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the three systems


By Bayla Keyes


Intonation is an acoustic reality based on the overtone series. The first important relationship was discovered by the Greek mathematician Pythagoras in around 530 B.C.E., who found that certain perfect harmonies were formed by mathematical ratios of frequencies:  the octave, whose ratio is 2:1, the fifth, which is 3:2, and the fourth, which is 4:3. These ratios, and the beautiful pure sonorities which are the audible proof of their perfection, are the basis for typical violin tuning, with “open” fifths, which ring so deliciously. Tuning with as many open fifths and their inverse, perfect fourths, as possible, will yield expressive intonation; whole steps will be wide and half steps narrow. This is the intonation system used by violin soloists, and to some extent, by cellists and even violists.

To use expressive intonation, slightly raise the third and seventh tones in major scales (most often the sixth as well); lower the third tones in minor scales. Slightly exaggerate all accidentals-- put sharps higher and flats lower. (In this system, an Eb will be lower than a D#!) Tune all notes possible to open strings using pure octaves and fourths and when necessary, combinations of these intervals, i.e., B on A string should be tuned to open E, G to G string, C to G to G string, etc.  Half steps should be as close as possible. This will bring an expressive, intense quality to your intervals, and will be particularly helpful in fast passagework, where the ear perceives half-steps differently.


In 1482 the Spanish theorist Bartolomeo Ramos de Pareja introduced just intonation, a system in which the major third is calculated by the ratio of 5:4, and other intervals are arrived at by using the Pythagorean ratios of the octave, fifth and fourth.  Tuned in this system, 3rds and 6ths will have a “third tone,” an acoustic phenomenon which greatly enhances the pleasure of the listener and the player. This intonation is only used today while playing 3rds and 6ths, either by themselves or in relation to others in a chord. 

Equal Tempered

Unfortunately, intervals do not add up over distance or are not consistently sized in the Pythagorean ratios and just system; a high C will not be the same note as a low C, and playing pieces in different keys (modulating) is impossible.  In order for Western classical music to be born, a compromise system had to be devised in the sixteenth century. This system is equal temperament, and it divides the octave into twelve equal half-steps. The only interval in this system which rings is the octave; all others are slightly squeezed.

Using equal temperament is advisable in many if not most situations, e.g. orchestral playing, matching the piano or in combination with winds.  However you can expect to be using Pythagorean intervals as well, in order to make perfect ringing fourths and fifths with other musicians. Equal tempered intervals will not ring because, except for octaves, they are all equally out of tune!

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