The Development of the Equal Temperament Scale

Evolution or Radical Change?

What are Harmonics?

Both mathematicians and musicians use the word “harmonic”.  Many people do not know the connection between the musical and mathematical meaning of this word.  However, the concept underlying both of the definitions is identical.  From Ancient Greece through the Renaissance, the study of harmonics was not compartmentalized nearly as much as it is today.  Music and mathematics were treated as one.

The harmonic series in mathematics is the pattern: ( ).  When an instrument creates a sound, this sound does not contain a single frequency.  Many (theoretically infinite but in practice, fifteen at most) overtones are created above the fundamental pitch that contribute to the instrument’s characteristic “voice” or timbre.  Different instruments have different volume configurations for the set of overtones, but they are the same set of frequencies if each instrument is playing the same fundamental pitch.  The frequencies of the fundamental tone and all possible overtones are known as harmonics.  For a given frequency, m, the harmonics of a note tuned to a fundamental pitch m would be: (m, 2m, 3m, 4m, …).

Once again, a vibrating string is the best way to describe what creates these sounds.  When a string vibrates, it does so as a whole.  However, it also vibrates as if it were two strings each half the original’s length, three strings of each a third the original’s length, and so forth.  The pivot points for these sub-vibrations are nodes.[i]  The graph below represents a two dimensional cross-section of the first six vibrations in this series.

Click this picture to hear the first seven harmonics

The sum of all these vibrations resembles the lines used to represent sounds in digital audio recording and editing software.  The graph below represents the sum of all the waves in two cycles of the previous graph.

This naturally occurring pattern is the same as the harmonic series in mathematics.  The equation for frequency of a string in relation to its physical characteristics is: , where l is length in centimeters, T is tension in dynes and m is density (grams per centimeter of string).[ii]  Since the tension and density are theoretically constant, frequency is inversely proportional to the sub-lengths of a string.  This explains the pattern for harmonic frequencies outlined above.

It should be noted that this representation of harmonics assumes a string is perfectly flexible, has a unison shape and density for its entire length, and has no restraints at its endpoints.  Realistically, these conditions are never met; some imperfections are seen as positive characteristics of an instrument.  Special tuning techniques are required to compensate for these imperfections (such as stretching octaves on a piano).  These techniques are not in the scope of this paper because the tempered scale was developed as a mathematical solution for tuning one octave.  Due to the relatively small range of one octave, ignoring these imperfections has little impact on the study of this tuning method.

As will be discussed in later sections, these harmonics became the foundation for rules of harmony and the development of the tempered scale.  To understand our current tuning system, one has to understand the connection between harmonics and music.

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Last modified: November 20, 2005