The Development of the Equal Temperament Scale

Evolution or Radical Change?

What Role do Harmonics Play in Music?

Melody is an element of music shaped by harmonics.  Melody is a series of pitches (frequencies) played one after another.  Each has specific length and starting point (i.e. the series has a specific rhythm).  Since an infinite number of frequencies exist, a limit must be present for the choice of pitches.  In order to communicate, the performer must be “speaking a language” that is familiar to the listener.  Harmonics influenced these choices, but they are still an infinite set.  Also, the range covered in the beginning of the harmonic series would be taxing on a single human voice and therefore sound unnatural to the listener if used as a basis for melody.

The most important overtone in music is, not surprising, the first one.  This harmonic creates the interval called an octave in music.  Any two frequencies that have a ratio equal to  are called octaves of the same pitch.  This is because their harmonic components are so similar, they sound as one when played in unison.  Notes with this ratio are designated the same letter name in modern music notation.  The other advantage of an octave is its range, which is comfortable enough to be sung.  It is also wide enough to contain a variety of distinguishable pitches.

The number of frequencies within an octave is something that is less perfect in its development.  We as humans have imitated the naturally occurring sounds that were pleasing and familiar.  Unfortunately, the harmonics series that forged these sounds do not perfectly divide the octave range.  Harmonic intervals higher than an octave can be moved down to the first octave by multiplying them by some power of two.  Since the two notes will have a ratio of , the resulting pitch will be a different octave of the same note.  For example, the seventh harmonic can be multiplied by four (two to the power of two) resulting in the ratio of four sevenths.  Since this fraction falls between one and one-half, the resulting pitch is less than an octave above the fundamental frequency.  Since the ratio between one-seventh and four sevenths is four (or ), the frequency is a lower octave of the same pitch as the harmonic. 

Repeating the process above creates a set of notes based on the natural harmonics within a reasonable range.  Placing the more audible harmonics in the lower octave, people noticed that having twelve pitches seemed to divide the octave more evenly than adding more or less.  In the table below, the simpler ratios that became part of our current chromatic scale (every pitch from one octave played from lowest to highest) are shown with their corresponding pitches.  Many of these pitches cannot be found in the more audible harmonics with ratios consisting of numbers less than ten.  The ratios (in decimal form) between each of these pitches are listed in the next row of the chart.  The bottom row shows what the final ratio would be if twelve steps were made using each ratio.  In this example, C is the fundamental pitch.

 Dividing the Octave with Natural Harmonics

Although these more important harmonics do not perfectly fit into a scale divided into twelve equal parts, it can be seen why the tonal system scale evolved to contain this many pitches.  The extension of these ratios by twelve come very close to the desired ending ratio of one half (or one octave).  The mean of the extended ratios is (.499) with a standard deviation of (0.037).  There are cultures that use quarter-tones (notes directly between two chromatics) in their music.  The mean and standard variation for this scale (with twenty-four parts) is the same as the above example.  However, quarter-tones are most often used as embellishments and are difficult for most listeners to distinguish.

While highlighting the strengths of a twelve-note chromatic scale, the calculations above also reveal the difficulty in tuning fixed pitched instruments.  Since the ratios between tones derived by harmonics are not consistent, the scale is completely dependent upon the starting pitch.  The most common scale in western music is called the major scale.  It does not use all twelve notes of the chromatic. The chart below illustrates how a tuning method used before and during the early Renaissance would be used to create this scale.  The “Just Intonation” scale is almost completely derived from the harmonics of a single pitch.  The first row of values shows the tuning when applied to the tonic “C.”  The second row shows what values would be needed to create another major scale using the “D” from the first scale as its tonic.  Notes left blank were not needed to play these scales.

Ratios Needed to Play in the Key of C and D

Since four of these notes are different, it is impossible to tune a fixed pitch instrument for both of these major keys at once.  To further illustrate this problem, the chart below uses decimal approximations for the ratios and adds the scales “E” through “B.”

Ratios (decimal equivalents) Needed for Multiple Major Keys

Every one of these scales has some variance from the ratios needed to play “C” Major.  The “A#” or “Bb” has the most distinct ratios of all the pitches because it is the only one that is used as a sharp and a flat.  Our equal tempered scale has conditioned us to think of them as one pitch with two names, but they are actually two separate pitches.  Adding the remaining major scales would create even more problems with more sharps and flats sharing the same fixed pitches. 

One can guess the most commonly used keys and harmonies for a given period by looking at the imperfections of the prominent fixed pitch tuning methods of the time.  Those scales with the least imperfections are likely the scales most used.  In the next section, the most important tuning methods leading to and including Equal Temperament will be put in a historical context and explained.  These are also the same tuning methods analyzed for this research.

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Last modified: November 19, 2005