Of all these methods, the shortest, easiest and most accurate is that which employs fourths and fifths only. It is used in such a manner that, by tuning a circle of fifths and fourths, the last sound tuned provides the octave to the first, thus completing the circle and the octave of tempered sounds.

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In this method we proceed as follows:

  1. Pitch C is tuned by the tuning fork.
  2. F below pitch C is tuned, being a tempered fifth.
  3. G below pitch C is tuned, being a tempered fourth.
  4. D above G is tuned, being a tempered fifth.
  5. A below D is tuned, being a tempered fourth.
  6. E above A is tuned, being a tempered fifth.
  7. B below E is tuned, being a tempered fourth.
  8. F sharp below B is tuned, being a tempered fourth.
  9. C sharp above F sharp is tuned, being a tempered fifth.
  10. G sharp below C sharp is tuned, being a tempered fourth.
  11. D sharp above G sharp is tuned, being a tempered fifth.
  12. A sharp below D sharp is tuned, being a tempered fourth.
  13. F above A sharp is tuned, being a tempered fourth.

This last F is the octave to the first F tuned, and should coincide exactly with the latter.

The reader will, of course, realize that the tempering of these various intervals must be tested by means of the generated beats. Helmholtz, in “Die Lehre der Tonempfindungen,” calculates that the tempered fifths should average .6 of a beat per second at standard pitch within the octave that we are treating. This is equivalent to three beats in five seconds. But it is impracticable to measure the generated beats upon the pianoforte in this manner. The tone of the instrument is too evanescent and fleeting. We may, however, attain to a very fair approximation. If, for example, each fifth be tuned so that two distinct beats are heard before the sound dies away, it will be found that the beat-rate is a near approximation to the calculated average. The two beats that we speak of occur in about three seconds, while the Helmholtz rate is three and one-third seconds for two beats.

Again Helmholtz gives an average beat-rate for tempered fourths in the same octave; namely, one per second. If we tune the fourths so that we hear three distinct beats we shall likewise obtain a very fair approximation to the calculated beat-rate.

We showed above that the last sound produced by the building up of a progression of twelve fifths is sharper than the sound produced by the piling up of seven octaves from the same tonic. The two sounds thus produced ought to coincide, for the compass of twelve fifths and of seven octaves is the same. We concluded, therefore, that the Equal Temperament required the flattening of all the fifths.