Cent (music)

The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each. Typically, cents are used to measure extremely small finite intervals, or to compare the sizes of comparable intervals in different tuning systems, and in fact the interval of one cent is much too small to be heard between successive notes.

Alexander J. Ellis based the measure on the acoustic logarithms decimal semitone system developed by Gaspard de Prony in the 1830s, at Robert Holford Macdowell Bosanquet's suggestion. Ellis made extensive measurements of musical instruments from around the world, using cents extensively to report and compare the scales employed,^[1] and further described and employed the system in his edition of Hermann von Helmholtz's On the Sensations of Tone. It has become the standard method of representing and comparing musical pitches and intervals with relative accuracy.

Use

Comparison of equal-tempered (grey) and Pythagorean (blue) intervals showing the relationship between frequency ratio and the intervals' values, in cents. The curve shown is the plot of the equation, at left.

Like a decibel's relation to intensity, a cent is a ratio between two close frequencies. For the ratio to remain constant over the frequency spectrum, the frequency range encompassed by a cent must be proportional to the two frequencies. An equally tempered semitone (the interval between two adjacent piano keys) decidedly spans 100 cents. An octave -- two notes that have a fequency ratio of 2:1 -- spans twelve semitones and therefore 1200 cents. Since a frequency raised by one cent is simply multiplied by this constant cent value, and 1200 cents doubles a frequency, the ratio of frequencies one cent apart is precisely equal to 2^1/1200, the 1200th root of 2, which is approximately 1.0005777895.

If one knows the frequencies a and b of two notes, the number of cents measuring the interval from a to b may be calculated by the following formula (similar to the definition of decibel):

$n = 1200 \cdot \log_2 \left( \frac{b}{a} \right) \approx 3986 \cdot \log_{10} \left( \frac{b}{a} \right)$

Likewise, if one knows a note a and the number n of cents in the interval from a to b, then b may be calculated by:

$b = a \times 2 ^ {n/1200}$

To compare different tuning systems, convert the various interval sizes into cents. For example, in just intonation the major third is represented by the frequency ratio 5:4. Applying the formula at the top shows this to be about 386 cents. The equivalent interval on the equal-tempered piano would be 400 cents. The difference, 14 cents, is about a seventh of a half step, easily audible. The just noticeable difference for this unit is about 6 cents.

Piecewise linear approximation

As x increases from 0 to 1/12, the function 2^x increases almost linearly from 1.00000 to 1.05946. The exponential cent scale can therefore be accurately approximated as a piecewise linear function which is exact at semitones and whose maximum error occurs halfway between semitones. That is, n cents for n from 0 to 100 may be approximated as 1+.0005946n instead of 2^n/1200. The error is zero when n is 0 or 100, and is about 0.72 cents high when n is 50, where the correct value of 2^1/24 = 1.02930 is approximated by 1 + .0005946*50 = 1.02973. This error is well below anything humanly audible, making this piecewise linear approximation suitable for most practical purposes while being conceptually and numerically easier to work with in many situations.

Human perception

It is difficult to establish how many cents are perceptible to humans; this accuracy varies greatly from person to person. One author stated that humans can distinguish a difference in pitch of about 5-6 cents.^[2] The threshold of what is perceptible, technically known as the just noticeable difference, also varies as a function of the timbre of the pitch: in one study, changes in tone quality reduced student musicians' ability to recognize as out-of-tune pitches that deviated from their appropriate values by ±12 cents.^[3] It has also been established that increased tonal context enables listeners to judge pitch more accurately.^[4]

When listening to pitches with vibrato, there is evidence that humans perceive the mean frequency as the center of the pitch.^[5] One study of vibrato in western vocal music found a variation in cents of vibrato typically ranged between ±34 cents and ±123 cents, with a mean variation of ±71 cents; the variation was much higher on Verdi opera arias.^[6]

Normal adults are able to recognize pitch differences of as small as 25 cents very reliably. Adults with amusia, however, have trouble recognizing differences of less than 100 cents and sometimes have trouble with these or larger intervals.^[7]

Sound files

The following audio files play various intervals. In each case the first note played is middle C. The next note is sharper than C by the assigned value in cents. Finally, the two notes are played simultaneously.

Played separately, the notes may not show an audible difference, but when they are played together, a beat may be heard. At any particular instant, the two waveforms reinforce or cancel each other more or less, depending on their instantaneous phase relationship. A piano tuner may verify tuning accuracy by timing the beats when two strings are sounded at once.

See also

Music portal

• Interval (music)
• Musical tuning
• Savart

References

Footnotes

1. ^ Alexander Ellis: On the Musical Scales of Various Nations HTML transcription of the 1885 article in the Journal of the Society of Arts (Accessed September 2008)
2. ^ http://etd.gatech.edu/theses/available/etd-04102006-142310/ D.B. Loeffler, Instrument Timbres and Pitch Estimation in Polyphonic Music. Master's Thesis, Department of Electrical and Computer Engineering, Georgia Tech. April (2006)
3. ^ http://links.jstor.org/sici?sici=0022-4294%28199922%2947%3A2%3C135%3AEOTCOI%3E2.0.CO%3B2-9 J. M. Geringer; M.D. Worthy, Effects of Tone-Quality Changes on Intonation and Tone-Quality Ratings of High School and College Instrumentalists, Journal of Research in Music Education, Vol. 47, No. 2. (Summer, 1999), pp. 135-149.
4. ^ C.M. Warrier; R.J. Zatorre (February 2002), "Influence of tonal context and timbral variation on perception of pitch" (PDF), Perception & Psychophysics 64 (2): 198–207, doi:10.3758/BF03195786, http://www.zlab.mcgill.ca/docs/Warrier_and_Zatorre_2002.pdf, retrieved 2008-09-27 
5. ^ J.C. Brown; K.V. Vaughn (September 1996), "Pitch Center of Stringed Instrument Vibrato Tones" (PDF), Journal of the Acoustical Society of America 100 (3): 1728–1735, doi:10.1121/1.416070, PMID 8817899, http://www.wellesley.edu/Physics/brown/pubs/vibPerF100P1728-P1735.pdf, retrieved 2008-09-28 
6. ^ E. Prame (July 1997), "Vibrato extent and intonation in professional Western lyric singing", The Journal of the Acoustical Society of America 102 (1): 616–621, doi:10.1121/1.419735, http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JASMAN000102000001000616000001&idtype=cvips&gifs=yes 
7. ^ I. Peretz; K.L. Hyde (August 2003), "What is specific to music processing? Insights from congenital amusia" (PDF), Trends in Cognitive Sciences 7 (8): 362–367, doi:10.1016/S1364-6613(03)00150-5, PMID 12907232, http://www.brams.umontreal.ca/plab/downloads/PeretzHyde03.pdf, retrieved 2008-09-27 

Notations

• Ellis, Alexander J.; Alfred J. Hipkins (1884), "Tonometrical Observations on Some Existing Non-Harmonic Musical Scales", Proceedings of the Royal Society of London 37 (232–234): 368–385, doi:10.1098/rspl.1884.0041, JSTOR 114325. 

External links

• Cent conversion: Whole number ratio to cent
• Cent conversion: Online utility with several functions