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Tuning Sofware That Shows 3Rd And 5Th Overtone Frequecny


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I understand that the pitch of the 3rd and 5th overtones can be adjusted to match correctly to the fundamental when tuning a reed (the inharmonicity). This is achieved by adjusting the profile of the reed rather than just filing the base or the tip to lower or raise the overall pitch. The reed tongue vibrates in a figure 8 pattern if not filed flat. All this shows up in the overtones.

 

Problem is... there are plenty of tuners that show the root frequency and some that have built in spectrum analyzers like TonalEnergy and others that show multiple notes at once like Dirks but what about the actual frequency of the 3rd and 5th overtones!

 

Does anyone know of a tuner that show the frequencies of the fundamental, 3rd and 5th? Amplitude of the overtones would be nice as well.

 

I was once told that 6 out of 10 concertina suffer from reed vandalism! I want to be part of the solution, not part of the problem.

 

Thanks,

John

Edited by 4to5to6
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Would be interesting to hear of a tuner that is capable of shoing 3rd and 5th, you are defnately right that the usal amount of Inharmonicity is a question of incitent. Normaly nobody taks care of this circomstancy. In my opinion not the 3rd and the 5th are the real problem- is more the higher asolutly out of tune tranversal rotation of the reed. filing the reed to a piramide or thinner at the ages my change this as well. Best regards, Johann

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While the reed's can vibrate in multiple harmonic modes that can be varied in pitch by adjusting the profiles, in a flat reed, these natural frequencies rise at a significantly greater rate than the musical harmonic series of an ideal string ( one with no stiffness ) what would be the second harmonic is significantly sharper than the octave, and it gets worse fast from there. I have plotted these ( natural harmonics for both reed's with no profile, and for reed's with a variety of profiles ( pulled from accordions and good concertinas) and none of them approach the straight line graph of the musical harmonic series. They all look like one side of a steep parabola.

In the instrument, however, with the combination of reed driven by airflow, an excellent musical series is produced. Under a strobe, the reed exhibits only it's simplest vibrational mode. There is some disagreement about how the overtone series is produced, but it is not due to the reed itself vibrating in multiple modes. My personal opinion is that it is a result of the varying velocity of the reed when it is moving with the airflow on the down swing compared to the velocity curve moving against the airflow on the up swing. This not quite sinusoidal motion is divided up ( probably in our brains or ears ) into the normal harmonic series in strengths for each harmonic so their sum matches the overall wave form. If you look at the wave forms of a note generated while playing. You will see it is nearly sinusoidal at low volumes, and becomes drastically less so at higher volumes.

My instruments all share a fairly strong musical fifth overtone ( nicely in tune ) and while I spent a long time refining my profiles, and love the sound of these instruments. I made zero effort to try to tune harmonics, and instead chose to go for a very smooth stiffness gradient from one reed to the next so each reed behaves very much like it's neighbor.

I am not disputing that it is possible to tune a few natural resonances of a reed much like the center is carved away on a vibraphone bar for a nice pure sound. ( tuning more than one you will quickly find tuning the next spoils the first) But sound production in free reeds is a very different process than in stringed instruments or in struck bars, basically being a periodic shaped interruption of air flow. Getting the reed to vibrate just up and down may be a good thing if the figure eight or elliptical tip motion is sufficient to cause the reed to strike the window, but you are likely to do as much damage to the reed as the bad filing or tuning job did in the first place.

There are things local to the reed that influence its tone possibilities, like clearance, length to width proportions, and spring rate due to profile ( how quickly the reed tension builds when deflected ). Other factors in the instrument have a much larger effect on the reed sound, like materials and thicknesses, chamber dimensions, sound path etc.

Studying reed's can be fun. They are less well understood than most other sound generators. Just watch out for assumptions ( which we all have ). I have managed to disprove most if mine, and now don't a assume I am right ( no matter how opinionated ) about anything. I have worked with the ideas that have given good results over 70+ instruments. That doesn't mean they are right, only that they don't spoil a good concertina.

