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#2311753 - 08/06/14 12:03 AM Prout's and Chris Leslie's Beaters  
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I am interested in how one might predict beating from two sinewaves. The two examples by Prout and Chris Leslie present a challenge, at least to me, because the only trigonometric identity I can find is

sin(at)+sin(bt)=cos((at+bt)/2)cos((at-bt)/2)

which does not do a good job on predicting the beating. The line spectrum gives no insight - just two spikes at the appropriate frequencies.

But, the auto-correlation does a nice job if you look at the correct time lag range.

Consider Prout's example of 440 and 883 Hz. The following graph shows the time domain plot and then two auto-correlations over two time lag ranges. The short time lag range corresponds to the dominant frequency (or pitch, if you believe Professor Eric Heller of Harvard) of 440 Hz. The long time lag range, which includes the short time lag range, picks out the beating frequency of 3 Hz (time lag peak at 0.333 s). (Note that the peak corresponding to 440 Hz appears in the bottom graph at 0.0023 s or 2.27 ms but it is lost in the other neighboring peaks because the time scale is so compressed.)


[Linked Image]

Chris Leslie's example of 440 and 663 Hz shows a dominant frequency (pitch) of 220 Hz and a beating frequency of 6 Hz (time lag peak at 0.167 s.

[Linked Image]

Keep in mind that I did these examples with two pure sinewaves. There are no higher harmonics present, by design. So, one of the sine waves is not beating against a harmonic of the other one because it is not present. smile

Last edited by PaintedPostDave; 08/06/14 12:33 AM.

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#2311760 - 08/06/14 12:16 AM Re: Prout's and Chris Leslie's Beaters [Re: PaintedPostDave]  
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I am lying here listening to the katydids outside chirruping at slightly different frequencies, listening to how they sometimes coincide, sometimes seem to be completely out of phase, and imagining the combinatorial nature of the interaction of the periodic sounds that they make.


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#2311777 - 08/06/14 01:09 AM Re: Prout's and Chris Leslie's Beaters [Re: BDB]  
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What about the co-efficient drag, considering the absorption of signals either from or pertaining to, 2 entities not related in scope, and wanting of realistic data which is sometimes so convoluted as to what makes sense and what does not...this is clearly one of those situations, no?

Last edited by Grandpianoman; 08/06/14 01:10 AM.
#2311783 - 08/06/14 01:16 AM Re: Prout's and Chris Leslie's Beaters [Re: PaintedPostDave]  
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BDB,
Where does one find a tutorial on combinatorics describing wave functions. The cursory investigation I made turned up nothing I could comprehend.

I do think if you want to establish a new paradigm of discussing piano waves over the currently accepted description of partials contained within a tone-that you should do us the courtesy of some comparative demonstation of the superior utility of combinatorics over partials.


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#2311788 - 08/06/14 01:36 AM Re: Prout's and Chris Leslie's Beaters [Re: PaintedPostDave]  
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I am not certain where the combinatorial method of looking at the interference of waves is written up, but it is easy enough to work out thinking about the katydids. Think of their chirrups as square waves. One chirrups at one rate, while the other does it at a slightly different rate. There will be times when they are both making the noise at the same time, and others when one is making the noise and then the other is making it between the other's noise, and then there are all the times in between, where it is transitioning from one of those extremes to the other. It is simply a matter of figuring out the rate at which that happens.

(I would do more, but that should be enough to let people who only want to think about it think about it, and those people who want to do the math do the math.)


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#2311789 - 08/06/14 01:38 AM Re: Prout's and Chris Leslie's Beaters [Re: Ed McMorrow, RPT]  
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From Wikipedia:

"Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size (enumerative combinatorics), deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria (as in combinatorial designs and matroid theory), finding "largest", "smallest", or "optimal" objects (extremal combinatorics and combinatorial optimization), and studying combinatorial structures arising in an algebraic context, or applying algebraic techniques to combinatorial problems (algebraic combinatorics).

Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry,[1] and combinatorics also has many applications in mathematical optimization, computer science, ergodic theory and statistical physics. Many combinatorial questions have historically been considered in isolation, giving an ad hoc solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics is graph theory, which also has numerous natural connections to other areas. Combinatorics is used frequently in computer science to obtain formulas and estimates in the analysis of algorithms.

A mathematician who studies combinatorics is called a combinatorialist or a combinatorist."

A piano tuner who claims to use combinatorics is called a _____.


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#2311796 - 08/06/14 02:00 AM Re: Prout's and Chris Leslie's Beaters [Re: PaintedPostDave]  
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I did not actually combine the two sine waves, or do any math, but my thought was exactly as you describe, that is, there is a combined wave with long wave intensity undulations at a set frequency. These are not the same as "beats" when nearly coincident frequencies are combined because there is not the same degree of cancellation and addition. They are however clearly audible.

