Pythagorean Tuning & Just Tuning

One of the most basic and essential methods of tuning is called pythagorean tuning. It relies on a 3:2 frequency relationship between the fifth and the root of the scale. Starting with an initial frequency, for example middle C, the rest of the notes are tuned by multiplying the frequency of C by 3/2. This multiplication process is continued by treating the new note as the root note until each note has a specific frequency. This does create a problem, however, as the resulting notes do not fit into a scale within the same octave range. One essential relationship between notes is that the relationship between an octave is 2:1; this relationship is found in essential every method of western tuning. The octave relationship is applied to all of the highly tuned frequencies, and they are essentially halved until they form an ascending scale. This process is known as reducing the octave (The Physics of Music and Color).

Another important tuning method is called just tuning. Similarly to pythagorean tuning, just tuning relies on the 3:2 relationship between the fifth and the root of the scale. It also relies on a 5:4 relationship between the major third and the root of the scale. By applying these two relationships to the root note of a scale, and through the process of reducing the octave, a full diatonic scale can be defined through just tuning. After the process is completed, each note ends up with a fractional relationship to the root:

(The Physics of Music and Color)

Because of the reliance on the ratio of the third, just tuning creates a scale with relationships that differ from pythagorean tuning. 

Works Cited:

1.  Gunther, Leon. The Physics of Music and Color. New York, New York: Springer, 2012.