Some Harmonic Problems with Just Intonation

We noticed in the last Just Intonation Post that we encounter 2 different perfect fifths: between C to G, E to B and  G to D are all the Pythagorean ratio of 3:2, but the perfect fifth from D to A is 40:27, or one syntonic comma flatter than 3:2.  This leads to some problems with basic harmony.  If we only consider I, IV and V, everything works out fine, but as soon as we add some more interesting chords to the mix, namely ii, iii and vi minor triads, we encounter a phenomenon known as drift.


Let's consider this basic harmonic cadence:

I  vi  ii  V  I

This cadence occurs in thousands, probably millions of songs throughout popular and classical styles.  Here is a clip played in equal temperament:

Now, if we were to try this same cadence with just temperament, we would have to be mindful of the following:

• all major thirds have ratio 5:4
• all minor thirds have ratio 6:5
• any note that is repeated in two successive chords retains the same tuning.

I chord: C E G in the 4:5:6 Just ratio,  meaning E is one syntonic comma flat and the G is a 3:2 Pythagorean G.

vi chord:  need the C and E to remain.  If we use the A from the just scale (one syntonic comma flat) all our ratios work out.  All good so far.

ii chord: need the A to remain.  F to A is a major third, so our regular (Pythagorean 4:3) F works, but D to F must be a just minor third, or 6:5. 

This gives us a 10:9 D which is one syntonic comma flatter than the D in the Just Scale (9:8).  Not ideal, but everything is still OK.

V chord: Now we need the D (10:9) to remain.  The B is a minor third below D, so the same calculation yields

The arrow indicates an octave adjustment

The B in our Just scale is 15:8.  The B we need for our cadence is one syntonic comma flatter.
To calculate the G for this chord, we need to find a note a just major third below 50:27

which is a syntonic comma flatter than the 3:2 value for G we started with.  Now we have problems because the final I chord needs to keep this value for G, which means the C and E must shift down a syntonic comma to maintain the 4:5:6 relationship.

Here is how the cadence sounds:

Now listen to the drift.  The audio below is just the first one chord and the last played back and forth.