Mathematics in Musical Intonation

Craig Morgan

Mathematics has an influence on just about everything.  Musical intonation is no different.  In fact, mathematics has been part of music since its very beginning.  This site is designed as a means of stepping through various stages of intonation history, and finding the importance of mathematics within its development.  There are hundreds of different scaling systems, but I focus on five of the most common and most used: Just, Pythagorean, Mean Tone, Well-Tempered, and Equal-Tempered.  

It was interesting to see how these different styles related to each other, and surprisingly, how they still are related.  For example, the 12-tone equal-temperament scale is the most common to our ears.  This is what we hear the vast majority of the time.  However, an orchestra does not play in equal-temperament.  The brass instruments often tend to slightly raise or lower certain notes such as the major 7th and the major 2nd to anticipate the move to the tonic (root).  The stringed instruments, such as the violin and cello, are tuned purely.  A violin has four strings based on the perfect fifth interval.  Without frets, these instruments can be played in a number of different ways.  Only the harp, piano, and tunable percussion instruments are tuned in equal-temperament.  The remaining instruments are tuned in some variation of just intonation.

Unaccompanied choirs do not even use an equal-tempered scale.  Each member of the chorus is constantly listening to the other members, and slightly changing his or her pitch to match.  When one sings alone, there is a natural tendency to go slightly flat.  This is because there is nothing to use as a checking device.

It is important to remember that there have been many different types of musical intonations used throughout the world and throughout history.  Compositions are intended to be played in the system in which they were written in.  Because there are at least slight differences among many different scales, there will be a different sound (whether it is detectable by the human ear or not).  Therefore, by playing a number in a different intonation in which it was intended runs the risk of losing its character.