Twelfth root of two

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

A musical interval is a ratio of frequencies and the 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.{{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}}

Applying this value successively to the tones of a chromatic scale, starting from A above middle C (known as A₄) with a frequency of 440 Hz, produces the following sequence of pitches:

The final A (A₅: 880 Hz) is exactly twice the frequency of the lower A (A₄: 440 Hz), that is, one octave higher.

Other tuning scales use slightly different interval ratios:

• The just or Pythagorean perfect fifth is 3/2, and the difference between the equal tempered perfect fifth and the just is a grad, the twelfth root of the Pythagorean comma ($\backslash sqrt[12]\{531441/524288\}$).
• The equal tempered Bohlen–Pierce scale uses the interval of the thirteenth root of three ($\backslash sqrt[13]\{3\}$).
• Stockhausen's Studie II (1954) makes use of the twenty-fifth root of five ($\backslash sqrt[25]\{5\}$), a compound major third divided into 5×5 parts.
• The delta scale is based on ≈$\backslash sqrt[50]\{3/2\}$.
• The gamma scale is based on ≈$\backslash sqrt[20]\{3/2\}$.
• The beta scale is based on ≈$\backslash sqrt[11]\{3/2\}$.
• The alpha scale is based on ≈$\backslash sqrt[9]\{3/2\}$.

{{See also|Audio time stretching and pitch scaling}}

File:Monochord ET.png

File:Pitch class space star.svg depicts equal distances between notes (logarithmic)]]

Since the frequency ratio of a semitone is close to 106% ($100\backslash sqrt[12]\{2\}\; \backslash approx\; 105.946$), 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 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 shifting to achieve similar results, ranging from cents up to several half-steps. Reel-to-reel adjustments also affect the tempo of the recorded sound, while digital shifting does not.

Historically this number was proposed for the first time in relationship to musical tuning in 1580 (drafted, rewritten 1610) by Simon Stevin.{{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}} In 1581 Italian musician Vincenzo Galilei may be the first European to suggest twelve-tone equal temperament. 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, calculated circa 1605 by Flemish mathematician Simon Stevin, in 1636 by the French mathematician Marin Mersenne and in 1691 by German musician Andreas Werckmeister.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.

• Fret
• Just intonation § Practical difficulties
• Music and mathematics
• Piano key frequencies
• Scientific pitch notation
• Twelve-tone technique
• The Well-Tempered Clavier

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{{Reflist}}

• {{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