Dana

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Would be interesting to hear of a tuner that is capable of shoing 3rd and 5th, you are defnately right that the usal amount of Inharmonicity is a question of incitent. Normaly nobody taks care of this circomstancy. In my opinion not the 3rd and the 5th are the real problem- is more the higher asolutly out of tune tranversal rotation of the reed. filing the reed to a piramide or thinner at the ages my change this as well. Best regards, Johann

It is my understanding that the twisting mode of a reed can come close enough to the fundamental when the length to width proportions are too close. ( a short but wide reed) I don't remember what the proportions are where this can happen. Normally this twisting mode is well below the reed's pitch. I can see unpleasant things happening if the modes are too close. Is this the transversal rotation you are speaking of? I can never keep clear what transverse means in these contexts since we aren't talking earthquakes and seismology.

I always love your posts. You are so much better at saying the right things.

Dana

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Interesting and helpful discussion. Thanks! I am curious to know what tuner or tuner software you use so I can check the overtone harmonic's frequencies and amplitudes.

John

Edited by 4to5to6
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Sometimes the electronic tuner will refuse to show the fundamental, locking instead onto a harmonic, and I have always imagined this is because that harmonic is particularly loud. It may be possible to take the tuner out of auto note selection and lock it to the known note value of a fifth or third. Amplitude of the overtones is represented in some computer based tuners.

I take it your "figure eight" shape is in the vertical plane, not sideways where it would soon lead to issues in any reed with tight clearances. I check all reeds along their length for sideways thickness variations a number of times during filing and when I find a variation I cancel it by filing the other side, creating a triangular shape in the sideways reed profile (we are only talking about 1-1.5thou here) as referenced by Johann earlier in the thread.

My thinking on complex vibration patterns (I think of these as containing secondary vibrations) is they are less likely to be generated if the reed naturally assumes a pleasing curve when bent for examination ie. watch out for stiff or over flexible patches.

As a simple way to understand the production of secondary vibrations, if you were to imagine a reed with neither a belly at the root or a tapering at the tip, a reed which is a straight bar. This occurs in all concertinas (I think!), in mine it happens at around F# above middle C. If this reed was made from a piece of rolled steel at exactly the right thickness then it would curve perfectly if bent by hand. In practice, when it is filed to thickness it will have undulations. Imagine if the reed was to be 10 thou (.25mm) all along its length. Now, after filing it to size lets check the thickness. Yes, it is roughly 10thou but it in fact undulates between 9.5 and 10.5. This is 10% of it's total thickness! Now lets imagine there are two major "thin" spots (9.5thou), one near the root and another three quarters of the way to the tip. The first thin spot will cause the reed to bend relatively more at that point and lower the pitch. The second weak spot will not lower the pitch but it will mean the area of the reed between it and the tip will be able to vibrate in a separate rhythm to the rest of the reed. Imagine the reed starting and bending at the weak spot near the root. From the weak part three quarters to the tip the reed will lag behind but as the reed opens further it will catch up and then overtake the main part of the reed. It then may flex back downwards as the main part of the reed is still rising etc. And when the main part of the reed begins to return the whiplash on the tip could be considerable, cancelling or enhancing its separate speed.


4 (4to5to6 seems so formal), you don't mention where your information came from, is this from a scientific or informal source? I would be interested to hear your prescription for shifting, say, a fifth, should you be able to measure it. Where would you file the reed? And also, I think 6 out of 10 suffering from poor reed work is way too low, 9.5 out of 10 might be getting closer.


Johann, as ever your post contains tantalising thoughts placed in view but just out of my reach due to my language limitations!

Edited by Chris Ghent
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I think Wim Wakker mentioned that sort of tuning once. The tuner I used to use that showed the peaks at each harmonic is found at shaku6.com as the tuner, sound color analyzer. Unfortunately it doesn't appear to have been updated to a modern OS. Tuner apps are really variable and most are geared for specific instruments, but there should be apps that will display any audio wave form as seperate harmonics. I stopped using it since it ceased to be telling me anything new and I retired my windows 98 toshiba laptop in favor of a Mac for my microscope and other shop duties. Who knows though, it might work on a newer Windows machine. Anything is better than the Hewlet Packard Wave Analyzer ( prior to their PC days ) I started with. ( weighs 30 pounds! )

Dana

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There are a whole bunch of iOS spectrum analysers at the Apple App Store. I don't know about Android. Look for ones that lable the spikes with their frequency so you can better tell how they are mathematically related. They are only a few bucks so you might want to try different ones.

