 The Clarinet BBoard
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Author: tonyscarr
Date: 2012-02-10 05:10
How fast does the clarinet reed go? Does its speed change, and if so, in what manner?
Tony Scarr
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Author: oca
Date: 2012-02-10 08:24
Excellent question. Perfect time for a science based answer
We know that Velocity=Wavelength*Frequency
We also know that Velocity of sound in air is approximately 343 meters per second.
Conclusion
The reed vibrates differently for different notes played. Specifically, there is an inverse relationship between wavelength and frequency.
This means that the higher the sound, the faster the vibration.
Science is amazing
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Author: Mark Charette
Date: 2012-02-10 10:56
oca wrote:
> Science is amazing
Especially when you read the question.
What is the SPEED of the reed (let's assume tip) (both maximum and average) for a given frequency (using a Bb clarinet and let's say middlle C as written to start) and how do the curve and length of the table affect those scalar values? Does reed stiffness affect the tip speed because of outside variables? How about reed velocity over time and how it varies vs. pressure differential for a given note? Or how it velocity over the range of notes?
The original question (what is the speed of he reed?) may have been incomplete, but it's a lot more interesting.
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Author: Luuk ★2017
Date: 2012-02-10 14:34
Let's make some semi-educated guesses to get to an indicative answer.
Let's say the tip opening at rest is 1 mm, and the reed opens 2 mm at max during playing. During one cycle the reed then travels 4 mm (2 up and 2 down). At 440 Hz that would make 4 mm per 1/440 sec making about 1.8 m/s average.
(Note: the real average speed will be 0, unless you are walking around while playing...).
Now, in reality the reed will travel somewhat faster because some time is lost while hitting the mouthpiece and, at the other end, while turning around at the extreme opening. Also, I expect that at higher frequencies the reed will not open as much as at lower freq's so this may not be linearly extrapolated to other frequencies.
If you need an order of magnitude estimation, I would say 'max speed of the reed tip will be about 2 to 3 m/s at mid range tones'.
Anyone having real measurements to put in?
Regards,
Luuk
Philips Symphonic Band
The Netherlands
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Author: Bennett ★2017
Date: 2012-02-10 14:55
Is the frequency of the reed's vibration really (almost) proportional to the frequency of the note played? When we overblow ![[E6]](http://test.woodwind.org/clarinet/BBoard/smileys/e6.gif) is the reed really vibrating 4x as fast as when we play ![[E3]](http://test.woodwind.org/clarinet/BBoard/smileys/e3.gif) ? What about notes in the clarion range? Does mid-staff C cause the reed to vibrate twice as fast as middle C?
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Author: alto gether
Date: 2012-02-10 18:19
I'd hate to think I'm the only almost-a-physicist-once in this group, but I'll answer as one.
Average *velocity* is zero, since it moves in both directions.
Something freely vibrating has displacement = amplitude * sin(t*2pi*f) and velocity = amplitude*2pi*f*cos(t*2pi*f). So, picking A440 and 1mm as amplitude(I.e., midpoint 1mm, opening to max of 2mm and closing to 0mm) the velocity would vary from 0 to 2.77 meters/sec, with an average *speed* (undirected velocity) of that * 2/pi, or 1.76 m/sec.
Clarinet reeds are NOT freely vibrating objects. They spend a variable part of the cycle stopped against the mouthpiece waiting for the next pressure pulse to arrive. Therefore the peak speed is somewhat greater than the oversimplified calculation above. Amplitude drops off with frequency, so doing much better than "about 3 meters/second max" would require measurements.
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Author: alto gether
Date: 2012-02-10 18:23
Yes, Bennett. The reed vibrates at the frequency being played. For a high note, it moves back and forth much oftener per second than for a low note. However, it doesn't move as *far*, so the actual speed it moves is *not* simply proportional to the frequency of the note being played.
Post Edited (2012-02-10 18:24)
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Author: Mark Charette
Date: 2012-02-10 19:59
alto gether wrote:
> However, it doesn't move as *far*,
It might or might not not, depending on rebound & stiffness & table curve
> so the actual speed it moves is *not* simply proportional to
> the frequency of the note being played.
Most certainly true - the average & peak speeds (at tip) are going to be dependent on - other than frequency - reed stiffness, pressure differential, table curve, possibly reed harmonic modes (I've seen pictures of the reed under vibration where the reed does not act as a plane but vibrates at higher frequencies in both lateral and longitudinal modes - i don't know what order effect these would have), and more I'm sure. An awful lot of non-linearities to confound a simple answer :D
It was/is an interesting question. 2 or 3 meters/sec already is a good clip ...
