such as a Pennywhistle

by N. Drozdoff

Nicholas Drozdoff ndrozdoff@hotmail.com

aka The Harmonic Series

Before we move on to closed end resonance or closed end pipes (a very important discussion because we are then getting closer to brasswinds) it is important to understand what the harmonic or overtone series means in musical terms. Let's consider a series built on an open ended pipe which has as its lowest note (fundamental) a Low C with a frequency of 130.81 hz (recall that a hz is a cycle per second or the number of times the wave travels back and forth down the pipe in one second).

The series for this pipe will go as follows:

- Low C at 130.81 hz (1X fundemental or first harmonic)
- Middle at 261.63 hz (2X fundamental, second harmonic or first overtone)
- G over middle C at 392.00 hz (3X fundamental, third harmonic or second overtone)
- C above middle C at 523.25 hz (4X fundamental, fourth harmonic or third overtone)
- E5 at 659.26 hz (5X fundamental, fifth harmonic or fourth overtone)
- G5 at 783.99 hz (6X fundamental, sixth harmonic of fifth overtone)
- B-flat-5 at 915.67 hz (7X fundamental, seventh harmonic or sixth overtone)
- C6 at 1046.5 hz (8X fundamental, eighth harmonic or seventh overtone)
- D6 at 1177.29 hz (9X fundemental, ninth harmonic or eighth overtone)
- E6 at 1308.1 hz (10X fundamental, tenth harmonic or ninth overtone)
- F6 at 1438.91 hz (11X fundamental, eleventh harmonic or tenth overtone)
- G6 at 1569.72 hz (12 X fundamental, twelfth harmonic or eleventh overtone)

and so on.

I dragged the series out like this to make a point of some relevance to trumpeters. Notice how the notes get progressively closer together as we get higher in the harmonic series. From C = 1046.5 hz we have a major scale. This is the effect that allows natural trumpet and natural horn players to produce nice scales in spite of the lack of valves. Now, beware! This scale is not exactly what we are used to calling "in tune", but, for the purpose of our discussion here, it we should now have a better understanding of how open ended resonance works.

Now you should be screaming bloody murder! All of this overtone stuff is about open ended resonance but brasswinds are inherently closed ended!

If I have bugged you about this, then I'm glad! We'll dispel those misconseptions caused by errant physics teachers and musicians yet!

This protest is correct. Our next discussion will be on closed end resonance. We'll see that closed end resonance doesn't work the same way. However, brasswinds are designed using geometric trickery in order to contrive this open ended series, or rather, one very close to it.

For now, however, simply enjoy the simple harmonic structure of an open ended pipe. The rest will become clear in our next couple of installments.

As an aside before I close this discussion out: notice the octaves are all in powers of 2: 2X, 4X, 8X, 16X, etc. As I mentioned in the pysics of open ended tubes (Essay #1A) this is a "sure sign" of open ended resonance.

What, then, would consititue a "sure sign" of a closed ended resonance?

Tune in next time.

*P.S. I have adjusted the numbers shown above to reflect
the proper value for middle C. Originally I had succumbed to the
old physics teachers plight; I used 256 = C because it was shown
this way on the tuning forks that I had used in my high school
physics classes. As a professional musician by night, I wanted to
make sure that this oversight was corrected. Much of the info. used here came from the John Backus book cited in the "Suggested Reading" section (see index page).*

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