Equal Temperament

 If we consider what happens when the sequence of perfect fifths is extended until 13 different pitches are produced. Starting from 1, the second perfect fifth has the relative frequency 3/2, the third pitch has the relative frequency (3/2)² = 9/8, and so on. 

The thirteenth pitch obtained in this manner is equal to:

                   (3/2)¹² = 531,441/4096 = 129.7463......

which is close to, but not quite equal to, the starting pitch raised by seven octaves, or 2⁷ = 128.  The pitch discrepancy between the two (just a little over 1 percent) is the Pythagorean comma, which is the amount by which the sequence of perfect fifths misses being "closed" after the sequence is carried out past twelve pitches.

One way to arrive at the basic idea of equal temperament is to consider the small size of the Pythagorean comma.  If the perfect fifth were to be replaced by another slightly smaller interval, called the tempered fifth then the discrepancy of the Pythagorean comma could be spread equally among twelve intervals, each of which would be very close indeed to the perfect fifth.

In other words, we arrive at equal-tempered tuning by noting the closest approximation of a solution to the previously mentioned fundamental problem of musical pitch:

                                      (3/ 2)^m =(2/1)ⁿ

occurs when M = 12 and N = 7, the discrepancy being the Pythagorean comma. We then relax the constraints on the problem slightly by substituting the tempered fifth for the perfect fifth.

Because the Pythagorean comma is small, we expect the discrepancy between the tempered fifth (T5) and the perfect fifth (P5) to be even smaller.

TOM SCARFF
1 Martello Court
Portmarnock
Dublin
Ireland.

Email: tscarff@eircom.net