The Development of the Equal Temperament Scale

Evolution or Radical Change?

Just Intonation

When musicians play an instrument with a flexible pitch system (e.g. voice or violin), it can be argued that they perform in “just” intonation.  Pythagoras was correct in stating we prefer simple intervals.  It does not matter to the human brain when these intervals are not the same throughout all possible key signatures.  Just like a driver making slight adjustments to a steering wheel to keep a car between the lines, musicians can adjust the frequencies of a pitch when facing different contexts.

The Pythagorean focus on fourths and fifths was shared by early European musicians in terms of harmony (more than one note played at once).  The more simple an interval, the less tension is heard when this interval is played at once.  Starting in the middle ages, monks in the Roman Catholic Church sang Gregorian chants during worship.   The melodic structure of these chants was based on ancient writings about the Greek tetrachords (sets of four notes paired to make an eight note scale).  To this day, Western musicians use the Greek words for these “modes” (scale configurations).  However, the names do not match the original intent of the Ancient Greeks because of interpretative mistakes.[i] 

These Gregorian Chants are still performed today, but they also acted as a starting point for developments in polyphony (more than one melody played at once).  As hundreds of years passed, church composers began experimenting with harmonic variations of the Gregorian chant.  Starting with simple one-note drones behind a more complex melody (organum) and eventually moving to parallel melodies with a consonant distance apart (discant).  By the thirteenth century, the increasing complexity of European sacred music was also influencing secular music[ii].

Many fixed pitch instruments were shunned during the early Middle Ages because of their connections to “pagan” traditions of Rome.  Versions of these instruments slowly came back however.  Bell sets could be found throughout the middle ages.  Large pipe organs were built in Cathedrals starting in the tenth century.  The percussive dulcimer was not a new instrument, but was introduced to Western Europe (invented in Persia) by the fourteenth century.  The psaltery (a small, hand plucked predecessor to the harpsichord) was another fixed pitch instrument introduced from the Middle East and commonly played during the later part of the Middle Ages.[iii]

The most common tuning in the first part of the Middle Ages was the Pythagorean.[iv]  The combination of increased harmonic complexity and the popularity of fixed pitch instruments made this a more difficult tuning to use.  Most combinations of notes from the Pythagorean result in complex length ratios.  Complex length ratios produce a discordant sound when played together.  The only combinations of notes producing pleasant harmonies are all but each of the fourths and fifths.  Having human voices demonstrate the ideal harmonies throughout the scale probably highlighted flaws for listeners.

Click here to hear the Pythagorean scale applied to Mozart's Sonata no. 11 in A major.

Writings by the Alexandrian philosopher Claudius Ptolemy (85-165 CE) influenced the development of a new tuning system during the sixteenth century in Europe.  In Harmonics, Ptolemy expanded the consonant string length ratios to contain whole numbers up to six.  Ptolemy’s tetrachords balanced the debate between those who believed scales should be based on mathematical proportions and those who thought they should be completely based on what is aesthetically pleasing.[v]  Although less symmetric than the Pythagorean tuning, his were based on simple ratios that were also pleasing to the ear. 

By the early Renaissance, musicians began to favor thirds and sixths as harmonies.  The Pythagorean explanation of consonance did not include these intervals, but Ptolemy’s did.  A major third (five semitones) and major sixth (ten semitones) can be produce by a string length ratio of (4:5) and (3:5) respectively.  Italian music theorist Gioseffe Zarlino (1517-1590) studied the intervals discussed by Ptolemy and found that the most consonant intervals were created from ratios derived from the expression: ( ).[vi]   Matching these types of ratios with intervals considered to be consonant during this time, Zarlino published a standardized system for tuning an octave with “just” intervals.[vii]  Although not the basis for its development, the diatonic notes (naturals) in Zarlino’s Just Intonation scale are simple harmonics found in the note “F,” and the accidentals (sharps and flats) are simple harmonics of various diatonic notes.  The following chart displays the diatonic ratios.

Just Intonation Diatonic String Ratios

Click here to listen to the "just" intonation scale.

With this configuration, there are two types of major seconds (two semitones apart) and one type of minor second (one semitone apart) in the major scale.  Further analysis would reveal more than one type of other intervals as well.  Adding the chromatic ratios highlights the unevenness of this scale.

Just Intonation Chromatic Ratios

The two tuning methods discussed so far became the extremes from which mathematicians and music theorists either tried to find compromises or to perfect.  The Pythagorean was developed using a systematic and uniform method.  Searching for pleasing harmonies and not using any sort of algorithm was the basis for just intonation.  These two philosophies were later brought closer together with the discovery of overtones.

The Greeks wrote about the harmonic series for interval string lengths, but not for multiple pitches simultaneously produced on one string.  The first publication of this knowledge was by French mathematician Marin Mersenne (1588-1648 CE).  Although he could never explain the physics behind them, Mersenne was the first to identify overtones found in a vibrating string.[viii]  The set of string lengths needed to produce these overtones as fundamentals is the harmonic series.  A mathematically predictable property of sound could be used to identify consonance of intervals.  Instead of identifying consonant ratios by trial and error, one could compare the frequencies of each string length and find out how many overtones are shared.  The fewer audible frequencies in common, the more dissonant the interval is for two string lengths.

While helping to identify consonance in a more scientific way, more questions were created by this discovery.  For example, the most consonant interval (other than an octave) is a perfect fifth.  This matches music theory from ancient Greece through the Middle Ages, but musicians began to favor major thirds by the early Renaissance.  This contradiction allowed continuation of the debate between those who sought a mathematical answer and those who favored a more observational approach.  Also, knowledge of the overtone series made it clear that any tuning of a fixed pitch instrument would be a compromise of ideal harmony.  Before equal temperament became the standard for Western music, two families of compromises existed: “mean-tone” and “well” temperament. 

Send mail to mgrenfell@charter.net with questions or comments about this web site.
Last modified: November 20, 2005