The Development of the Equal Temperament Scale

Evolution or Radical Change?

The “Pythagorean” Scale

As mentioned before, Pythagoras (569-475 BC) spent time in Egypt.  This further complicates the traditional view that he and his followers invented the mathematical tuning methods bearing his name.  Records in China indicate that a similar tuning method was developed two thousand years before Pythagoras.  According to documents written about 240 BC, emperor Huang Ti (2700 BC) told a music master named Ling Lun to build a set of 60 bells.  Ling Lun came up with a mathematical method for creating pitch pipes to tune this large set of bells.  The “spiral of fifths” produced by Ling Lun’s system is almost identical to Pythagoras’s solution.[i] 

An elaborate set of bells has been discovered in China.  Dated at 433 BC, these can play twelve tones and were tuned using perfect intervals. This seems to confirm a deeper understanding of music theory and chromatic tuning than even Chinese scholars had anticipated.  Unfortunately, the Chinese emperor Qin Shiuangdi (ruling 221-210 BC) destroyed many music documents and instruments.  Much of the evidence of China’s advancements in chromatic music was probably lost at this time.[ii]  Since then, Chinese music has traditionally been based on the pentatonic scale.  Even after the bells were discovered in 1979, the Chinese government was slow to disclose the findings that would contradict their view of traditional music. 

The mathematics described in this section accurately depicts both Pythagoras’ and Lun’s tuning methods (discrepancies between the two will be outlined).  The system for this method of tuning is based on one interval.  In music, there is a “circle of fifths.”  Two notes five diatonic steps apart (counting the bottom note as “one”) are called a fifth.  The actual number of chromatic steps is always eight.  If you create another fifth from the top note of the previous fifth and continue this pattern, you will get the following series of notes:

C, G , D, A, E, B, F#, C#, G#, D#, A# (or Bb), F, C

It is called the circle of fifths because it ends where it began after the twelfth note.  Using the string length ratio “2:3,” a perfect fifth can be tuned very easily.  By tuning either up a fifth or down a fourth (inverted fifth with the ratio 3:4), all twelve notes could theoretically be tuned in one octave.  Ling Lun described this as add and subtract one-third.  In other words, the next length in the sequence would be:

    with x being the length of the lower note of the interval

                                    or

    with x being the length of the higher note of the interval

Both Pythagoras’ and Lun’s methods produced relatively the same results.  The following chart shows the string length intervals of the chromatic scale produced by these tunings:

Comparison of Ling Lun’s and Pythagoras’ Tunings  

Click here to listen to the Pythagorean scale.

The differences found between these two representations are only due to the starting and ending points of the “circle.”  Lun’s version starts at “C” and continues to “E#” (not “F” since it was calculated as a fourth below “A#”) and the Pythagorean version starts at “C” going in two directions and ending at “Eb” and “G#.”  It must be noted that this comparison is difficult to present since it involves using modern terms for note values and comparing two theories with their own representations of pitch.  Hence, this comparison is very likely skewed by modern interpretations.  Regardless of any possible notation discrepancies, both systems are remarkably similar.

As well as sharing many of the same ratios, both of these scales also contain the same flaw.  The point where the “circle” closes does not form a perfect fifth.  This is located between “E#” and “C” in the Chinese system and “Eb” and “G#” in the Greek system.  The imperfect fifth was later named a “wolf” interval and became the focus of future musicians and mathematicians from the Middle Ages until today. 

Evidence suggests the Chinese understood this to be a spiral of fifths and that this method would produce an infinite number of pitches.  They chose to use only the first twelve tones from this spiral.  Unlike the Greeks, they built instruments with all twelve notes.  The Greeks created eight note scales from the twelve tones and simply avoided the “wolf” interval.   These scales existed prior to the tuning method being made public; they were only enhanced by the Pythagorean’s work. 

The Pythagoreans believed that important intervals were based on string length ratios with integers one through four.  The ratios within one octave are: 1:1 (unison), 1:2 (octave), 2:3 (fifth), and 3:4 (fourth).  Although the Pythagorean scale is built using these ratios, the simplicity in harmonic intervals is lost in the final product.  As instrumental music became more complex, this scale did not have a sufficient number of “consonant” intervals (based on simple ratios).  With help from rediscovered writings of a scholar from Alexandria, a less systematic but more “perfect” scale was created during the Middle Ages in Europe.

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Last modified: November 21, 2005