Just Intonation - Diatonic Scale

Diatonic Scale

It is possible to tune the familiar diatonic scale or chromatic scale in just intonation in many ways, all of which make certain chords purely tuned and as consonant and stable as possible, and the other chords not accommodated and considerably less stable.

The prominent notes of a given scale are tuned so that their frequencies form ratios of relatively small integers. For example, in the key of G major, the ratio of the frequencies of the notes G to D (a perfect fifth) is 3/2, while that of G to C (a perfect fourth) is 4/3. Three basic intervals can be used to construct any interval involving the prime numbers 2, 3, and 5 (known as 5-limit just intonation):

• 16:15 = s (Semitone)
• 10:9 = t (Minor tone)
• 9:8 = T (Major tone)

which combine to form:

• 6:5 = Ts (minor third)
• 5:4 = Tt (major third)
• 4:3 = Tts (perfect fourth)
• 3:2 = TTts (perfect fifth)
• 2:1 = TTTttss (octave)

A just diatonic scale may be derived as follows. Suppose we insist that the chords F-A-C, C-E-G, and G-B-D be just major triads (then A-C-E and E-G-B are just minor triads, but D-F-A is not).

Then we obtain this scale (Ptolemy's intense diatonic scale):

The major thirds are correct, and two minor thirds are right, but D-F is a 32:27 semiditone. Others approaches are possible (see Five-limit tuning), but it is impossible to get all six above-mentioned chords correct. Concerning triads, the triads on I, IV, and V are 4:5:6, the triad on ii is 27:32:40, the triads on iii and vi are 10:12:15, and the triad on vii is 45:54:64.