Dana

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  • 5 weeks later...
I understand that the pitch of the 3rd and 5th overtones can be adjusted to match correctly to the fundamental when tuning a reed (the inharmonicity). This is achieved by adjusting the profile of the reed rather than just filing the base or the tip to lower or raise the overall pitch. The reed tongue vibrates in a figure 8 pattern if not filed flat. All this shows up in the overtones.

 

 

Hi John, I understand you to mean that the 3rd and 5th overtones you refer to are the overtones in the musical tone, and not the frequencies of the higher modes of vibration that are possible with the free reed, which normally vibrates as a bar in its first vibrational mode. Thus, there is no inharmonicity in the sound spectrum, or adjustment to be made in order to “match correctly” to the fundamental, during the sustained musical tone. There is, however, an inharmonicity in the start transient of the tone, in which the steady periodic first mode is developing, and such a transient can occur for several dozen to tens of dozens of cycles (up to hundreds of milliseconds), until only the sustained, periodic musical tone remains. During this sustained tone, the tongue vibrates virtually in sinusoidal fashion, and contributions to this musical tone from higher mode tongue vibration is extremely small, most often probably below detection by human hearing. Such non-harmonic contributions have been measured by experimenters, and I can provide references, but the conclusion is that such contributions to the sustained musical tone occur only at relatively high blowing pressures. I also add that such contributions can also occur when the natural reed vibration is not properly supported by the reed cavity, resulting in screeches from these higher modes that resemble the screech caused by beginning clarinet players, and the physical mechanism is basically the same.

Thus the overtones in the musical tone must be for practical purposes harmonic (integer multiples of frequency), without which, periodic motion would not be possible, resulting in noise (as detected by human hearing).

I don’t understand why you say it’s possible to “match correctly” the musical tone spectrum, by means of adjusting the reed profile. Is this “reed profile” the same thing as builders refer to in describing a change in tongue cross section along the long axis of the tongue? If so, where did you learn of this procedure? Do you know of any builders that practice this procedure of matching correctly the musical tone spectrum? What would constitute a “correct match”?

I also don’t understand what you mean by a “figure 8 pattern” in the vibration of the tongue. Is this while looking edge on towards the tip of the vibrating reed? Might it be a twisting oscillation? As other here have mentioned, the tolerances between the tongue and slot are very small, approaching less than 0.001 inch, and although I’m skeptical, I do think it’s possible. But it’s the phrase “if not filed flat” that intrigues me, and can you please elaborate?

Problem is... there are plenty of tuners that show the root frequency and some that have built in spectrum analyzers like TonalEnergy and others that show multiple notes at once like Dirks but what about the actual frequency of the 3rd and 5th overtones!

Does anyone know of a tuner that show the frequencies of the fundamental, 3rd and 5th? Amplitude of the overtones would be nice as well.

 

Audacity is a free sound file software, and it offers a Fast Fourier Transform to calculate sound spectrum. The version I have doesn’t allow you to see a real time display of the transform as the musical tone is generated, but there are other FFT software that does. But I don’t see a pressing need for real time, at least if you want to make adjustments to the tongue profile, which cannot be done as the musical tone speaks.

While the reed's can vibrate in multiple harmonic modes that can be varied in pitch by adjusting the profiles

 

Hi Dana, I don’t understand what you mean by “multiple harmonic modes.” You explain later on that the higher modes of a vibrating bar do not have frequencies that are in harmonic ratio, so was this just a typo? But another interpretation is that you think it possible to tune the higher modes of a reed tongue (no longer a “bar”) to harmonic ratios by profiling. Well, it may be possible, but barely, if we consider practicality. However, a determined maker might possibly succeed in doing so for one or two higher modes, and learning the skills to make it easier. Then, we’d have to ask why do it? There might be a fortuitous case that the self-excitation mechanism might possibly excite a higher harmonic mode along with the fundamental, in which case some higher harmonic (say near the 5th) of the musical tone would be emphasized. For the bar, these higher modes tend not to be excited, probably because they are non-harmonic. But I can see that a higher mode with harmonic relationship to the first might be excited. But finally, would it be worth all the work, when all you can do is emphasize at most just a few higher harmonics in the musical tone?

