To prove the enharmonic pitches of B# and C are not the same pitch requires some background information:

1. Pitches are created by a medium vibrating at a specific number of times per second.
2. Identical intervals have a consistant ratio of vibrations per second.
3. The interval of two pitches a P8 apart is 2 to 1 (A 880 is a P8 higher than A 440).
4. The interval of two pitches a P5 apart is 3 to 2 (A 660 is a P5 higher than A 440).

This information comes from Pythagorean tuning and is based on the physics of natural acoustics.

To calculate the enharmonic dilemma, begin with a very low C (in the example below, 4 octaves below middle C, whch vibrates about 16 times per second). Each P8 above that pitch is twice as many vibrations per second, ending 3 octaves above middle C (which vibrates about 2048 times per second).

The next calculation begins with the same low C. Each P5 above that pitch is 1.5 times (3 divided by 2) as many vibrations per second, ending 3 octaves above middle C on a B# (which vibrates almost 2076 times per second).

Thus, B# is more than 1% higher than its enharmonic equivalent of C. Do the math.

Clearly, if keyboard instruments maintained this discrepancy, many intervals would sound terribly out of tune. To avoid this problem, the standard modern system of tuning keyboards is called equal temperament, in which the octave is divided into 12 half-steps, all equally out of tune, but are so close that the effect is barely noticable. All octave intervals are still tuned, however, in the 2 to 1 ratio.

Go back to Chapter 2. Vocabulary.