Robert Scott insisted that there were partial leakages or generation due to distortion and we were hearing those instead. That could also be the case, but surely those partials would be too weak to be of any consequence.


Last edited by Chris Leslie; 08/06/14 02:03 AM.

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#2311798 - 08/06/14 02:07 AM Re: Prout's and Chris Leslie's Beaters [Re: PaintedPostDave]  
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Tunatorialist. f

BDB, is probably on to something, but if he won't go into details, it's difficult to figure out a method of testing his hypothesis. It doesn't need to be math, but it does need to be more detailed in the observation so that it can be tested. There are some crazy things going on in the sound of the piano that are not fully understood.


Masters degree in piano technology, +factory(s) training, etc., blah, blah, yada, yada, yada...[uncensored break-out in song]..."it don't mean a thing, if you aint got that swing."
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#2311807 - 08/06/14 03:36 AM Re: Prout's and Chris Leslie's Beaters [Re: PaintedPostDave]  
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It is combinatorics, finite math. Draw a picture. Use small numbers. I have, but I do not have time to post it now.


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#2311812 - 08/06/14 04:00 AM Re: Prout's and Chris Leslie's Beaters [Re: BDB]  
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wanna hear beats ? take any flight and make sure your seat's between the engines... That'll give you beats you'll never forget !


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#2311813 - 08/06/14 04:02 AM Re: Prout's and Chris Leslie's Beaters [Re: PaintedPostDave]  
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is called a "combinatunerist "


We meet upon the level, and..you don't fear someone whose story you know!
#2311814 - 08/06/14 04:04 AM Re: Prout's and Chris Leslie's Beaters [Re: PaintedPostDave]  
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It may make life easier if you transfer your analysis to the frequency domain instead of keeping it in the time domain.


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#2311890 - 08/06/14 10:08 AM Re: Prout's and Chris Leslie's Beaters [Re: PaintedPostDave]  
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Is not air elasticity enough to make us perceive those fluctuations.

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#2311899 - 08/06/14 10:39 AM Re: Prout's and Chris Leslie's Beaters [Re: PaintedPostDave]  
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Here is the rather crude drawing:

[Linked Image]

You can imagine it cleaned up, but it shows how adding something which is periodic in 6 to something that is periodic in 9 results in something which is periodic in 3, with a bunch of other things happening.


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#2311908 - 08/06/14 11:17 AM Re: Prout's and Chris Leslie's Beaters [Re: BDB]  
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Originally Posted by BDB
Here is the rather crude drawing:

[Linked Image]

You can imagine it cleaned up, but it shows how adding something which is periodic in 6 to something that is periodic in 9 results in something which is periodic in 3, with a bunch of other things happening.


Thanks BDB. Well shown. People tend to forget that phase is so important.

#2311910 - 08/06/14 11:29 AM Re: Prout's and Chris Leslie's Beaters [Re: PaintedPostDave]  
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Using combinatoric graphing to illustrate the beats we hear tuning unisons, octaves and tempered intervals would still entail drawing a picture of each partial. I don't see how you can draw the sum total waveform conbinatoricaly to illustrate the beats.


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#2311911 - 08/06/14 11:36 AM Re: Prout's and Chris Leslie's Beaters [Re: tonyster220763]  
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"It may make life easier if you transfer your analysis to the frequency domain instead of keeping it in the time domain."

Actually, that is where I usually start but in this case, as I mentioned in my first post, the frequency domain analysis only shows the spikes at the frequencies and nothing about the low frequency beating.


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#2312036 - 08/06/14 05:39 PM Re: Prout's and Chris Leslie's Beaters [Re: PaintedPostDave]  
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BDB, how much have you thought about this? Can you go into any more detail? If you can, the more science-inclined people here might be able to help with the math and/or acoustics part.

I agree that sounds/tones combine to form other sounds/tones, especially in the case of the piano: A+B=A+B+C (and maybe even D, E, and F). In other words, I don't think partials explain all the "overtones," mainly because I've observed some sounds to behave differently that they really should if they are true partials.

You've seen the high speed videos and know that there are partials involved. How do you incorporate that information into your thought process. That is what I haven't understood from you yet. Partials exist, right? Are you just saying that you don't use them to tune? Or, what's the deal?

I think it would be an interesting conversation. I want to know how a combinatunist processes sound in order to tune the piano.


Masters degree in piano technology, +factory(s) training, etc., blah, blah, yada, yada, yada...[uncensored break-out in song]..."it don't mean a thing, if you aint got that swing."
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#2312062 - 08/06/14 06:37 PM Re: Prout's and Chris Leslie's Beaters [Re: A454.7]  
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Originally Posted by A443
BDB, how much have you thought about this? Can you go into any more detail? If you can, the more science-inclined people here might be able to help with the math and/or acoustics part.