…in a flat reed, these natural frequencies rise at a significantly greater rate than the musical harmonic series of an ideal string ( one with no stiffness ) what would be the second harmonic is significantly sharper than the octave, and it gets worse fast from there. I have plotted these ( natural harmonics for both reed's with no profile, and for reed's with a variety of profiles ( pulled from accordions and good concertinas) and none of them approach the straight line graph of the musical harmonic series. They all look like one side of a steep parabola.

 

Okay, I see that you are using the term “natural harmonics” to describe what are really “higher modes” of vibration. “Harmonics” probably goes back to Greek times and was related to the harmonious tones they found in musical sounds, using lengths of string that have whole number ratios in their lengths. The term then got combined into other, related mathematical concepts (such as in the harmonic functions, sine and cosine), so there are multiple meanings in common speech. But in science there is no doubt. A harmonic series in mathematics I believe is defined as a series having integer multiples of each other, and this rule is followed strictly when talking about sound spectrums.

I’m curious, how did you excite these higher modes? As you verified, the mathematical solution for the vibrational modes of a bar (strip of metal with significant thickness) can be found in standard vibration texts, such as the old, still excellent, classic one by Phillip Morse, “Vibration and Sound” (or the later version, written by Morse and Ingard). In the idealized bar, the frequency of the second transverse mode is 6.27 x f1, where f1 is the frequency of the first transverse mode. The third mode vibrates at 17.55 x f1, and the fourth at 34.4 x f1, which as you say, are rapidly increasing in nonlinear fashion. Incidentally these ratios are proportional to the inverse square of the bar’s length, unlike with the idealized string, where the allowed frequencies depend on the inverse first power. If you still have your data, you can check this, though there could be differences since you were probably not working with idealized bars.

Let’s recall that the tuning fork also vibrates more or less as a vibrating bar. When you strike a tuning fork, you first get a high frequency ping, a pronounced start transient (attack) which rapidly dies out to the sinusoidal fundamental cantilever-type vibration (as described above). This is basically what the free reed does, only the attack is not nearly as sharp, or as energetic, since it’s caused by much gentler aerodynamic forces

In the instrument, however, with the combination of reed driven by airflow, an excellent musical series is produced.

 

When I measure the harmonic frequencies calculated by Audacity, I most often find integer ratios within very small percent errors.

Under a strobe, the reed exhibits only it's simplest vibrational mode.

 

As I mentioned above, this result, together with the result of distortion due to a contribution from the second beam mode at high blowing pressures have been well documented in the literature.

There is some disagreement about how the overtone series is produced, but it is not due to the reed itself vibrating in multiple modes.

 

Agreed, with restrictions due to attack, high blowing pressure, and mismatched cavities.

My personal opinion is that it is a result of the varying velocity of the reed when it is moving with the airflow on the down swing compared to the velocity curve moving against the airflow on the up swing.

 

This is part of the “self-excitation” mechanism, which is periodic, and any periodic function describing that motion can be represented mathematically by an infinite series of sinusoidal functions (Fourier’s Theorem). Physically, the tongue’s motion is caused by an airflow (forcing function), and in feedback, the airflow is caused by the motion of the tongue. The asymmetric nature of these aerodynamic forces that you point out allow for a net transfer of energy from the steady airstream to the vibrating tongue. The sinusoidal response, which is the major response that interests us, is a result of the fact that, in periodic motion and at sufficiently low amplitudes, the governing equations are linear. With much natural phenomena and with small amplitudes of response (usually up to 10 % of their mean values), the effects are closely linear, but as amplitudes become large (around 20% mean value), linearized solutions lose accuracy. The study of acoustics illustrates the close relationship between physics and mathematics.

My instruments all share a fairly strong musical fifth overtone ( nicely in tune ) and while I spent a long time refining my profiles, and love the sound of these instruments. I made zero effort to try to tune harmonics, and instead chose to go for a very smooth stiffness gradient from one reed to the next so each reed behaves very much like it's neighbor.

