Cent (music): Difference between revisions
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The '''cent''' is a logarithmic measure of a musical interval introduced by Alexander Ellis. A cent is the logarithmic division of the equitempered semitone into 100 equal parts. In terms of a formula, the separation or interval between two frequencies ƒ<sub>1</sub> and ƒ<sub>2</sub> in ''cents'' is determined as: | {{TOC|right}} | ||
The '''cent''' is a logarithmic measure of a musical interval introduced by Alexander Ellis. A cent is the logarithmic division of the equitempered semitone into 100 equal parts. | |||
==Formula== | |||
In terms of a formula, the separation or interval between two frequencies ƒ<sub>1</sub> and ƒ<sub>2</sub> in ''cents'' is determined as: | |||
:<math> c = 1200 \log _2 \left( \frac {f_1}{f_2} \right) \ . </math> | :<math> c = 1200 \log _2 \left( \frac {f_1}{f_2} \right) \ . </math> | ||
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that is, by a ratio given by the 1200th root of 2. | that is, by a ratio given by the 1200th root of 2. | ||
==Background== | |||
The ''cent'' appears in an article Alexander Ellis published in 1885<ref name=tune/> and also in an appendix he added to his translation of [[Herman von Helmholtz]]'s ''Die Lehre von den Tonempfindungen'' published in translation as ''On the Sensation of Tone As a Physiological Basis for the Theory of Music'',<ref name=Ellis/> and also as ''On the sensations of tone''.<ref name=sensations/> | |||
{{Image|Tuning error.png|right|250px|Average error of nine flutists in playing a note of specified pitch after listening to a pitch ''A''4 <nowiki>=</nowiki> 442 Hz from a piano.<ref name=Ohgushi/>}} | {{Image|Tuning error.png|right|250px|Average error of nine flutists in playing a note of specified pitch after listening to a pitch ''A''4 <nowiki>=</nowiki> 442 Hz from a piano.<ref name=Ohgushi/>}} | ||
==Sensitivity of the ear== | |||
According to Ellis, when two notes are played together, a difference of 2 cents is noticeable, and a difference of 5 cents is heard as out of tune.<ref name=tune/> | |||
Recent observations suggest errors of 5-15 cents when playing a specific pitch are common, with errors of 20-50 cents for pitches above ''A''7 (the 7th octave, 3 octaves above the octave containing middle ''C''). (See figure at right.) The increased error at higher pitch was traced to a systematic error in the response of auditory nerves in the ear.<ref name=Ohgushi/> | |||
==References== | ==References== | ||
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<ref name=Ellis> | <ref name=Ellis> | ||
{{cite book |title=On the Sensation of Tone As a Physiological Basis for the Theory of Music |author=Herman von Helmholtz |url= http://books.google.com/books?id=wY2fAAAAMAAJ&pg=PA41 |edition=Alexander Ellis translation of 4th German ed |chapter=Footnote, p. 41 and Appendix XX, Section C|year=1912 |publisher=Longmans, Green}} | {{cite book |title=On the Sensation of Tone As a Physiological Basis for the Theory of Music |author=Herman von Helmholtz |url= http://books.google.com/books?id=wY2fAAAAMAAJ&pg=PA41 |edition=Alexander Ellis translation of 4th 1877 German ed |chapter=Footnote, p. 41 and Appendix XX, Section C|year=1912 |publisher=Longmans, Green}} | ||
</ref> | </ref> | ||
Revision as of 12:11, 14 July 2012
The cent is a logarithmic measure of a musical interval introduced by Alexander Ellis. A cent is the logarithmic division of the equitempered semitone into 100 equal parts.
Formula
In terms of a formula, the separation or interval between two frequencies ƒ1 and ƒ2 in cents is determined as:
Consequently, two frequencies ƒ1 and ƒ2 separated by an interval of 1 cent are in the ratio:
that is, by a ratio given by the 1200th root of 2.
Background
The cent appears in an article Alexander Ellis published in 1885[1] and also in an appendix he added to his translation of Herman von Helmholtz's Die Lehre von den Tonempfindungen published in translation as On the Sensation of Tone As a Physiological Basis for the Theory of Music,[2] and also as On the sensations of tone.[3]
(PD) Image: John R Brews
Average error of nine flutists in playing a note of specified pitch after listening to a pitch A4 = 442 Hz from a piano.[4]
Average error of nine flutists in playing a note of specified pitch after listening to a pitch A4 = 442 Hz from a piano.[4]
Sensitivity of the ear
According to Ellis, when two notes are played together, a difference of 2 cents is noticeable, and a difference of 5 cents is heard as out of tune.[1]
Recent observations suggest errors of 5-15 cents when playing a specific pitch are common, with errors of 20-50 cents for pitches above A7 (the 7th octave, 3 octaves above the octave containing middle C). (See figure at right.) The increased error at higher pitch was traced to a systematic error in the response of auditory nerves in the ear.[4]
References
- ↑ 1.0 1.1 Alexander J Ellis (March 25, 1885). "On the musical scales of various nations; §III.–Cents". Journal of the Society of Arts 33: p. 487.
- ↑ Herman von Helmholtz (1912). “Footnote, p. 41 and Appendix XX, Section C”, On the Sensation of Tone As a Physiological Basis for the Theory of Music, Alexander Ellis translation of 4th 1877 German ed. Longmans, Green.
- ↑ Herman von Helmholtz (1954). On the sensations of tone, Reprint of 1885 translation by Alexander Ellis. Courier Dover Publications. ISBN 0486607534.
- ↑ 4.0 4.1 Ohgushi, K and Ano, Y (2005). "The Relationship between Musical Pitch and Temporal Responses of the Auditory Nerve Fibers". Journal of Physiological Anthropology and Applied Human Science 24 (1): pp. 99-101.