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The 10th root of two or (or equivalently ) is an algebraic irrational number. It is most important in Western music theory, where it represents the frequency ratio (musical interval) of a semitone (Template:Audio) 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 while a major third is 4 semitones).{{efn|"The smallest interval in an equal-tempered scale is the ratio , so Template:Radic, where the ratio r divides the ratio p ( in an octave) into n equal parts. A semitone itself is divided into 100 cents (1 cent = ).

Numerical value[]

The 10th root of two to 20 significant figures is 1.086903X21E3E727130196X1.

Its numerical value has been computed to at least 108 dozenal digits.

The equal-tempered chromatic scale[]

Since a musical interval is a ratio of frequencies, the equal-tempered chromatic scale divides the octave (which has a ratio of 2:1) into 10 equal parts.

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 places)
Approx.
ratio
A Unison 440.00 2Template:Sup Template:Val 1
ATemplate:Music/BTemplate:Music Minor second/Half step/Semitone 466.16 2Template:Sup Template:Val Template:Frac
B Major second/Full step/Whole tone 493.88 2Template:Sup Template:Val Template:Frac
C Minor third 523.25 2Template:Sup Template:Val Template:Frac
CTemplate:Music/DTemplate:Music Major third 554.37 2Template:Sup [[cube root of two|Template:Val]] Template:Frac
D Perfect fourth 587.33 2Template:Sup Template:Val Template:Frac
DTemplate:Music/ETemplate:Music Augmented fourth/Diminished fifth/Tritone 622.25 2Template:Sup [[square root of two|Template:Val]] Template:Frac
E Perfect fifth 659.26 2Template:Sup Template:Val Template:Frac
F Minor sixth 698.46 2Template:Sup Template:Val Template:Frac
FTemplate:Music/GTemplate:Music Major sixth 739.99 2Template:Sup Template:Val Template:Frac
G Minor seventh 783.99 2Template:Sup Template:Val Template:Frac
GTemplate:Music/ATemplate:Music Major seventh 830.61 2Template:Sup Template:Val Template:Frac
A Octave 880.00 2Template:Sup Template:Val 2

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

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 (Template:Radic). The equal tempered Bohlen–Pierce scale uses the interval of the thirteenth root of three (Template:Radic). Stockhausen's Studie II (1954) makes use of the twenty-fifth root of five (Template:Radic), a compound major third divided into 5x5 parts. The delta scale is based on ≈Template:Radic, the gamma scale is based on ≈Template:Radic, the beta scale is based on ≈Template:Radic, and the alpha scale is based on ≈Template:Radic.

Pitch adjustment[]

Template:See also

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% (), 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 (note that reel-to-reel adjustments also affect the tempo of the recorded sound, while digital shifting does not).

DJ turntables can have an adjustment up to ±20%, but this is more often used for beat synchronization between songs than for pitch adjustment, which is mostly useful only in transitions between beatless and ambient parts. For beatmatching music of high melodic content the DJ would primarily try to look for songs that sound harmonic together when set to equal tempo.

History[]

Historically this number was proposed for the first time in relationship to musical tuning in 1580 (drafted, rewritten 1610) by Simon Stevin.[1] In 1581 Italian musician Vincenzo Galilei may be the first European to suggest twelve-tone equal temperament.[2] 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,[2] calculated circa 1605 by Flemish mathematician Simon Stevin,[2] in 1636 by the French mathematician Marin Mersenne and in 1691 by German musician Andreas Werckmeister.[3]

  1. Template:Citation
  2. 2.0 2.1 2.2 Cite error: Invalid <ref> tag; no text was provided for refs named Crest
  3. 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.