Equal Temperament

                I recently had a discussion with someone about the use of the term “perfect” when labeling the intervals of a fourth and a fifth.   As mentioned in my last post, these intervals are perfect consonances.  However, unlike octaves and unisons, fourth and fifths can be altered.  The label “perfect” is used to distinguish the consonant form of these intervals from the diminished or augmented forms. 

                This discussion caused me to think about the nature of perfect fifths and fourths in today’s music compared to music of the past.  Technically, today’s fifths and fourths are not exactly perfect.  To understand why, we need to have a brief discussion on tuning systems.

                A tuning system is a method or formula for obtaining the correct distances between musical intervals on an instrument.  The Pythagorean tuning system (created by the mathematician Pythagoras) was used till the beginning of the 16th century.  The system was based on a scale that was composed of actual perfect fifths which measure to be 702 cents in distance.  Unfortunately, this system results in uneven interval distances for across the pitch spectrum.  Unisons and octaves are perfect, but there is one fifths (the wolf fifth) within the sequence that is a different size.  This causes the other intervals within the sequence to have two different sizes throughout the series. 

                People experimented with other tuning systems throughout the years in order to have more consistent intervals.  Eventually the system of equal temperament was accepted as the dominant tuning system.   In this system octaves are subdivide into halve steps of equal distance.  This results in fifths that are slightly flat when compared to a pure perfect fifth.  However, the mathematical inconsistencies that resulted from Pythagorean tuning (and other systems) are eliminated.  The distances between intervals are equal across every key and register of the pitch spectrum. 

                This equal temperament system is the one that we are used to hearing now when we listen to music.  Most people do not even realize that the fifth they hear on an equal tempered piano is not pure.  Piano tuners, however,  are very aware of this.  They are trained to hear the proper beating sound of an equal tempered fifth. 

                A classic example of the impact of tuning systems is Bach's Well-Tempered Clavier.  This is a collection of preludes and fugues written in all 24 major and minor keys for solo keyboard.  This collection was composed before equal temperament, and was originally played on instruments that were tuned with other systems.  The mathematical inconsistencies of these systems caused the different keys to poses different sonic qualities and characters.  This sonic variety is lost when the pieces are performed on keyboard instruments tuned with an equal temperament.  So, we have gained symmetry in our intervals by sacrificing the individual character of the different key signatures and the pure fifth.