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overblowing wind instruments

 Quote by bcrowell OK, I have a recorder I can use at work, and a convenient setup for measuring frequencies for a student lab, so I'll try it. When you claim "close to zero" "correlation," what exactly are you claiming? Are you claiming that even for a C scale (i.e., essentially just opening the holes one at a time, starting from the bottom), there will be close to zero correlation in the statistical sense of correlation (i.e., a small R^2 value)? I'd definitely bet a six-pack against that. Or are you just claiming that my simple explanation fails to explain stuff like cross-fingerings (which I specifically said it wouldn't do)? Or are you claiming that a C scale will just not give detailed quantitative agreement with my simple explanation (which wouldn't surprise me)? I suspect that we just have differing ideas of how much simplification is too much.
The holes in the recorder will have different sizes. That implies that someone has already taken into account the failure to follow a simple rule, I think. I think one reason that a recorder is designed so specifically may be to remove the skill required for accurate note production. That's why it works so well (adequately) in groups of relatively unskilled child players playing in 'simple keys'. Oh god - twinkle twinkle again!!!

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Here's my data from the plastic toy recorder we had in the physics stockroom. I don't think it's meant to be a real, usable musical instrument. (For one thing, it's virtually impossible to get the damn thing to speak on the high notes unless you start on a low note and run up a scale to the high one. This is kind of interesting in its own right, since it seems physically similar to overblowing -- an effect that clearly can't be explained by any simple, linear model.)

f (Hz),open holes,b (cm),note,v/2b (Hz)
590,0,29.6,1,574
655,1,25.1,2,677
730,2,23.1,3,736
780,3,21,4,810
870,4,18.9,5,899
995,5,16.7,6,1018
1020,6,14.4,b7,1181

This is in CSV (comma-separated value) format, which people should be able to read into a spreadsheet program, but I think it's more or less readable as is. The first column is the measured frequency. The second is the number of open holes. The third is the distance from the tip of the mouthpiece to the center of the first open hole (or to the end of the tube). The fourth column is the frequency v/2b predicted by the simple model I proposed, which is that the frequency is determined by simply truncating the length of the sound column to the first open hole and taking that as half a wavelength.

I would say that the agreement between theory and experiment is reasonably good. The agreement is not good enough that you could use v/2b as your sole mathematical rule for building a good, practical instrument that plays in tune, but I think it's good enough to say that the theory is basically right as a first approximation, and further corrections can be taken as perturbations on top of that.

The first-open-hole model predicts that it doesn't matter what holes you close or open below the first open one. This seems to be more or less true on this instrument, but not exactly true. I observe that if I finger a note near the top of this scale and then close and open various holes that are farther from the mouthpiece, I sometimes get no change in pitch and sometimes get a drop of a quarter-tone or a semitone.

 Quote by sophiecentaur The holes in the recorder will have different sizes. That implies that someone has already taken into account the failure to follow a simple rule, I think. I think one reason that a recorder is designed so specifically may be to remove the skill required for accurate note production. That's why it works so well (adequately) in groups of relatively unskilled child players playing in 'simple keys'. Oh god - twinkle twinkle again!!!
On this piece-of-**** instrument, the holes that your index fingers rest on are slightly smaller than the others.
 Recognitions: Gold Member Science Advisor Staff Emeritus dlgoff's #17 is interesting, because it shows how a nonlinearity enters in the case of a reed instrument. However, it's not obvious to me how that nonlinearity produces overblowing. I found a book that discusses overblowing very clearly in instruments such as recorders, bamboo flutes, and whistles: Backus, The acoustical foundations of music, Norton, 1969, pp. 184-186. These instruments work based on edge tones at the mouthpiece. Here's a brief online description of an edge tone: http://hyperphysics.phy-astr.gsu.edu...usic/edge.html To me this example is very transparent. The edge tone occurs because of a flow of air that is binary in nature. The air either flows past one side of the knife-edge or the other. This binary object "wants" to do its 010101... oscillation at some frequency, but it has to accomodate itself to the air column, which only has certain resonant frequencies. In overblowing, this square-wave vibration occurs at a harmonic of the air column's fundamental frequency. The square wave has Fourier components at all the multiples of that, and some or all of these components match up with the resonances of the air column.
 Recognitions: Science Advisor There are two strange things about your experiment. With all holes closed, the discrepancy in frequency is the opposite way to all the other results. And the discrepancy for the highest note is much bigger than the rest (15% compared with 2 or 3%). I'm not sure what "the tip of the mouthpiece" means. The start of the vibrating air column is the "window" or rectangular slot before the finger holes. The bevelled edge of the "window" is the lip that created the edge tone, if you believe in edge tones. Did you see the comment by Benade in your hyperphysics link on edge tones? It's easy to measure the edge tone frequency of an organ pipe. If you put some cotton wool in the pipe to damp its resonance you can hear the edge tone. As Benade says, the observed edge tone frequencies are much too high for the coupled system to "pull" the edge tone frequency to match a resonance of the pipe. For some styles of organ pipe voicing, you can hear the edge tone as a transient sound at the start of the note. If it is at a harmonic of the pipe frequency, it is more like the 5th or 6th harmonic than the first. FWIW the usual problem with recorders being reluctant to playing some notes is just internal dirt. A quick blast with a high pressure air line might fix that problem! Plastic recorders come in all qualities, from complete junk up to instruments used by professionals. For folk music, they have advantages over wood - the sound is louder and brighter, for outdoor use they are much less temperamental about humidity or even rain, and they are a lot more resistant to accidental damage.

