Cent (music)

The cent is a logarithmic unit of measure used for musical intervals. Typically cents are used to measure extremely small 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, and introduced it 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

1,200 cents are equal to one octave — a frequency ratio of 2:1 — and an equally tempered semitone (the interval between two adjacent piano keys) is equal to 100 cents. This means that a cent is precisely equal to 2^1/1200, the 1,200th root of 2, which is approximately 1.0005777895065548592967925757932, or Failed to parse (Missing texvc executable; please see math/README to configure.): \tfrac{1}{17.3}

of one percent.

If you know the frequencies a and b of two notes, the number of cents measuring the interval between them may be calculated by the following formula (similar to the definition of decibel both formally as well as in its purpose to linearize a physical unit which is exponential but perceived logarithmically by humans):

Failed to parse (Missing texvc executable; please see math/README to configure.): n = 1200 \log_2 \left( \frac{a}{b} \right) \approx 3986 \log_{10} \left( \frac{a}{b} \right)

Likewise, if you know a note b and the number n of cents in the interval, then the other note a may be calculated by:

Failed to parse (Missing texvc executable; please see math/README to configure.): a = b \times 2 ^ \frac{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.

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.^[1] The threshold of what is perceptible also varies as a function of the timbre of the pitch: in one study, changes in tone quality negatively impacted student musicians' ability to recognize as out-of-tune pitches that deviated from their appropriate values by +/- 12 cents.^[2] It has also been established that increased tonal context enables listeners to judge pitch more accurately.^[3]

When listening to pitches with vibrato, there is evidence that humans perceive the mean frequency as the center of the pitch.^[4] 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.^[5]

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.^[6]

Sound files

The following .ogg files play various cents intervals. In each case the first note played is middle C. The next note a C which is sharper by the assigned cents value. Finally the interval is played.

The notes may not have had a perceivable difference, but when played together, they are comparably out of phase. The sum of the two waveforms either adds to or diminishes from the loudness of the sound wave; there are instances when the waves are of equal magnitude in opposite directions, canceling each other out and producing moments of "silence". A given note on the piano is tuned not only to pitch, but also so that the strings are in phase with one another. A piano tuner may verify this by playing that note against a common interval (4th, 5th, octave).

References

• Ellis, Alexander J.; Alfred J. Hipkins (1884). "Tonometrical Observations on Some Existing Non-Harmonic Musical Scales". Proceedings of the Royal Society of London 37: 368–385. 

See also

• Interval (music)
• Musical tuning

External links

• Cent conversion: Frequency ratio to cents and cents to frequency ratio
• Cent conversion: Whole number ratio to cent
• How to convert a ratio to centscs:Cent (hudba)

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