Pythagorean Tuning part 1

I am teaching a class at Cal State Monterey (C-scum) on the Mathematics of Music.  We are currently studying the evolution of our modern tuning, starting with Pythagoras.  (The same guy with the right triangles.)  Pythagorean tuning is based solely on perfect fifths and octaves.  Given a starting pitch, all of the remaining scale degrees are arrived at by going up or down by a fifth (which corresponds to multiplying or dividing the fundamental frequency by 3/2) or going up or down by an octave (which corresponds to multiplying or dividing by 2.)  This came about seemingly because the fifth is the first overtone which differs from the fundamental pitch.


If you lightly touch a vibrating string one-third of its length, the string will vibrate at 3 times the frequency in three equal sections.  If you play guitar, this is the same as gently touching an open string right above the seventh fret.  This produces a note an octave and a perfect fifth above the tone of the open string.

The ratio of the frequency of the harmonic with the fundamental is 3:1.  The note produced is an octave too high to be a useful part of the scale starting with the fundamental pitch, so we drop it an octave by dividing by 2.  The scale now contains {C,G,C'}  where the octave C' comes from the 1st harmonic.  The ratio of frequencies is {1:1, 3:2, 2:1}

Stay tuned for part 2.