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The language Benade uses is that the reed "collaborates" with the natural resonant frequency of the length of the vibrating air column.* As the length of the "tube" gets shorter and shorter, the natural resonant frequency weakens and has less control over the oscillation of the reed which then allows the player's oral cavity to take charge of the reed's vibrations. This is what makes playing in the altissimo range possible. This is also why an entire octave scale can be played on the mouthpiece alone and why players can play a palm D and lower the pitch as much as a step and a half or more just by changes in the oral cavity. On lower notes with a longer body tube this becomes much more difficult, if not impossible.

In a study entitled "Some Aspects of Tuning and Clean Intonation in Reed Instruments" J. P. Dalmont, B. Gazengel, and others found that the natural resonance frequency of a given note determined by the impedance is somewhat higher than the "played frequency" of the note. Their conclusion was that the mouthpiece combined with the player's embouchure produced a sort of "end correction".

* In my understanding of the term, this is not the same as "sympathetic vibrations" in which an object whose natural resonant frequency matches the frequency of soundwaves coming from another source and is thereby set into vibration. I used to demonstrate this effect to my class by pressing the sustain pedal on a piano and playing a note on a trumpet loudly into the open top. Once the trumpet stopped, one could hear the strings that matched the fundamental and harmonics of the played note sounding only those pitches.
 

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Given this is true, which I believe it to be, what is the significance of blowing a particular pitch on a mouthpiece alone?
This has been a much debated topic on SOTW for quite a while. This is just my opinion:
  • Practicing on the mouthpiece alone and learning to play tunes and scales helps to develop control of the oral cavity which is essential to playing overtones and altissimo.
  • Playing the mouthpiece apart from the saxophone (without changing the oral cavity) is a tool to gauge how tight or "pinched" the embouchure is by what note it produces.
 

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I disagree. It might make sense if all mouthpieces were the same, but they aren't.
I make mouthpieces with different chambers and shank lengths, the pitch varies.
Sidestepping the "argument" that the alto mouthpiece must produce an A concert or the tenor mouthpiece must produce a G concert, can we agree that tightening the embouchure to play a pitch that is significantly above the "natural" pitch of the mouthpiece as determined by the chamber and shank length produces a tight and pinched tone on the instrument? Conversely can we agree that loosening the embouchure to play a pitch that is significantly below the "natural" pitch of the mouthpiece as determined by the chamber and shank length produces a tone on the instrument that is flat and flabby? This is what I mean by using the pitch of the mouthpiece as a teaching tool.

Note: This is not the same as raising or lowering the played pitch on the mouthpiece by changes in the "oral cavity" while keeping the tightness of the embouchure the same. Also in my experience, I can "lip" notes much farther up and down on the mouthpiece alone than on the neck or the instrument itself. This leads me to conclude that on the mouthpiece alone, the embouchure of the player has at least as much, if not more control over the pitch sounded that the natural resonant frequency of the length of the mouthpiece shank + the size of the chamber.
 

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The discussion appears to have drifted from the topic of which frequency the reed vibrates at to the pitch played on the mouthpiece alone, although they are somewhat related.

As far as I can tell the idea of the mouthpiece pitches can be traced back to Santy Runyon and is still used by classical teachers and players today, notably Eugene Rousseau.

Santy's fable about having the "Theramin" provide the input pitch into the saxophone has no basis in science and has since been thoroughly debunked. However there are parts of Santy's story that have a ring of truth about them. He claimed that after he learned what mouthpiece pitch to use on his alto sax, his tone sounded better so much that other saxophone players he played with wanted to know his secret. He ended up giving many of them lessons to supplement his income from playing.

I did not come up with this hypothesis, and can't remember where I first read it, but to me it sounds entirely plausible. It goes like this:

In the late 20's and early 30's there were lots of clarinet players who noticed that saxophone players were getting a lot more work. They decided to get a saxophone and teach themselves to play in an effort to get "a piece of the action" and capitalize on in the popularity of the relatively new instrument. Many of them having no training and being unfamiliar with the saxophone tried to use their clarinet embouchure at first and play near the top of the mouthpiece pitch as is done in producing a "classical" clarinet sound. The result on the saxophone is a thin, pinched, tone, that is sharp in the higher register. Santy had learned to open up his sound by playing an A concert on the alto sax mouthpiece and shared that "secret" with those players who came to him for help.
This makes perfect sense to me because I played next to saxophone players in my university jazz ensemble who were "clarinet players" who switched to saxophone to fill in the empty slots in the band. As lead alto for four years trying to match pitch with someone who was always sharp in the upper register and get a full "sax soli" sound when players in the section had weak, thin tones was frustrating to say the least. Unfortunately I hadn't yet learned about how mouthpiece input pitch can affect the tone quality and intonation, so I was unable to offer any help. Following graduation I played tenor in a small "society band" next to a clarinet player playing lead alto who was the leader of the group and had several more years of the same experience. By then I knew the problem and the solution, but the individual was not open to help or advice since he sounded "fine" to his ears.
 

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I don't understand how they are related although the reed is associated with the mouthpiece so a good opportunity to bring up the topic of mouthpiece alone pitch yet again. Maybe it used to be relevant in the old days? (And I won't try to deny I mention it in my book/site, but I think things have moved on since Santy Runyon, however much respected he is.
The point that I wish to make is this: Just as the natural resonant frequency of the sounding length of the body tube of the saxophone determines the frequency of the reed's vibration, so does the the natural resonant frequency of the shank of the mouthpiece. The difference is that the length of the body tube has more control over the frequency the reed vibrations the longer it gets which narrows the range a player can "lip" notes up and down to change the pitch. The shank of the mouthpiece on the other hand has far less "command" of the frequency of the reed vibrations due to its shorter length. This allows the player to "take charge" of the frequency of the reed's vibrations and permits a much wider range of pitches both by tightening and loosening the embouchure, and by "voicing" changes in the oral cavity.

This topic was "fleshed out" in detail in this thread on Cafe Sax 7 years ago: Mouthpiece Pitch Revisited
 
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