 

How large are your fifth overtones (which would be 6th harmonic) compared to the fundamental? Please make the units clear in your results (amplitude as a length ratio, or in decibels). I’ve sometimes noticed that free reeds vibrate with a second harmonic (first overtone, or second partial) that is larger than the fundamental. I’ve never seen one with a larger 6th harmonic. Might your result be typical with traditionally made concertina reeds? I think it would be extremely difficult to appreciably alter the sound spectrum in a predictable way by modification of the tongue profile. Is “stiffness” the same thing meant by “spring constant”?

The second weak spot will not lower the pitch but it will mean the area of the reed between it and the tip will be able to vibrate in a separate rhythm to the rest of the reed.

 

Hi Chris, I don’t think the actual vibration is as you imagine. There cannot be different parts of the tongue that vibrate in “separate rhythms” during the sustained musical tone. Such behavior would preclude an overall periodic motion, and thus, air would not be chopped with a decided frequency, which in turn would result in noise. However, during a start transit, the different modes caused by these different parts could be excited, and in the special case when the frequency of a higher mode is harmonic to the fundamental, I could believe it perhaps possible that the higher mode could be somewhat excited during normal operation, as I explained above. But more generally, some spring/mass/dissipation systems do have more than one degree of freedom, as does the tongue you describe, and each degree of freedom will result in its own modes of vibration; however - when we consider sustained vibrations (periodic motion) - each of these modes needs to be continually excited by an appropriate periodic forcing function of the right frequency. Take a metal plate as an example, or a bell, or very commonly, a violin string. These each have many modes of vibration, most all of different frequencies, yet you can excite any mode in a sustained manner by applying a periodic force at the antinode of the mode’s vibration. The violin is distinguished in this group in that it’s made (along with its bow) to produced sustained vibrations, as is the free reed. A bell is made for its start transient; when hit by its clapper, most all modes are excited, but none are sustained. The piano is also primarily an attack instrument. Even though it seems to produce sustained tones, its musical tone is really a prolonged start transient. With a bell, there are so many modes with non-integer ratios in frequency that its sound is between noise and music. These higher modes, however, die out faster than the lower modes and we’re left with a low musical hum that goes on for a relatively long time in a big bell (a few tons). The higher frequencies in the higher modes produce larger particle velocities and higher cycle rates on which dissipative forces act and are thus shorter lived. For similar reasons, we only hear the low pitched rumbling of distant thunderstorms, fireworks, or bombs by the time the sound reaches our ears.

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Boy! Way to difficult to cut and paste to address all your comments. I'll do my best to respond. First, I was trying to draw a parallel between the natural vibration frequency series of the ideal string versus the series developed by cantilevered bar vibrations which are far from linear. I was under the impression that " harmonic" referred to regularly repeating ( in time ) phenomenon. The various bar modes of vibration are certainly regular, they just do not occur at integer multiples of the base mode. If harmonic only applies to integer multiples, sorry if I confused you. What would you call other non linear but regularly repeating series?

I measured the natural frequencies of reeds by driving them with sound from a nearby speaker ( from my viola plate tuning rig ) and a variable oscillator. I measured the response with a small magnetic pickup near the lower third ( near the clamped end ) of the reed which was held in a vise. I measured both flat reed stock with no window, starting sigh six inch long x 1/32" thick bars. And working my way down to both Wheatstone and accordion reeds in their shoes, but not in the instrument. While the numbers changed because of the length and thickness changed, the pattern was always the same with each higher mode rising much faster than a linear series, looking perhaps parabolic. That sort of shape anyway, though I don't know what order it might have been since I didn't have any data for the negative side of the graph and wasn't really concerned with more than it's decidedly non linear character.

