [[alpha scale]] is based on ≈<math display=inline>\sqrt[9]{3/2}</math>.

==Pitch adjustment==
{{See also|Audio time stretching and pitch scaling}}
[[File:Monochord ET.png|thumb|One octave of 12-tet on a monochord (linear)]]
[[File:Pitch class space star.svg|thumb|The [[chromatic circle]] depicts equal distances between notes (logarithmic)]]

Since the frequency ratio of a semitone is close to 106% (<math display=inline>100\sqrt[12]{2} \approx 105.946</math>), 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 [[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 similar results, ranging from [[cent (music)|cents]] up to several half-steps. Reel-to-reel adjustments also affect the tempo of the recorded sound, while digital shifting does not.

==History==

Historically this number was proposed for the first time in relationship to musical tuning in 1580 (drafted, rewritten 1610) by [[Simon Stevin]].<ref>{{citation|first=Thomas|last=Christensen|title=The Cambridge History of Western Music Theory|year=2002|page=[https://archive.org/details/cambridgehistory0000unse_t8n5/page/205 205]|publisher=Cambridge University Press |isbn=978-0521686983|url=https://archive.org/details/cambridgehistory0000unse_t8n5/page/205}}</ref> In 1581 Italian musician [[Vincenzo Galilei]] may be the first European to suggest twelve-tone equal temperament.<ref name="Crest"/> The twelfth root of two was first calculated in 1584 by the Chinese mathematician and musician [[Zhu Zaiyu]] using an abacus to reach twenty four decimal places accurately,<ref name="Crest"/> calculated<!--less accurately--> circa 1605 by Flemish mathematician [[Simon Stevin]],<ref name="Crest"/> in 1636 by the French mathematician [[Marin Mersenne]] and in 1691 by German musician [[Andreas Werckmeister]].<ref>Goodrich, L. Carrington (2013). ''[https://books.google.com/books?id=ofVAAQAAQBAJ&q=%22twelfth+root+of+two%22&pg=PT182 A Short History of the Chinese People]'', {{unpaginated}}. Courier. {{ISBN|9780486169231}}. Cites: Chu Tsai-yü (1584). ''New Remarks on the Study of Resonant Tubes''.</ref>

==See also==
* [[Fret]]
* [[Just intonation#Practical difficulties|Just intonation § Practical difficulties]]
* [[Music and mathematics]]
* [[Piano key frequencies]]
* [[Scientific pitch notation]]
* [[Twelve-tone technique]]
* ''[[The Well-Tempered Clavier]]''

==Notes==
{{notelist}}

==References==
{{Reflist}}

==Further reading==
* {{cite journal |last=Barbour |first=J. M. |author-link=James Murray Barbour |title=A Sixteenth Century Chinese Approximation for {{pi}} |journal=[[American Mathematical Monthly]] |volume=40 |issue=2 |year=1933 |pages=69–73 |jstor=2300937 | doi = 10.2307/2300937}}
* {{cite book |last1=Ellis |first1=Alexander |author-link=Alexander John Ellis |first2=Hermann |last2=Helmholtz |author-link2=Hermann von Helmholtz |title=[[On the Sensations of Tone]] |publisher=Dover Publications |year=1954 |isbn=0-486-60753-4 }}
* {{cite book |last=Partch |first=Harry |author-link=Harry Partch |title=[[Genesis of a Music]] |publisher=Da Capo Press |year=1974 |isbn=0-306-80106-X }}

{{Algebraic numbers}}
{{Irrational number}}

[[Category:Mathematical constants]]
[[Category:Algebraic numbers]]
[[Category:Irrational numbers]]
[[Category:Musical tuning]]