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CentsMusical intervals are often expressed in cents, a unit of pitch based upon the equaltempered octave such that one equal tempered semitone is equal to 100 cents. An octave

is then 1200¢ and the other equal tempered intervals can be obtained by addingsemitones:

Calculating CentsThe fact that one octave is equal to 1200 cents leads one to the power of 2 relationship:

This is convenient for calculating the frequency corresponding to a certain number ofcents. To calculate the number of cents for any two frequencies, the above relationshipmust be reversed. Taking the log of both sides gives:

Advantages of Cents NotationExamining the semitone A to B-flat at different points in the range of the piano willillustrate the fact that if expressed in cents, every equal tempered semitone is the same.Expressed in Hz difference, every semitone is different. The interval value in centsexpresses the ratio of the frequencies, which is the same for every equal temperedsemitone.

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Included with the semitone intervals above is an evaluation of the deviation in Hzneeded to equal 5¢, the nomina l just-noticeable difference for these pitches. Note thatthe range represented by 5¢ increases from less than a tenth of a Hz at the low end ofthe piano to about 10 Hz at the top end of the piano.

Just Noticeable Difference in PitchThe just noticeable difference in pitch must be expressed as a ratio o r musical interval since the human ear tends to respond equally to equal ratios of frequencies. It isconvenient to express the just noticeable difference in cents since that notation wasdeveloped to express musical intervals. Although research reveals variations , areasonable estimate of the JND is about five cents. One of the advantages of the centsnotation is that it expresses the same musical interval, regardless of the frequency range.

You can hear abouta nickel's worth of

difference

Pitch Details Related to CentsEvaluating the just noticeable difference in pitch by the "nickel's worth" rule isconvenient, but as you might expect it is an oversimplification. Rossing describesmeasurements of pitch discrimination with pure tones at about 80 dB for frequencies

between 1 and 4 kHz. The jnd is found to be about 0.5% of the pure tone frequency,which corresponds to about 8¢ . He also states that the jnd has been found to dependupon the frequency, the sound level, the duration of the tone, the suddenness of the

frequency change, the musical training of the listener, and the method of measurement.The real world can rarely be accurately characterized by simple rules.

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Rossing also reports that the critical bandwidth (related to loudness perception) is about30 times the jnd for pitch, suggesting that both are related to the regions of excitationalong the basilar membrane of the inner ear .