I was always under the impression that a musical fifth interval was created by dividing an ideal string into three equal segments. So the musical series based on increasing integers would be the fundamental, the octave, then the fifth. ( above the octave? ) Whether I am right or not, the particularly strong harmonic for my reeds in my concertinas is the third peak in the integer series of the fundamental. If the fundamental was 100hz, this would be the peak at 300hz. The second peak ( octave ) on my concertinas is never more than half the amplitude of the fundamental, where this third peak is always higher than the second, and is usually more like 70% of the height of the fundamental, and sometimes actually can be a little stronger. I am not concerned with the units much, since values would be affected by anything from where the mic was placed to reed size and room acoustics, so I just call the fundamental amplitude 1 and measure the other peaks relative to it. This makes comparing the different reeds much simpler, if possibly not highly accurate.

I find that reeds in an instrument are heavily influenced by the body of the instrument. There is nothing remarkable about my reeds except that they are long scale patterned roughly off a set of Wheatstone ( parallel sided ) duet reeds. They are not as wide as a set of Wheatstone Linota reeds and while according to reed organ literature, tends to produce reeds with stronger fundamental relative to overtones, the profiles are similar to the original set, though I reduced the overall number of sizes slightly and adjusted all the profiles to get a set of reeds that varied very smoothly in stiffness from low reeds around 14 gms per initial 2 degrees deflection to mid range reeds at about 20, and high reeds around 8gms. They are very even in character and response.

It is much more likely that there is something about my concertina construction that somewhat suppresses the even harmonics in favor of odd harmonics. ( none of the even harmonics are very strong, where most of the odd ones are. ( given that the strength of all of them drops as frequency increases) After finding that the higher cantilevered bar mode series didn't appear at all in the spectrum of the actual instrument driven in the normal way by the bellows, ( measured on my ancient HP wave analyzer) I felt that trying to tune the natural modes of the reeds was a fools errand. What I do think counts, similar to Chris's comments , is that having a reed that bends smoothly and relatively evenly rather than having bending too focused in one area is important both to reed efficiency and longevity. I adjust my profiles to reach certain overall stiffness values at the appropriate pitches ( since many combinations of length weight distribution and stiffness can all produce the same frequency, but will not have the same feel or overtone character at a given playing pressure ). I tune to the fundamental only.

Stiffness often refers to The modulus of elasticity, but most people equate it with the spring constant, being how they experience it. That is how I am using the term. Obviously since all my reeds are the same material, they have the same modulus of elasticity. I measure the "Spring constant" in grams at 2 degrees of deflection. ( with a special little gizmo)

I am still completely unclear about how the final wave form coming out of the instrument is created. On a BFO sweeping type wave analyzer, all you get is integer resonance ( regardless of volume, though the spectrum changes in amplitude at different frequencies. Whatever the reed is doing, the acoustic output is not even close to sinusoidal at normal playing volume. Benade talks about heterodyne frequencies as the reason for the linear series. I don't know what to think anymore.

Best,

Dana

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The iStroboSoft tuning app for iOS has an optional add-in that shows a spectrum graph.

 

I've used it for checking the musette detune on accordions, upper harmonics, etc.

 

https://www.petersontuners.com/products/istrobosoft/

 

The graph can be zoomed for additional detail, set to linear or log scale.

 

It also has optional modes that allow for tuning of upper harmonics using the standard strobe tuning visual model.

 

scope2.jpg

Edited by eskin
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The iStroboSoft tuning app for iOS has an optional add-in that shows a spectrum graph.

 

I've used it for checking the musette detune on accordions, upper harmonics, etc.

 

https://www.petersontuners.com/products/istrobosoft/

 

The graph can be zoomed for additional detail, set to linear or log scale.

 

It also has optional modes that allow for tuning of upper harmonics using the standard strobe tuning visual model.

 

scope2.jpg

Thanks for the heads up. It is hard to keep track of all these things. Not up to my Sanderson Accutuner for things like isolating the fundamental ( or any other harmonic you choose ) or working from a tuning chart ( since concertina reeds do not show the same pitch in the concertina as on the tuning table ), but even though the tuning " tools" like the wave form analyzer cost twice as much as the basic app, it still costs a thousand dollars less than the SAT. and does a lot that the SAT doesn't.

I am still amazed at how many people would be dead in the water without an electronic tuner, unable to hear when things are in tune. ( read guitar players ). Tuning a piano in equal or some other temperment takes a good bit of knowledge and practice, but people seem to think when the machine says it is in tune, the guitar ( or mandolin etc) is in tune, (when they actually tuned to the out of tune harmonics) people believe their eyes and ignore their ears.