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Author: Bennett ★2017
Date: 2012-02-11 15:47
I'm puzzled about the reed frequency = note frequency. When we play very high notes, we're getting high order harmonics of a fundamental. Are not these harmonics coming from a reed whose fundamental vibration frequency is much lower than the note coming out of the clarinet?
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Author: LJBraaten
Date: 2012-02-11 16:41
Mine have reached a top speed of 72 mph. The farthest they've gone has been around 65 miles, but I'm not counting their speeds or distances before they reached me. I'm thinking about breaking my personal records by taking one of them with me the next time I fly back east to visit my daughter.
{sorry, someone had to do it}
Laurie
(Mr. Laurie J Braaten)
Post Edited (2012-02-13 01:49)
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Author: ww.player
Date: 2012-02-11 17:34
Bennett, the fundamental tone of a note is determined by the length and volume of the instrument to the first significant opening (tone hole or bell) and has nothing to do with the reed. The vibration rate (frequency) of the reed determines which harmonic of the fundamental is sounded.
Post Edited (2012-02-11 17:46)
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Author: alto gether
Date: 2012-02-11 20:43
ww.player is correct that the frequency is determined by the acoustics of the air column. The reed vibrates at the same frequency as the air because the air moves the reed, not vice versa.
That said, yes, the reed has its own physics and can affect which harmonic sounds.
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Author: Bob Phillips
Date: 2012-02-12 03:50
i like alto gether's math model. (Where did it go?)
To accommodate, Mark Charette's engineering insight, one might make some guesses at the spectrum contained in the reed's vibration to see what might happen.
A couple of years ago, I measured the frequency content of the noise made by my clarinet and shared the results here.
I was surprised to find that the spectrum came out to be darn near that of a square wave. That is for a frequency f, the sound spectrum consisted of the following Fourier Series:
Y = A*sin(2*pi*f*t) + A/3 *sin(2*pi*3*f*t) + A/5 * sin(2*pi*5*f*t) + A/7 * sin(2*pi*7*f*t) + A/9 * sin(2*pi*9*f*t) +...
where Y is the deflection, A the amplitude of the fundamental (the note being sounded), f is the vibration frequency in Hz (cycles per second), 2*pi, the factor needed to turn cycles/second into radians per second, and the sound is made up of only the odd harmonics (f, 3f, 5f, 7f, 9f...), and the amplitude of each harmonic is that of the fundamental divided by the harmonic number. That is close to what my data showed. t is the time.
To get the velocity, you differentiate with respect to the time; and that gives
V = A*2*pi*f[cos(2*pi*f*t) + 1/3*cos(2*pi*3*f*t) + 1/5*cos(2*pi*5*f*t) + 1/7*cos(2*pi*7*f*t) + 1/9 *cos(2**pi*0*f*t)... ]
Now pick a time when ALLof the cosine terms are at their maximum value of unity, to find the peak velocity to get
Vmax = 2*pi*A*f*[1+1/3+1/5+1/7+1/9+...]
Blast it! The infinite series in [] brackets sums to infinity, but gets there only slowly.
And, in the clarinet case, only the first few harmonics appear in the sound spectrum, so lets truncate it at the 9th harmonic to get the estimate:
Vmax ~2*pi*A*f*[1.79]
At altissimo A (A6), f = 1320 Hz, and Vmax for a 1-mm amplitude is
~14.8 meters/second, which is about 33.8 miles/hour.
Please check my math. Thanks
Bob Phillips
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Author: georgec
Date: 2012-02-12 04:23
Bob,
But why truncate at the 9th term? Why not the 11th? Or the 7th? As you observed, you can get as large a number as you wish by going far enough out in the series.
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Author: tonyscarr
Date: 2012-02-12 17:01
Thank you all! I hope I can sort out your various responses and see where the consistencies and inconsistencies are. This was my first posting on the bulletin board and I appreciate the level of interest.
Tony Scarr
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Author: Bob Phillips
Date: 2012-02-12 18:42
Attachment: D Livengood M3013.jpg (57k)
georgec.
OH, POO! I made a math error. When differentiating the sine, you end up with the frequency as a multiplier on the cosine, and that is n x f; so that n cancels the n (the harmonic number) in the denominator. The expression for velocity (corrected) is:
V = A*2*pi*f[cos(2*pi*f*t) + cos(2*pi*3*f*t) + cos(2*pi*5*f*t) + cos(2*pi*7*f*t) + cos(2**pi*9*f*t)... ]
It turns out that that series has a limit which gives the maximum value:
2*pi*f / sin(pi * f *t)
and, Oh Nuts. That goes to +/- infinity on every cycle --just like the wrong formula on the previous post.
