What is Pitch?

Those presenting the mathematics/music connection often use rhythm to make their case.  This makes sense because one is introduced to number values for note durations when receiving musical training.  Pitch is another example of mathematics in music that is less commonly discussed.  Pitch is taught completely by rote.  The theory behind what makes one pitch match another is not discussed, because knowing the mathematics will not cause one to suddenly sing in tune.  Although impractical as a teaching device, the application of pitch in music is purely mathematical.

Sounds are actually disturbances in air pressure caused by a vibration.  These disturbances are longitudinal waves as opposed to transversal waves.  The ripples created by a pebble dropped in calm water are transversal waves because the motion of each particle is perpendicular to the travel of the wave.  Sound waves are longitudinal because the air molecules are repeatedly moving away and then back towards the sound source.  Therefore, the particles are moving parallel to the direction of the wave.  Another distinction between sound waves and ripples in a pond is that the areas of compression in the air can be pictured as a series of spheres (three-dimensional) growing from the sound source as opposed to circles (two-dimensional) from the pebble in water.  Although distinct differences exist between sound and the pebble analogy often used to describe it, sound waves are often analyzed using two-dimensional drawings more akin to transversal waves.  This is acceptable because it is much easier to visualize and the sound sources themselves (such as a vibrating string) fall under the category of transversal waves.[i]

Most sounds encountered in every day life have waves with irregular frequencies.  Producing a sound wave with a regular frequency creates a single pitch in music.  The frequency is measured in Hertz (Hz), which is the number of cycles per second.  For example, the pitch A above middle C on a piano is usually tuned to a frequency of 440 Hz.  Graphing the wave produced by a pitch results in a sine wave if all harmonics are absent (harmonics will be discussed in the next section).  The vertical axis of this graph represents the level of compression in the air and the horizontal axis represents time.[ii]  This graph is identical in shape to a graph representing the physical motion of the sound source.  For this interpretation, the vertical axis represents the distance (positive or negative) a point on the object is displaced throughout the object’s vibration, with zero representing where this point would be if the object were at rest.  Studying the behavior of a string is an acceptable way to understand why the graph of a pitch forms a sine wave.

The center point of a vibrating string is affected by a force proportional to the distance it has traveled from its resting point.  This can be represented by the equation:

   

F is the force due to string tension, y is the distance from a resting point, and k is a constant of proportionality.  Knowing that force equals mass, m, times acceleration, a, and acceleration is the second derivative of the displacement, y, the following equation can also describe this force:

By substitution these equations arise:

           

Since this is a study of pitch, this representation of a musical sound does not include harmonics, which will be discussed in the next chapter.  The functions that satisfy the conditions for y in the second order differential equation above are in the form:

              

A and B are constants that depend on initial conditions of the vibration.  Using trigonometric identities and substitution, the function can be rewritten in the form:

 

With the function in this form, it is obvious that the wave produced by a string behaves exactly the same as a sine wave.[iii]  It can further be concluded that a sound wave of a fixed pitch also behaves as a sine wave.

The above equation can be manipulated to produce a sine wave that matches a specific frequency.  If A above middle C is the desired pitch, the frequency should be 440 Hz.  The period of a sine wave is , so the desired frequency can be made by setting   equal to , producing the equation .[iv]  Amplitude (volume of the string) is represented by c and phase (shifting of the wave from left to right) is represented by .  It can be seen that solving for  for any given k or m (i.e. tension or mass) will reveal many possible ratios of these two conditions needed to produce the pitch A 440 Hz with a string.

For the purposes of this thesis, amplitude and phase will be ignored.  Although they can have an effect on the perception of tuning (e.g. if one note is louder than another, the resulting beats may be less distinguishable), the theory of tuning is not affected by either of these elements.  If two strings both tuned to A 440 Hz are played and each have different phases, , and amplitudes, c₁ & c₂, their combined waves can be expressed as:

Using the trigonometric identities, y can be expressed the following ways:

Since  and  are both functions of constants, they will be set to new constants, m₁ and m₂ respectively.  With these new constants the following can be ascertained: 

This new representation of y shows that the resulting sound has the same pitch (A 440 Hz).  When musicians tune their instruments, they do so by listening for beats while comparing two sounds.  These beats are created by their pattern falling in and out of phase due to the two wavelengths being slightly different.  As the beats slow down and eventually vanish, the two wavelengths are made the same and the two instruments are tuned to the same pitch.  Since the above equation shows that amplitude and phase have no effect on the resulting pitch of two identical frequencies, a musician would hear no beats for this scenario because there is no discrepancy in wavelength.

The tuning method described above only works for unisons and octaves (frequencies that have a ratio that is ).  The next section will explore the natural properties of sound that influenced our choices in the relationships between different pitches.