Originally posted by Samejame:

When tuning their instruments to the keyboard before starting, I listen for beats, or overtones as I approach the same pitch. lets say I'm tuning the G string (no jokes please) of a violin, which is G below middle C on piano. Am I right in assuming that the two are properly in tune when I can no longer hear beats, or should there be beats for a certain pitch? Also is there a relationship between "cents" and the frequency of the beats you hear for a certain off pitch? Say the tuning is a bit flat, and I'm hearing an overtone of about two beats per second. What would the relationship of that frequency be to "cents".

I don't know whether I can explain this to your satisfaction without getting too technical or mathematical. But here it goes:

Beats are pulsations of sound that one perceives when two sound waves that do not vibrate at the same frequency OR whose frequencies are not INTEGRAL multiple of each other interact in successive constructive and destructive interferences. Theoretically, one should hear no beats when two notes are in tune (i.e., when they are vibrating at the same frequency), or when they are octaves (i.e., when the frequency of the higher note is an integral multiple of the lower note).

As far as frequency and cents are concerned, there is a relationship between the two, but this relationship is not linear. This nonlinearity arises because cents follow a linear progession, but frequencies follow a geometrical progession. For instance, the DIFFERENCE in cents between any two semitones is a constant and is equal to 100, but the DIFFERENCE in frequencies between the same two semitones is NOT a constant. The constant is in the RATIO of their frequencies (it is equal to 1.059463094 for an equal tempered scale).

As an example, consider the theoretical frequencies for C and C# in three octaves as given below:

Middle C - 261.626 Hz, Middle C# - 277.183 Hz

C an octave higher - 523.251 Hz, C# an octave higher - 554.365 Hz

C two octaves higher - 1046.502 Hz, C# two octaves higher - 1108.731 Hz.

Note that regardless of which pairs of semitones we are looking at, if we take the RATIO of the frequency for C# to that of C, we will always get 1.05946... However, if we take the DIFFERENCE of the frequencies for C# and C, we will get 15.557 Hz, 31.114 Hz, and 62.229 Hz, respectively for the three octaves. Note that the difference 'expands by a factor of two' for every octave going up (or 'contracts by a factor of 1/2' for every octave going down). This expansion or contraction is NOT reflected in the difference in cents. There are 100 cents between C and C# (and all other semitones) regardless of which octaves we are looking at. As a result, from middle C to middle C#, each cent represents a frequency difference of 15.557/100=0.15557 Hz. One octave higher, each cent represents a frequency difference of 31.114/100=0.31114 Hz. And two octaves higher, each cent represents 62.229/100=0.62229 Hz. Because of this nonlinear relationship, one needs to specify which two notes on the scale one is looking at before a valid relationship between cents and frequency can be established.

Eric