Musical Intervals and the Equal-Tempered Scale

Intervals in music are perceived to be "equal" if their frequency ratios are equal. For example, if you double the frequency of any sound you hear the pitch go up by one octave. If you increase the frequency by a factor of 3/2 (or 1.5), it goes up by a "perfect fifth". A factor of 4/3 gives you a perfect 4th, a factor of 5/4 is an un-tempered major third, and a factor of 6/5 is an un-tempered minor third.

The equal-tempered scale is best illustrated by the frets on a guitar. The frets divide the octave into 12 equal intervals, called "semitones". We use twelve because that way we find notes that are pretty close to the perfect fifths, fourths, and thirds described above. Going up one fret increases the frequency by a certain factor which I shall call "S". Going up two frets increases the frequency by two consecutive factors of S, or by S squared. Going up 12 frets increases the frequency by a factor of S to the 12th power. Since that's supposed to be one octave, we conclude that S to the 12th power must be equal to 2. Therefore S must be the 12th root of 2. With a calculator we find that S is approximately 1.0594631.

Lowering the pitch of any note in an equal-tempered scale by one semitone is equivalent to dividing its frequency by S. We accomplish this in wind instruments by increasing the effective length of the instrument by that same factor. If "Lo" represents the overall effective length of a horn or string, then we can make it play one semitone lower by increasing its length to S × Lo. The amount of extra length required to do this is the difference between those two lengths. I call that extra length "L2" because it is normally the length of the second valve tubing. Therefore:

L2 = S × Lo - Lo . . . OR, . . . L2 = (S - 1)Lo = 0.0594631 × Lo.

It's a good idea to use the equal-tempered scale as a starting point in adjusting your valve tubing lengths, but one must remember that it is a compromise, and is not a substitute for listening and adjusting to lock into the harmony of the ensemble.

Next: Tuning the First and Second Valves

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