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All these descriptions are models. All models are limited in scope. The reason specific models get adopted by people interested in solving problems is in the utility to make predictions from them.
I have yet to see how the combinatoric model is useful.
In a seemingly infinite universe-infinite human creativity is-seemingly possible.
According to NASA, 93% of the earth like planets possible in the known universe have yet to be formed.
Contact: toneman1@me.com
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Saying partials don't exist is like saying the earth is not a sphere.
Kees
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This is science and technology and not democratie nor religion. Here there are some who know and some who don't know.
Here the knowledge is the master and the ignorance is the slave.
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BDB, I am willing to think further about your concepts. But, the main problem I have is with your understanding of sine waves and how they look and sound when combined. Have you played around with them on a computer?!? Are you sure this is what you are listening to? The way they combined can be well seen, in addition to heard.
There are differences--I played around training myself to hear those differences for a few months--but for me, this was a different aural process altogether. If you have trained yourself to hear intervals in that way, I think it would be a fascinating conversation.
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Saying partials don't exist is like saying the earth is not a sphere.
Kees Glad to have another 'believer' on board, being that the earth is clearly not a sphere.
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Nice slow motion of the bass string. Interesting to see the damper twist. Looks like the damper still hits the string in places.
Seiler 206, Chickering 145, Estey 2 manual reed organ, Fudge clavichord, Zuckerman single harpsichord, Technics P-30, Roland RD-100.
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It's also pretty clear from the slow motion that string sound is made up of a lot of messy things besides just nice clean harmonics. Also the string gallops in three dimensions. This of course is what gives color to the sound. No mathematical model is likely to give such a messy picture. Perhaps this is why the computer modeled pianos are rather fatiguing to hear after a while.
If you are going to hear partials or harmonics by adding sine waves, you will need some kind of non-linear element to get mixing. A piano string has plenty of non-linearity as shown in the slow motion picture. If you are just dealing with sine waves on the screen you either need to multiply them or put them through some non linear device.
Also, earth is not a sphere, it's an oblate spheroid, as was proven in the 18th(?) century by comparing solar noon on pendulum clocks near the equator and comparing them to clocks at higher latitudes. This is due to the rotation of the earth. There is also a distinct wobble in the motion as well as a lot of ringing. More partials!
Seiler 206, Chickering 145, Estey 2 manual reed organ, Fudge clavichord, Zuckerman single harpsichord, Technics P-30, Roland RD-100.
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As I posted into another thread, just put graph sin(4.4*pi*x)+sin(4.41*pi*x) into google, select horizontal zoom and zoom out. The "beats" that you hear are an amplitude modulation effect. If you put graph sin(4.4*pi*x)+sin(6.61*pi*x)- essentially a major 12th that's off by 1Hz, you will still see an amplitude modulation (that you can also hear as beats). So you can tune any interval with pure sinewave. Of course sin(A) + sin(B) is "combinatronics". You don't need partials to tune, you just need to remove any amplitude modulation to zero.
Paul. After much thought and experimentation, I agree with you and BDB. It is quite possible to tune, for example, a major third made up of two pure sine waves. The math, and the physics, are trivial. While piano strings are messy affairs, whether or not they consist of partials does not affect the ability to tune, just the choice of what to listen to when tuning. Amplitude modulation rules!
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This is science and technology and not democratie nor religion. Here there are some who know and some who don't know.
Here the knowledge is the master and the ignorance is the slave.
So far, I have not read any post that shows ignorance, only different approaches.
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Saying partials don't exist is ignorance, is denying the existence of iH, is denying the existence of different types of octaves.
This very thread would not exist!
This is not only ignorance but also non sense!
