In this exercise we build upon the interval arithmetic done in Exercise 24.
1. Derive arithmetically the difference between a major second (II) and a minor second (2) in the naturally-tuned diatonic scale which we constructed.
That micro-interval is called a syntonic comma.
2. The standard tuning of a violin is in straight (i.e. pure 3:2) fifths (g - d' - a' - e'').
Note: In the natural scale in c which we constructed in class (NumLe23.html), the interval d - a was not a pure fifth. In tuning our violin, however, the strings are tuned a natural fifth (3:2) apart as if we were tuning a transposed c - g each time.
This forcing of fifths, in fact, lies at the heart of the tuning complications which we are examining and hope to untangle.
If I transpose the e'' thus obtained down one octave to e', the sixth (a pure VI = 5:3, corresponding proportionally to the c-a you calculated in Exercise 24) is too broad.
2a. Calculate and compare this sixth (derived from fifths) with the pure sixth of our original derivation.
2b. Compare the fourth e' - a' derived from fifths (using the transposed e' given above) with a pure fourth (4:3). How do these fourths compare?
3. A major third derived from fifths in the same manner will have a proportion of 81:64. This is the so-called Pythagorean third.
3a. In what proportion does a Pythagorean third differ from our pure major third?
3b. Can you characterize the difference between a major third and a Pythagorean third in terms of intervals? (Refer back to question 1 if you are having trouble.)
fmwww.bc.edu/MT/gross/NumEx26.html
cnnmj 8318c