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 Quote by AlephZero There are two strange things about your experiment. With all holes closed, the discrepancy in frequency is the opposite way to all the other results.
This doesn't surprise me, because b is being defined differently in that case than in the others. That one is measured to the end of the tube rather than to the center of an open tone-hole.

 Quote by AlephZero And the discrepancy for the highest note is much bigger than the rest (15% compared with 2 or 3%).
Yeah. Actually, looking at standard recorder fingerings on the web, nobody seems to use the one with all tone holes open. Also, the set of tone holes on this instrument is different than on a normal recorder. It only has 6, not 7 or 8.

 Quote by AlephZero I'm not sure what "the tip of the mouthpiece" means. The start of the vibrating air column is the "window" or rectangular slot before the finger holes. The bevelled edge of the "window" is the lip that created the edge tone, if you believe in edge tones.
"Tip" means the very end, the point farthest from the bottom of the tube. I'm sure you're right that the window would be more appropriate than the tip. (Backus calls this the "mouth.")

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 Quote by bcrowell "Tip" means the very end, the point farthest from the bottom of the tube. I'm sure you're right that the window would be more appropriate than the tip. (Backus calls this the "mouth.")
That would explain the lowest note result, because your "measured" tube length is clearly longer than the acoustic length, unless there is a huge end correction factor for some reason.

Whatever model of the physics you use, the acoustic length of the tube should always be longer than the actual length. That's what made me curious about your lowest note.

 Yeah. Actually, looking at standard recorder fingerings on the web, nobody seems to use the one with all tone holes open. Also, the set of tone holes on this instrument is different than on a normal recorder. It only has 6, not 7 or 8.
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So it's probably not a recorder, but a "tin whistle" made from of plastic. Actually that would explains why all the frequencies looked strange. The most common recorders have lowest note C (523 Hz) and the only other likely possibility is F. Tin whistles usually have the lowest note D (587 Hz) which is fairly close to what you measured.

Your hypothesis is that a tone hole acts like open end of a tube. I don't see that it matters that you are using non-standard fingerings. You are comparing the measured frequency with a mathematical model of the pipe, not with any particular musical scale.

