In this exercise we build upon the Pythagorean scale generated in class on 20 March by interleaving pure fifths transposed down octaves as needed. The resulting diatonic scale has only 'major' seconds (9:8) and semitones (256:243) narrowed by a syntonic comma (81:80).
1. How does a Pythagorean major third (IIIp) compare with a naturally tuned major third? (III)
2. Compare a Pythagorean minor third (3p) with a naturally tuned minor third (3).
3. Compare the fifth d - a of the naturally tuned scale (V) with the same fifth in a Pythagoran scale (Vp).
Cf Exercise 24, question B4.
4. Calculate, for a Pythagorean scale, the ratios for the intervals
c - d
c - e
c - f
c - g
c - a
c - b
c - c'
and compare these side by side with the same intervals from a naturally tuned scale as you have already done in an earlier interval arithmetic exercise (Ex24).
5. Calculating the Pythagorean comma.
5A. Derive the ratio for a macrointerval of seven successive octaves (e.g. _C_ - C - c - c' -....-c'''').
5B. Derive the ratio for a macrointerval composed of twelve successive fifths (e.g _C_ - _G_ - D - A - e - ....- b#''' (= c'''').
Theoretically these ratios should be the same, because after twelve fifths in the circle of fifths we come to a B#, which is enharmonic with the C where the circle began seven octaves earlier.
5C. Calculate the difference between the result of 5A and that of 5B. This is the so-called Pythagorean comma.
So that you know you are in the right degree of maginitude, it approximates 74:73, but that is not the exact result, which you must derive, and for which you must demonstrate the derivation.
fmwww.bc.edu/MT/gross/NumEx28.html
cnnmj 8323d