With 7/12 as approximation to frac(log_{2}3), iterating the
"third harmonic" construction yields exactly twelve pitch classes.
If "0" corresponded to C on a piano, then the pitch classes would run as follows:
Since the interval from each note to the next is a (not quite perfect) fifth, this sequence of pitch classes is referred to as "the circle of fifths." |

The beauty of equal tempering is that the whole arrangement has a rotational symmetry of order 12. A piano piece written with starting pitch C can be "transposed" up so as to start with F by shifting each note five steps clockwise. No new notes are required. All the relationships between the notes of the piece will be the same: it will be the same piece of music, just set higher.

The price paid for this convenience is that none of the intervals, except the octaves, sounds quite right. In the New Grove Dictionary of Music and Musicians there is a 14-page entry for "Temperament" which explains the history of the problem and the pros and cons of the various solutions. In particular it is still not known for sure whether the "well-tempering" consecrated by Bach was equal tempering or not. See also the Grove entry for "Well-tempered clavier." Jim Loy's web-page The Well-Tempered Scale comes down squarely on the "or not" side.

- 1. Natural harmony
- 2. Frequencies and notes
- 3. The equal-tempered keyboard
- 4. The wave equation and bird song

© copyright 2000, American Mathematical Society.