C and G

This combination is an interval known as a fifth, and is considered consonant. It is a very stable note combination, as indicated, perhaps, by the simple repetitive mathematical pattern in the graph of its sound energy. The fifth interval is an inversion of the fourth interval, and thus very similar in nature as to sound and mathematical pattern. (See C & F.) It is very easy and natural for our voices to sing these notes consecutively.

Perfectly tuned, the higher pitched note would vibrate at a rate exactly 50% faster than the lower. Piano tuners use this interval when tuning pianos, or at least they did before electronic tuning devices were available. Starting with note A above middle C (tuned at 440 Hz), tuners proceed, accounting for octaves, through the cycle of fifths: A, E, B, F#, C#, G#, D#, A#, F, C, G, D. This accounts for all the chromatic notes, but the next note in the sequence would be A again, and due to the relatively prime nature of numbers 2 and 3--specifically, no non-zero integer power of 2 can equal a non-zero integer power of 3--that A would not sound the same as the A the tuner began with. In fact, the frequency of the new note A would be slightly higher than 446 Hz, significantly different from 440 Hz, as far as our ears could tell. Some of the other notes would be noticeably out of tune and would sound bad.

Due to this unavoidable tuning problem, musicians began tuning keyboard instruments using the well-tempered system, which means the ratio of frequencies of half-tone intervals are the twelfth root of 2, or about 1.05946. All notes are slightly out of tune in order to even out the discrepancies from natural tuning. It is not easy to hear this difference, so no one really seems to mind. Johann Sebastian Bach was so delighted with this new development in 1720--some suggest that Bach himself devised the new tuning system--that he wrote The Well-Tempered Clavier in celebration, a set of preludes and fugues in all the possible major and minor keys. This system of tuning liberated music composers to to experiment with new harmonies which before would have sounded awfully out of tune.

For a formal proof that pianos cannot be perfectly tuned click here: Why Pianos Are Always Out Of Tune. (This is in pdf format.)

Click to hear this sound as played on a piano: C & G