The 12-Note Octave is Too Good to be True

[Playing notes on a piano] One... Two... Three... Four... Five... Six... Seven... Eight... Nine... Ten... Eleven... Twelve...

12 different pitches, and then back to where we began. Incredible! Fantastic! The mystical number 12. There are 12 hours in the A.M. and 12 hours in the P.M. The new day begins at 12 midnight. There are 12 months in a year. Both the Western and Chinese Zodiac have 12 signs. Further, the Chinese use a 12-year cycle for reckoning time. There are 12 eggs in a dozen. 12 dozen in a gross, and 12 ounces in a troy pound. There were 12 tribes in ancient Israel. Jesus had 12 apostles. There are 12 days of Christmas. My friends, Eastern Orthodoxy observes 12 great feasts. In Shia Islam there are 12 Imams. In Ancient Greece the principle gods of the Pantheon were the 12 Olympians. There are 12 ribs in the human body. 12 labors of Hercules, and, in the United States, 12 people on a jury.

The five new pitches were added as the black keys on the keyboard, and there you have it, neat as a pin. A closed, 12-pitch Universe generated through a circle of fifths!

Except... Oh I just hate having to be the killjoy here, but there is one little problem. Isn't there always one little problem? You see, if we move upwards, through this circle of perfect fifths of three to two sonic ratios, when we get to the 13th pitch, we get to a pitch that's actually about an eigth of a tone sharper than the one on which we began. Oh! So close!

Well, the promise of an octave divided into 12 different pitches a semitone apart was too good so the solution to this little tuning problem was to temper or shrink all or some of these fifths so that the 13th pitch would indeed be the same as the first.

The number 12 seems so perfect, but a 12-note scale made of 3/2 ratios brings the circle around to a point a little sharper than the the next octave.