I agree that sounds/tones combine to form other sounds/tones, especially in the case of the piano: A+B=A+B+C (and maybe even D, E, and F). In other words, I don't think partials explain all the "overtones," mainly because I've observed some sounds to behave differently that they really should if they are true partials.

You've seen the high speed videos and know that there are partials involved. How do you incorporate that information into your thought process. That is what I haven't understood from you yet. Partials exist, right? Are you just saying that you don't use them to tune? Or, what's the deal?

I think it would be an interesting conversation. I want to know how a combinatunist processes sound in order to tune the piano.

If I can answer: According to my interpretation of "combinatronics" in the context of piano tuning as we have been discussing, I think I use it quite often. For example, I usually don't think of particular partial combinations when listening to beats, but rather just that beats happen that can be used to advantage. I use FBI progression as a tool for all fine tuning, but because the reality is that different partials ebb and flow unpredictably across the range, and that falseness is very often confounding the sense of beats, I just listen to whatever beats emerge and get them as progressive as possible while at the same tim keeping the SBIs as beat-less compromised as possible. I am not thinking about what the actual responsible partials are.

Last edited by Chris Leslie; 08/06/14 06:57 PM.

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#2312098 - 08/06/14 08:07 PM Re: Prout's and Chris Leslie's Beaters [Re: PaintedPostDave]  
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You really think that example explains the combinatronics of tunatorialists? Hummm...

I still don't understand how that differs from partial theory. I get that you don't count beats--you feel/sense them--but they are still beats of coincident partials, no? As far as I've experienced so far, beats are coincident partials, and waverings from combinations don't produce the same "pulse" like that of the coincident partial beat. They are there, and are perceived, but their interactions are more like auras with less/no sonic mass behind them. Combination tones don't beat the same way coincident partials do: they shimmer.


Masters degree in piano technology, +factory(s) training, etc., blah, blah, yada, yada, yada...[uncensored break-out in song]..."it don't mean a thing, if you aint got that swing."
--Klavierbaukuenstler des Erwachens--
Email: klavierbaukuenstler@gmail.com
#2312102 - 08/06/14 08:13 PM Re: Prout's and Chris Leslie's Beaters [Re: PaintedPostDave]  
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BTW, if you were cool with the open 6-string tuning of Olek, then I get how combinatronics would apply. But that doesn't seem to sit well with you, hence my confusion. When all those strings aren't in perfect DOA unison, yet one is still able to tune without clearly hearing coincident partials beat, a different listening skill is being applied, IMHO: combinatronics!


Masters degree in piano technology, +factory(s) training, etc., blah, blah, yada, yada, yada...[uncensored break-out in song]..."it don't mean a thing, if you aint got that swing."
--Klavierbaukuenstler des Erwachens--
Email: klavierbaukuenstler@gmail.com
#2312226 - 08/07/14 03:55 AM Re: Prout's and Chris Leslie's Beaters [Re: Ed McMorrow, RPT]  
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Originally Posted by Ed McMorrow, RPT
Using combinatoric graphing to illustrate the beats we hear tuning unisons, octaves and tempered intervals would still entail drawing a picture of each partial. I don't see how you can draw the sum total waveform conbinatoricaly [combinatorially] to illustrate the beats.


The whole point of PaintedPostDave's graphs is that partials do not need to be there to generate beats. Once you accept that, then you need to find another way to explain them. Combinatorial mathematics does that. The addition of square waves, which I graphed rather crudely, is a simple, basic method of understanding the way it works.

Those of you who have taken integral calculus will recognize the relationship between the square waves and more complex waves. But even if you have not, there is a lot that can be done with this more basic math. In fact, you cannot truly understand what Fourier did without it.


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#2312251 - 08/07/14 05:38 AM Re: Prout's and Chris Leslie's Beaters [Re: PaintedPostDave]  
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Those aren't "beats" on that graph, and that does not represent how pianos are tuned! Whatever that is needs to be called by a different name, as it applies to the piano.

It doesn't mean, however, that those "shimmers" can't be tuned, but the tuning of that effect has no bearing on the tuning of the piano. If PaintedPostDave were to offset the sine waves slightly, as it would naturally happen in the piano, the shape/pattern would change dramatically. If PaintedPostDave were to do that for 3+3 sine waves, with slight different offsets, and again the pattern would be completely different. Change the offset/starting-point and those "shimmers" diminish.

The tuning of that effect is not changed via frequency, so it doesn't explain the "tuning" process, as it relates to piano, at all.


