Just Intonation, part II

In the last section, we noticed that the Just Major Third, 5:4, is one syntonic comma (81:80) flatter than the Pythagorean Major Third (81:64).  Observe that the 3 notes that differ between the two scales are the major third, the Major Sixth (5:3 for Just vs. 27:16 for Pythagorean) and the Major seventh (15:8 for Just and 243:128 for Pythagorean).  If we perform the division, we notice that each of the Just Intervals is exactly one syntonic comma (81:80) flatter than its Pythagorean counterpart.  This gives rise to some new issues.





Although the Just triads sound much more in tune than the Pythagorean triads, the price we pay is that the syntonic comma creates unequal seconds and fifths.  C to D, F to G and A to B all have he ratio of 9:8 regardless of whether Just or Pythagorean intonation is used.  But, look at the major second between D and E as well as between G and A.

Two of the major seconds in the Just scale differ by a syntonic comma from the other 3 major seconds in the scale

Similarly, the perfect fifths from C to G, E to B, G to D and A to E all have ratio 3:2, but the fifth from D to A has ratio 40:27 which differs by ... wait for it ... one syntonic comma from the Pythagorean fifth.

When we try to transpose to a different key, this creates a new problem.  If we start with G having a ratio of 3:2 and build major triads using the 4:5:6 ratio from I, IV and V (G, C, D),  we see the following

G        B        D

3:2     15:8     9:8

C         E         G

1:1      5:4      3:2

D       F#         A

9:8    45:32   27:16

We expected the new note F#, but we also got an A with a ratio of 27:16, not 5:3 which is the ratio for a major sixth in the Just Scale.  Every time we transpose up or down a fifth, one pitch goes "out of tune with itself".  Guess how much it goes out of tune by .....   one syntonic comma.   The more you transpose, the worse it gets.

Stay tuned ...  more to come.