I'll attach a figure to this post showing a measurement of the terms in the series. From it, you can get an idea of the amplitude of the harmonics. Here, the fundamental is D5, which is 587.33 Hz. The data were sampled at 128,000/second, limiting the measured bandwidth to 64KHz; and other the tone was held for one second, giving a frequency resolution of 1Hz, and you can see from the spread in the peaks that I didn't keep the pitch constant during the test.
The plot is in dB = 20 * natural log of the amplitude ratio, and every peak is down about 9dB because I couldn't blow loud enough to get the fundamental up to 0dB
This table compares the peaks:
1 3 5 7 9 11 13 (harmonic number)
-9 -23 -31 -37 -41 -44 -47 (for a square wave)
-9 -33 -26 -30 -47 -46 -53 (read from chart)
0 -10 +5 +7 -6 -2 -6 (difference)
Significantly, the 3rd harmonic is weak, the 5th and 7th are loud, and it looks like all of the rest are down. For example, the 13th is down 6-dB, or about half.
(The even harmonics are actually absent.)
Taking the measured data gives the amplitudes:
1 3 5 7 9 11 13 (Harmonic number)
1 .192 .73 .223 .095 .0100 .071 (Amplitude of harmonic)
Add them all up, and get ~2 (1.954) Assuming that the reed tip is slapping against the mouthpiece tip rail, this needs to be scaled by the tip opening of about 1-mm, so each of these amplitudes must be divided by 2 (assuming that on the other half cycle, the reed doesn't move further our than the tip opening) (and assuming that our sound pressure level reflects what the reed is doing..)
Finally, we multiply each of these amplitudes by 2*pi*the harmonic number * the fundamental and add them up to get:
1 3 5 7 9 11 13
537 1075 1612 2149 2687 3224 3761 (Frequency)
.5 .096 .136 .112 .048 .05 .035 (Reed excursion. millimeters)
1688 648 1380 1507 805 1015 835 (tip speeds mm/sec)
Total speed about 8,000 mm/sec, or 8 meters per second, or 17.6 miles/hour.
And it looks like more harmonics should be included, as the 13th harmonic's contribution to reed tip speed is still about 10% of the total.
Bob Phillips
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Author: alto gether
Date: 2012-02-12 21:01
The time derivative of a series that sums to a square wave will always sum to infinity, since a square wave consists of instantaneous changes in position. Since a mass of air can not accelerate instantly to infinite speed, it follows that no real sound is actually a square wave.
(I just removed several paragraphs of me attempting to explain the physics of a clarinet reed/air-column interaction. Google "single reed acoustics" to find the good stuff.)
I conjecture that boils down to this: take the simplest mathematical model and multiply peak reed tip speed by somewhere between 1.5 and 2.
It would be feasible to mount a laser/mirror/light receiver system in an actual clarinet or saxophone and actually measure reed speed. Since looking for reed tip speed measurement already gets this thread, it may be that nobody has done it yet.
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Author: BRNman
Date: 2012-02-13 00:11
Whoa... musicians are really good at math... or aerodynamics... or both. cool
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Author: Bob Phillips
Date: 2012-02-13 16:01
OOOPPPS, it did, and I skipped over it. Sorry.
Alto gether. Please do that in-laboratory reed speed measurement.
I've got doubts that the "away from the mouthpiece" part of the cycle looks like anything other than a "curving diving board," and that the interesting stuff happens when it slaps against the lay of the mouthpiece.
(and how hard do you have to blow to get it to close completely?)
Good MS ME thesis project! Particularly if you figure out what happens at the edges of an unbalanced reed, ...
... and, ...
Bob Phillips
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Author: alto gether
Date: 2012-02-13 19:34
Bob Phillips: I'm a songwriter and retired computer geek with a 40-years-obsolete MSc in physics and no access to a lab. The measurement I suggest would have been impossible way back then, but the equipment now available makes the experiment doable for anybody with about a work-year of time, a couple of thousand dollars of budget, the enthusiasm to do it, and either moderate mechanical skills or access to a damn good lab assistant. It sounds obvious as a Masters-level musical acoustics project but a bit much for a hobbyist with my skill set.
In my former field, there is a saying that the two most dangerous things are a programmer with a screwdriver and a technician with a password. I was that programmer.
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