About Just Intonation

There are many ways to tune musical instruments. Different musical cultures (in different times and places) have used many different tunings. The modern Western method of 12-equal divisions of the octave is but one possibility. An alternative to 12-tone equal temperament is called Just Intonation. This uses intervals that are defined by ratios of small integers. As long ago as Pythagoras, people had observed that such simple integer ratios correspond to many of the consonant intervals. If you feel like digging deeper into this, there are several great resources on the web (see below).

Why do simple integer ratios sound special? The simplest of the simple ratios is 2/1, the octave. One way to understand why the octave is special is due to coinciding partials between the overtones of harmonic sounds. The nth overtone of the octave occurs at the same frequency as the 2nth overtone of the root. Thus the octave and root tend to meld together because they are, in fact, composed of very similar overtones. If this seems odd, you can read a bit more about it.

In fact, the same line of reasoning can be used to explain that fifths (which are defined as a ratio of 3/2) sound smooth or consonant because every second overtone of the fifth coincides with every third overtone of the root. Similarly, in a third (a ratio of 5/4), every fifth overtone of the third coincides with every sixth overtone of the root. More information about this.

The Just Intonation scales focus on intervals such as these that are composed of ratios of small integers. The ratios 2/1, 3/2, 4/3, 5/4, and 6/5 represent the octave, the just fifth, the just fourth, the just major third, and the just minor third, respectively. These intervals are close to (but not exactly the same as) the 12-tone equal tempered intervals with similar names. Read more about 5-limit JI.