Dana

Edited by Dana Johnson
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The various bar modes of vibration are certainly regular, they just do not occur at integer multiples of the base mode. If harmonic only applies to integer multiples, sorry if I confused you. What would you call other non linear but regularly repeating series?

 

 

Hi Dana. Thanks for your replies. The frequency of the various modes of vibration of an idealized bar are not related, in principle, to each other in any simple way, as they are, for instance, with an idealized string. They occur mathematically as a solution to transcendental equations, without any other special meaning, except, only in an approximate way (see below). But each mode does exhibit regular, periodic vibrations, when excited by a forcing function that matches the mode’s vibration frequency. The periodic motion that results in each case can be represented mathematically by a Fourier series with its own fundamental and harmonic overtones, and in this sense we can call the mode’s response “regular.” The fact that a given mode, when compared to a different mode, has a different frequency of vibration that is not related in any simple way to that of the other mode, doesn’t have anything to do with the fact that we call each mode’s own vibration “regular.” I don’t know of any term that would describe the group of frequencies that characterize the series of modes in an idealized bar, except to say that their frequencies are the result of a mathematical solution.

While the numbers changed because of the length and thickness changed, the pattern was always the same with each higher mode rising much faster than a linear series, looking perhaps parabolic.

 

Yes, the frequency of each mode goes as (beta,i)^2, where beta,i is the ith term in an infinite series of solutions to a transcendental equation - in particular, one involving the functions cosh and cos. And for i greater than 2 (first overtone), beta,i is practically equal to (i – ½). Thus, the frequency goes as (I - 1/2)^2, or i^2 when i is greater than about 4, which is then approximately parabolic.

I find it interesting that in your instruments the second harmonic (octave) is suppressed and your third harmonic (fifth) is strong. Moreover, the even harmonics are in general suppressed, compared to the odd harmonics. In my own experience with free reeds, I find that I like more a musical tone that emphasizes the odd harmonics, which is characteristic of (one end) closed tubes, and which the clarinet approximates. Similarly, of the Asian free reeds, I like most the sound of the bawu, which again approximates a closed tube. If I push to find a reason for this, perhaps it’s because the even harmonics are just octave tones - octaves of the fundamental, or octaves of lower odd harmonic tones. They thus add little to the musical tone, and it’s the odd harmonics that add color (other harmonies in the scale). Unfortunately, I never heard your instruments, but my guess they have more of what I would call a “hollow” sound to the tone, and my speculation leads me to believe that I would like it. Incidentally, you don’t need very many harmonics to produce a beautiful musical tone. Some of the tones of the clarinet and of the trumpet contain only a few, or a couple, overtones. Do you think your instruments have a characteristic sound that differs from most other traditionally made instruments?

…I am still completely unclear about how the final wave form coming out of the instrument is created… Whatever the reed is doing, the acoustic output is not even close to sinusoidal at normal playing volume. Benade talks about heterodyne frequencies as the reason for the linear series. I don't know what to think anymore.

 

I think it best here to keep a distinction between the physics of what’s going on and the mathematics that (for whatever reason) accurately represents the physics. It’s true that the wave form of the pressure pulses in the vicinity of the reed are difficult to intuitively understand. The important thing however is that these pulses are periodic events. With that physical fact, it then becomes straight forward to represent these pulses mathematically as a Fourier series. There is of course an uncanny relationship between physical events that happen in the so called real world and basic, purely ideological mathematical concepts. But the net result is that the Fourier description adequately represents these physical events. Then, when we also discover that our brains are also using such a description in making sense out of physical sound waves, the mystery grows further. I believe heterodyne frequencies are a mathematical phenomenon. When a signal contains two or more frequencies that are relatively close together, these extra frequencies occur within the mathematics, showing up in the Fourier series. But this is a much bigger topic, and maybe it would be worthwhile to first look at some experimental results of (physical) pressure wave forms produced by the free reed, and from these, try to explain their shapes with reference to the instantaneous position of the vibrating tongue. Perhaps in another thread.

 

Best regards,

Tom

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