It would be nice to know the tube lengths measured from a more correct end point. The pattern of the discrepancies from your hypothesis might then be more revealing.
 Recognitions: Gold Member Science Advisor This is certainly a complex topic. You haven't even got onto Timbre yet!
 The tube does not just choose the note within a wide noise spectrum or between several harmonics produced independently by the reed or blowhole. It determines how the reed or blowhole behaves to create just the emitted note. Easier to understand with a reed. During one oscillation period of the air column, when the pressure is minimum, it attracts the reed to close the mouthpiece, thus reducing the air throughput. When pressure is high, the reed is pushed open to increase the throughput. The reed that brings air when pressure is high brings the negative impedance that lets the air column oscillate by providing power to it. For this, the reed must act as a spring, not a mass. In other words, its resonant frequency must be quite higher than the emitted note. In a blowhole, the negative impedance relates with the time needed by air to cross the hole (hence the air pressure). It's similar to some power vacuum tubes: pressure deviates the air jet, but once the jets arrives at the bevel, the pressure has already changed, hence the negative resistance. Though, at a flute, we change also the thickness of the jet and its angle. ------------------------------------ You shouldn't try to understand pitch vs length on a recorder, because the bore is not cylindrical nor even conical. It's first straight, then converging, in order to overblow in tune despite other effects, first of them the inductance of the narrower blowhole. As well, narrow toneholes are inductive and produce a lower note. Just look where the overblow hole is located at the clarinet, but it produces a (bad) lower note on the fundamental because it's narrow and long hence inductive. One reason more against understanding through a recorder, or even worse, a bassoon, which works as an ocarina (Helmholtz resonator) in its short notes. Take a flute, you can explain the pitch then. Or a clarinet also, but its bore isn't really cylindrical, nor its toneholes big. The flute has a conical headjoint, but only to combine with the extra volume near the stop and the inductance of the blowhole, in order to tune the overblow. ------------------------------------ The inductance of narrow toneholes makes them less effective in stopping the wave there, so lower holes matter and can define half-tones. Narrow toneholes are also extremely important for sound quality. They define losses, frequency response, and even a very nonlinear behaviour at the bassoon. After wide holes improved the flute, all best manufacturers tried them on all instruments and consistently failed. The clarinet has gotten and kept holes more or less at the proper places, but not really big. The oboe and the bassoon had to keep very small holes to sound good, even at the Heckel design. Narrower holes explain most of the Tarogato's sound, different from a soprano saxophone. Notice the many small holes at its bellow. One nice bassoon attempt by Triébert can be seen in Brussels' museum. ------------------------------------ The part with negative conductance the creates the oscillation has also a susceptance. This is personal thought, not widely known. Again easier to grasp at a reed than a flute, and can be experimented convincingly when playing a saxophone, or even better a bassoon. As higher pressure in the tube pushes the reed away, the mechanical movement also increases the air volume in the tube. This is equivalent to a capacitance. Not a small one: similar to the tuning volume in a saxophone's mouthpiece, or to its missing volume at the truncated cone. This susceptance is in parallel with the reed's negative conductance and with the admittance of the air column. It influences the pitch AND the overblow. A harder reed (best combined with a less open mouthpiece) needs more pressure for the same movement hence produces a smaller susceptance. Same for a narrower reed and mouthpiece. They pitch the instrument higher. Little bit at the clarinet, important at the saxophone, huge and obvious at the bassoon, where musicians tune the reed by cutting it. It is this susceptance that prevents the instrument from overblowing when not desired. The susceptance must be big enough to short-circuit the negative conductance, so that the reed can't compensate the column's losses at the harmonic, and oscillation occurs at the fundamental only. Now, if you press the reed more firmly, first the pitch rises because the free-moving part of the reed is shorter hence the reed's susceptance decreases, and then the note jumps to the overtone. Which means that the reed and the mouthpiece must match the instrument. With parts from an alto saxophone, you can't overblow a bassoon. ------------------------------------ Back to the wide toneholes: timber gets bad when the reed is too efficient, because the reed closes and opens too brutally then. Similar to a bagpipe then. The musician adjusts his lip pressure for that, but only if the instrument allows it. The losses at the air column must match approximately the reed's negative conductance. As the closer reed is needed to produce higher notes, the negative conductance changes, and the tube's losses must still match the reed and mouthpiece. This is true when bridging to the second mode, and needs shorter notes to have narrower holes than long notes - can be observed at real instruments. This equilibrium would be necessary when playing soft and loud. The clarinet achieves it, the saxophone doesn't. Narrow holes also de-tune the higher harmonics which then are attenuated as they don't resonate any more. Softer sound, emission less easy. Harmonics alignment is knowingly paramount to make an easy instrument, but because the reed's susceptance varies with the note, the manufacturer can't tune the overblow perfectly and align the harmonics. ------------------------------------ Octave holes help a lot by killing the resonance at the fundamental (or generally the unwanted mode. Early saxophones had independent octave keys that helped the third overtone if wanted). For that, the opening doesn't suffice, as it would only change the pitch. They must also bring losses big enough to prevent oscillation, even though the reed would provide the negative conductance. This part is better documented in books. ------------------------------------ Cross-fingerings can be partially understood as better octave holes, always at the optimum location, possibly combining several of them, to define strongly the allowed vibration mode. But they're more. As the frequency increases, tone holes get less efficient in producing a short-circuit, because of their inductance - more so if they're narrow. More well-situated open holes add their effects to produce a good sound reflection, while closed holes in between keep the vibration. At higher notes, the length corrections by the tone holes and the reed and mouthpiece are huge, so the distance between the cross-fingering holes is easier to compare with the note's pitch. My apologies for the long post, it's a passion...