Twelfth root of two

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The twelfth root of two or <math>\sqrt[12]{2}</math> (or equivalently <math>2^{1/12}</math>) is an 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 ({{#if:Minor second on C.mid|{{#ifexist:Media:Minor second on C.mid|<phonos file="Minor second on C.mid">Play</phonos>|{{errorTemplate:Main other|Audio file "Minor second on C.mid" not found}}Template:Category handler}}}}) 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).Template:Efn A semitone itself is divided into 100 cents (1 cent = <math>\sqrt[1200]{2}=2^{1/1200}</math>).

Numerical valueEdit

The twelfth root of two to 20 significant figures is Template:Val.<ref>Template:Cite OEIS</ref> Fraction approximations in increasing order of accuracy include Template:Sfrac, Template:Sfrac, Template:Sfrac, Template:Sfrac, and Template:Sfrac.

The equal-tempered chromatic scaleEdit

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^{Template:Frac} times that of the one below it.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

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:

Note	Standard interval name(s) relating to A 440	Frequency (Hz)	Multiplier	Coefficient (to six decimal places)	Template:Abbr ratio	Template:Abbr (± cents)
A	Unison	440.00	2^{Template:Frac}	Template:Val	1	0
ATemplate:Music/BTemplate:Music	Minor second/Half step/Semitone	466.16	2^{Template:Frac}	Template:Val	≈ Template:Frac	+11.73
B	Major second/Full step/Whole tone	493.88	2^{Template:Frac}	Template:Val	≈ Template:Frac	−3.91
C	Minor third	523.25	2^{Template:Frac}	Template:Val	≈ Template:Frac	+15.64
CTemplate:Music/DTemplate:Music	Major third	554.37	2^{Template:Frac}	[[cube root of two#In music theory\|Template:Val]]	≈ Template:Frac	−13.69
D	Perfect fourth	587.33	2^{Template:Frac}	Template:Val	≈ Template:Frac	−1.96
DTemplate:Music/ETemplate:Music	Augmented fourth/Diminished fifth/Tritone	622.25	2^{Template:Frac}	[[square root of two\|Template:Val]]	≈ Template:Frac	+17.49
E	Perfect fifth	659.26	2^{Template:Frac}	Template:Val	≈ Template:Frac	+1.96
F	Minor sixth	698.46	2^{Template:Frac}	Template:Val	≈ Template:Frac	+13.69
FTemplate:Music/GTemplate:Music	Major sixth	739.99	2^{Template:Frac}	Template:Val	≈ Template:Frac	−15.64
G	Minor seventh	783.99	2^{Template:Frac}	Template:Val	≈ Template:Frac	+3.91
GTemplate:Music/ATemplate:Music	Major seventh	830.61	2^{Template:Frac}	Template:Val	≈ Template:Frac	−11.73
A	Octave	880.00	2^{Template:Frac}	Template:Val	2	0

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 scalesEdit

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 (<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 adjustmentEdit

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File:Monochord ET.png

One octave of 12-tet on a monochord (linear)

File:Pitch class space star.svg

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 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.

HistoryEdit

Historically this number was proposed for the first time in relationship to musical tuning in 1580 (drafted, rewritten 1610) by Simon Stevin.<ref>Template:Citation</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 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). A Short History of the Chinese People, Template:Unpaginated. Courier. Template:ISBN. Cites: Chu Tsai-yü (1584). New Remarks on the Study of Resonant Tubes.</ref>

NotesEdit

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ReferencesEdit

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