Recently I got into an e-mail discussion on hand stopping. I had not thought about it in a while and some interesting points came out as a result.
Of particular interest was the fact that there are two apparent pitch changes that result from hand stopping. A question was posed to me as to whether or not the pitch went up or down. I commented that I had always heard it going down. This is attributable to the fact that the more the hand is inserted into the bell the more it is forced to behave like a pressure anti-node as opposed to a pressure node (sorry for the jargon -- I'll explain it in another page in more plain English). She wrote back that she sees two pitch changes; first a lowering of a semitone and then a sudden upward shift of a wholetone yeilding a net semitone shift above the open horn pitch.
Having thoroughly read John Backus' book, "The Acoustical Foundations of Music", I had read about this, but I had forgotten about it.
I attempted an "off the cuff" set of explanations to see if I could shed some new light on the subject. As an ex-engineer I like to use the old "engineer's axiom"; "If all else fails, read the instructions!" I assiduously avoided looking it up until I had thought it through concretely first. After that I did refer to Backus' and Rossing's books.
I must say that I have abandoned my explanation for theirs. After all, they are experienced PHDs in physics while I am slowly completing my physics masters. Learning is indeed an amazing and wonderful thing.
What follows is a discussion or interpretation of their ideas. I will try to avoid jargon as much as possible.
Pitch Changes in French Horn as a Result of Hand Stopping
I will begin with the assumption that most french horn players will readily agree the hand stopping will lower the pitch of the instrument a semitone. The question arises when we move the hand more tightly into the bell for what is termed a full stop. It is accepted that the pitch is then a semitone above the open horn. This is, in fact, only partially true. This can be readily explained by coming to an understanding of the harmonic series and thinking in ratios as opposed to whole and half steps. Note: almost no math skills are required here, so don't worry!
Let's review the overtone series for a horn in F:
These pitches were taken from the listing for the tempered scale in Backus' book, page 153. It should be noted that the note listed as the fundamental, F1, is in fact a false fundamental. The actual note that can be hit on the horn will be lower than that. It can be lipped up to the F1, but the actual resonance is lower. This effect is even more noticeable on trumpet (the instrument which I play). The point here is this: when one plays a pitch and in lipping up hits a note a perfect fifth higher, the original note was in fact the second harmonic (or first overtone, if you prefer). This should help us keep our bearings.
Now let's assume that by full stopping that we have created a new series which is such that each new note has been lowered to roughly a half step above that of the original. This is what Backus suggests happens, and this is indeed the case. It is quite easy to prove this to one's self.
Here are the two series side by side for comparison:
If you carry out the division for the other notes you'll notice some variation, but considering that tempering a scale is somewhat confilctory with the natural resonances, and that these numbers are moidified for temperment, the variance, at least to me, seems acceptable.
Now, some of my readers are experienced physicists. I invite them to contact me if they feel that I have not done a proper analysis of this situation. I don't like the fact that the the ratios aren't identical and feel that brushing this off by virtue of temperment seems a bit sloppy. I am sure that there is something missing. However, I haven't had time to dig in on this. I brought this up at a physics teachers conference in the spring, and no one commented, but, perhhaps they just didn't think of anything at the time. If you, dear reader, can see the flaw here, please let me know, and I will adjust this page. I think the qualitative discussion that follows, however, is accuarte and helpful.
Now if we try to examine this by musical interval, we can, if not careful, become confused. For example, the first two notes are as follows: lowering a C3 to an F#2 constitutes an interval of a tritone. However, lowering a C4 to an A#3 is an interval of a major second. You may wonder "how on earth is it that my horn can figure out what note that I'm playing and lower it just the right interval to cause a new series just a half step above the old one?" The mistake here is in analyzing this physical effect in musical terms. The physics insists on being a bit more mathematical. What in fact happens is that the entire series is lowered evenly across the spectrum by a somewhat specific ratio of frequencies that ends up just a half step above the original series.
Now to explain why it seems as though there are two pitch changes when moving from a stopped horn to a full stopped horn. With the new series so close, horn players naturally slip up to the next series without necessarily being aware of it. They are experienced in simply dealing with a new transposition and leaving it at that. It is simply assumed that there are two pitch changes. This is not the case, as can be seen with a little study.
You can prove this to yourself. In fact, I found it rather obvious, even though I am a professional trumpeter and am rather clumsy on horn. Play a note in the middle register of the horn. Note the feel. Now full stop the horn. You should notice that there is a "clicking into place" feel (or, if you prefer, a "dropping into a slot" feel). This feel can only occur if we are moving via our embouchure to a new harmonic! It may seem that you have drifted up a half tone (musically, this is the case), but what in fact happens is that you have drifted so low that the next higher harmonic is only a half tone above where you "just were" so you "go there."
Now try slowly stopping the horn without allowing your chops to click up to a higher harmonic. You should find it rather easy to feel the pitch drift down in a smooth and continuous fashion. If you hold the note you can then nudge it up to the next "stopped" harmonic, which is the one a semitone higher.
In conclusion, there is only one way the pitch moves via hand stopping a horn: DOWN!
Now as to why the pitch drops, I'll refer you to John Backus book (The Acoustical Foundations of Music) for now. I'll add another page with a layman's interpretation of this.
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