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The twelfth root of two or \sqrt 2/math> (or equivalently 2^) is an algebraic
irrational number In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...
, approximately equal to 1.0594631. It is most important in Western
music theory Music theory is the study of the practices and possibilities of music. ''The Oxford Companion to Music'' describes three interrelated uses of the term "music theory". The first is the "rudiments", that are needed to understand music notation (ke ...
, where it represents the
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
ratio In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...
( musical interval) of a
semitone A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically. It is defined as the interval between two adjacent no ...
() in
twelve-tone equal temperament Twelve-tone equal temperament (12-TET) is the musical system that divides the octave into 12 parts, all of which are equally tempered (equally spaced) on a logarithmic scale, with a ratio equal to the 12th root of 2 ( ≈ 1.05946). That resultin ...
. 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). A semitone itself is divided into 100 cents (1 cent = \sqrt 2002^).


Numerical value

The twelfth root of
two 2 (two) is a number, numeral and digit. It is the natural number following 1 and preceding 3. It is the smallest and only even prime number. Because it forms the basis of a duality, it has religious and spiritual significance in many cultur ...
to 20 significant figures is . Fraction approximations in increasing order of accuracy are , , and . , its numerical value has been computed to at least twenty billion decimal digits.


The equal-tempered chromatic scale

A musical interval is a ratio of frequencies and the
equal-tempered An equal temperament is a musical temperament or tuning system, which approximates just intervals by dividing an octave (or other interval) into equal steps. This means the ratio of the frequencies of any adjacent pair of notes is the same, w ...
chromatic scale divides the
octave In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...
(which has a ratio of 2:1) into twelve equal parts. Each note has a frequency that is 2 times that of the one below it. 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

Other tuning scales use slightly different interval ratios: * The
just Just or JUST may refer to: __NOTOC__ People * Just (surname) * Just (given name) Arts and entertainment * ''Just'', a 1998 album by Dave Lindholm * "Just" (song), a song by Radiohead * "Just", a song from the album ''Lost and Found'' by Mudvayne ...
or
Pythagorean Pythagorean, meaning of or pertaining to the ancient Ionian mathematician, philosopher, and music theorist Pythagoras, may refer to: Philosophy * Pythagoreanism, the esoteric and metaphysical beliefs purported to have been held by Pythagoras * Ne ...
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 In musical tuning, the Pythagorean comma (or ditonic comma), named after the ancient mathematician and philosopher Pythagoras, is the small interval (or comma) existing in Pythagorean tuning between two enharmonically equivalent notes such as ...
(). * The equal tempered
Bohlen–Pierce scale The Bohlen–Pierce scale (BP scale) is a musical tuning and scale, first described in the 1970s, that offers an alternative to the octave-repeating scales typical in Western and other musics, specifically the equal-tempered diatonic scale. T ...
uses the interval of the thirteenth root of three (). * Stockhausen's '' Studie II'' (1954) makes use of the twenty-fifth root of five (), a compound major third divided into 5×5 parts. * The
delta scale The δ (delta) scale is a non-octave repeating musical scale. It may be regarded as the beta scale's reciprocal, since it is "as far 'down' the ( 0 3 6 9) circle from α as β is 'up'".Taruskin, Richard (1996). ''Stravinsky and the Russian Trad ...
is based on ≈. * The
gamma scale The γ (gamma) scale is a non-octave repeating musical scale invented by Wendy Carlos while preparing ''Beauty in the Beast'' (1986) though it does not appear on the album. It is derived from approximating just intervals using multiples of a sin ...
is based on ≈. * The
beta scale The β (beta) scale is a non-octave-repeating musical scale invented by Wendy Carlos and first used on her album ''Beauty in the Beast'' (1986). It is derived from approximating just intervals using multiples of a single interval without, as is ...
is based on ≈. * The
alpha scale The α (alpha) scale is a non-octave-repeating musical scale invented by Wendy Carlos and first used on her album ''Beauty in the Beast'' (1986). It is derived from approximating just intervals using multiples of a single interval, but without r ...
is based on ≈.


Pitch adjustment

Since the frequency ratio of a semitone is close to 106% (1.05946\times100=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 shift Pitch shifting is a sound recording technique in which the original pitch of a sound is raised or lowered. Effects units that raise or lower pitch by a pre-designated musical interval ( transposition) are called pitch shifters. Pitch and tim ...
ing to achieve similar results, ranging from cents up to several half-steps (note that 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 Simon Stevin (; 1548–1620), sometimes called Stevinus, was a Flemish mathematician, scientist and music theorist. He made various contributions in many areas of science and engineering, both theoretical and practical. He also translated vario ...
. In 1581 Italian musician
Vincenzo Galilei Vincenzo Galilei (born 3 April 1520, Santa Maria a Monte, Italy died 2 July 1591, Florence, Italy) was an Italian lutenist, composer, and music theorist. His children included the astronomer and physicist Galileo Galilei and the lute virtuoso and ...
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 Zhu or ZHU may refer to: * Zhu (surname), common Chinese surnames *Zhu River, or Pearl River, in southern China *Zhu (state), ancient Chinese state, later renamed Zou *House of Zhu, the ruling house of the Ming dynasty in Chinese history *Zhu (str ...
using an abacus to reach twenty four decimal places accurately, calculated circa 1605 by Flemish mathematician
Simon Stevin Simon Stevin (; 1548–1620), sometimes called Stevinus, was a Flemish mathematician, scientist and music theorist. He made various contributions in many areas of science and engineering, both theoretical and practical. He also translated vario ...
, in 1636 by the French mathematician Marin Mersenne and in 1691 by German musician
Andreas Werckmeister Andreas Werckmeister (November 30, 1645 – October 26, 1706) was a German organist, music theorist, and composer of the Baroque era. He was amongst the earliest advocates of equal temperament, and through this advocacy was highly influential to ...
.Goodrich, L. Carrington (2013).
A Short History of the Chinese People
', . Courier. . Cites: Chu Tsai-yü (1584). ''New Remarks on the Study of Resonant Tubes''.


See also

* Fret * Just intonation § Practical difficulties *
Music and mathematics Music theory analyzes the pitch, timing, and structure of music. It uses mathematics to study elements of music such as tempo, chord progression, form, and meter. The attempt to structure and communicate new ways of composing and hearing mus ...
*
Piano key frequencies This is a list of the fundamental frequencies in hertz (cycles per second) of the keys of a modern 88-key standard or 108-key extended piano in twelve-tone equal temperament, with the 49th key, the fifth A (called A4), tuned to 440 Hz (refe ...
*
Scientific pitch notation Scientific pitch notation (SPN), also known as American standard pitch notation (ASPN) and international pitch notation (IPN), is a method of specifying musical pitch by combining a musical note name (with accidental if needed) and a number ide ...
*
Twelve-tone technique The twelve-tone technique—also known as dodecaphony, twelve-tone serialism, and (in British usage) twelve-note composition—is a method of musical composition first devised by Austrian composer Josef Matthias Hauer, who published his "law o ...
* ''
The Well-Tempered Clavier ''The Well-Tempered Clavier'', BWV 846–893, consists of two sets of preludes and fugues in all 24 major and minor keys for keyboard by Johann Sebastian Bach. In the composer's time, ''clavier'', meaning keyboard, referred to a variety of i ...
''


Notes


References


Further reading

* * * {{Irrational number Mathematical constants Algebraic numbers Irrational numbers Musical tuning