Masters degree in piano technology, +factory(s) training, etc., blah, blah, yada, yada, yada...[uncensored break-out in song]..."it don't mean a thing, if you aint got that swing."
--Klavierbaukuenstler des Erwachens--
Email: klavierbaukuenstler@gmail.com
#2312266 - 08/07/14 06:46 AM Re: Prout's and Chris Leslie's Beaters [Re: A454.7]  
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A443, I think you are assuming that the "shimmers" caused by the 440 and 663 combination, for example, are being used to tune pianos. That is not the case, for me at least. This post was only to demonstrate how the shimmers occur and not to imply that they are useful as a piano tuning tool.


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#2312279 - 08/07/14 07:26 AM Re: Prout's and Chris Leslie's Beaters [Re: PaintedPostDave]  
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Chris Leslie, that's just it: I believe there are other things going on with what we hear in the tuning process not explained by partial theory, and BDB seems to think he has an alternative theory, but the details are yet forthcoming--so, I am left with whatever I can piece together on my own accord.

Usually when people confidently state well-thought-out concepts, it usually has, at least, some basis in reality. Until, and unless, I can completely disprove the concept, I keep it in the back of my mind and reference it in my future observations/experiments.

It seems like BDB has argued his case before with other technicians and is not looking for another round. However, I still have no idea how his theory is applied to the process of piano tuning. To me, it seems like he got caught up in a theoretical/mathematical concept that exists on paper, and confused that process with how you described you/[our] tuning process.

BTW, is there an accepted, slightly more academic terminology for "shimmer?" This is what I perceive as an aura, it comes across to me as a beat with no body--in the piano as well. I also hear it more with my ears than I feel it with the body (i.e., vibrations)--which is why I say it seems to have less acoustic mass.


Masters degree in piano technology, +factory(s) training, etc., blah, blah, yada, yada, yada...[uncensored break-out in song]..."it don't mean a thing, if you aint got that swing."
--Klavierbaukuenstler des Erwachens--
Email: klavierbaukuenstler@gmail.com
#2312288 - 08/07/14 07:41 AM Re: Prout's and Chris Leslie's Beaters [Re: prout]  
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y'all realize the square waves have infinite harmonics...


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#2312294 - 08/07/14 07:53 AM Re: Prout's and Chris Leslie's Beaters [Re: tonyster220763]  
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Originally Posted by tonyster220763
y'all realize the square waves have infinite harmonics...


Odd-numbered harmonics, but who's counting.

Tony wins the prize! (and the prize is a clarinet)

Last edited by Chris Storch; 08/07/14 07:55 AM.

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#2312303 - 08/07/14 08:11 AM Re: Prout's and Chris Leslie's Beaters [Re: PaintedPostDave]  
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I'd like to hear from BDB or any other proponent of the combinatorial approach why...
a) F3-A3, when tuned pure, does not beat
b) F3-A3, when tempered, beats - neither at any of the fundamentals [edit: 176 and 220 Hz), nor at any difference tone [edit: and not at any sum tone either], but audibly and demonstrably at A5.

The question is serious, so please spare us from fairies and other such.

Last edited by Mark R.; 08/07/14 08:14 AM. Reason: as shown in post.

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#2312304 - 08/07/14 08:11 AM Re: Prout's and Chris Leslie's Beaters [Re: PaintedPostDave]  
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Montreal
For what it's worth, the ear cannot discern any difference in the static phase of various sinewaves - you can combine them in any arbitrary phase and the resultant sound is the same. Of course, if the phase is varying, you will hear it. If you don't believe me, try it out with audacity with a few partials at same 1f, 3f, 5f and offset the starting phase of each sine. The final result will sound identical, despite the waves looking quite different.

Also, summing arbitrary waveshapes is no different than summing their constituent partials - the final waveshape does depend on the phase, but, per the above, the sound will not (once again, assuming the phase relationships are fixed). The ear, is of course, sensitive to amplitude modulation, but is more sensitive to frequency change (i.e. a small amount of FM is more perceptible than a small amount of AM, particularly if the modulation frequencies are below the normal audio range).

Paul.


#2312328 - 08/07/14 08:51 AM Re: Prout's and Chris Leslie's Beaters [Re: PaintedPostDave]  
Joined: Apr 2012
Posts: 1,577
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A454.7  Offline
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Joined: Apr 2012
Posts: 1,577
Manywheres
Lol...ok, so "shimmer," acoustically speaking = amplitude modulation (AM)!

OK cool, then, those aren't beats, they are AMs!


Masters degree in piano technology, +factory(s) training, etc., blah, blah, yada, yada, yada...[uncensored break-out in song]..."it don't mean a thing, if you aint got that swing."
--Klavierbaukuenstler des Erwachens--
Email: klavierbaukuenstler@gmail.com
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