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As I posted into another thread, just put graph sin(4.4*pi*x)+sin(4.41*pi*x) into google, select horizontal zoom and zoom out. The "beats" that you hear are an amplitude modulation effect. If you put graph sin(4.4*pi*x)+sin(6.61*pi*x)- essentially a major 12th that's off by 1Hz, you will still see an amplitude modulation (that you can also hear as beats). I must disagree that you can hear the beat you see in the graph of (4.4*pi*x)+sin(6.61*pi*x) in google. When you zoom back far enough to see several of the 1Hz cycles, you do see what looks like amplitude modulation, but there is an important difference. True amplitude modulation is based on a single carrier frequency that is modulated by some lower frequency. The loudness of the resulting carrier (assuming the carrier is in the audio range) is the RMS value of the signal and in the case of a single modulated carrier the RMS value is directly proportional to the peak-to-peak amplitude. As the envelope goes up and down so does the loudness of the sound, and therefore you can hear the beat. But when you graph sin(4.4*pi*x)+sin(6.61*pi*x), the RMS value remains constant even though the peak-to-peak envelope of the waveform is changing. That is because the phase relationship between the two "carriers" is changing. As long as there is no non-linear distortion in the sound reaching your ears, you will not hear a 1Hz "beat" when listening to sin(4.4*pi*x)+sin(6.61*pi*x). If you do hear a slight beat it is because of some non-linearity in the sound system. Another way to look at it is to ask yourself which frequency will sound like it is beating? Will it be sin(4.4*pi*x) or sin(6.61*pi*x), or both of them, or will it be sin(13.2*pi*x)? If these were typical piano tones (not pure sinewaves) then sin(13.2*pi*x) is the one that will appear to be beating as the first coincident partial. But in the case of pure sinewaves there is no sin(13.2*pi*x) present. So it can't beat in the same way it does with piano notes. One more difference is that the waveform envelope that you want to call a beat is not actually sinusoidal. But it is sinusoidal in the case of graph sin(4.4*pi*x)+sin(4.41*pi*x). That should be a clue that something very different is going on.
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I asked a colleague, an electronics engineer, about combinatorics yesterday. He asked why and then his immediate suggestion was impulse waves. Here is the response of a concert hall to an impulse like a gunshot. ![[Linked Image]](http://www.acoustics.salford.ac.uk/acoustics_info/concert_hall_acoustics/images/impulse.gif) See the similarity to those plots of piano notes? More on concert hall acoustics.
Ian Russell
Schiedmayer & Soehne, 1925 Model 14, 140cm
Ibach, 1905 F-IV, 235cm
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As I posted into another thread, just put graph sin(4.4*pi*x)+sin(4.41*pi*x) into google, select horizontal zoom and zoom out. The "beats" that you hear are an amplitude modulation effect. If you put graph sin(4.4*pi*x)+sin(6.61*pi*x)- essentially a major 12th that's off by 1Hz, you will still see an amplitude modulation (that you can also hear as beats). I must disagree that you can hear the beat you see in the graph of (4.4*pi*x)+sin(6.61*pi*x) in google. When you zoom back far enough to see several of the 1Hz cycles, you do see what looks like amplitude modulation, but there is an important difference. True amplitude modulation is based on a single carrier frequency that is modulated by some lower frequency. The loudness of the resulting carrier (assuming the carrier is in the audio range) is the RMS value of the signal and in the case of a single modulated carrier the RMS value is directly proportional to the peak-to-peak amplitude. As the envelope goes up and down so does the loudness of the sound, and therefore you can hear the beat. But when you graph sin(4.4*pi*x)+sin(6.61*pi*x), the RMS value remains constant even though the peak-to-peak envelope of the waveform is changing. That is because the phase relationship between the two "carriers" is changing. As long as there is no non-linear distortion in the sound reaching your ears, you will not hear a 1Hz "beat" when listening to sin(4.4*pi*x)+sin(6.61*pi*x). If you do hear a slight beat it is because of some non-linearity in the sound system. Another way to look at it is to ask yourself which frequency will sound like it is beating? Will it be sin(4.4*pi*x) or sin(6.61*pi*x), or both of them, or will it be sin(13.2*pi*x)? If these were typical piano tones (not pure sinewaves) then sin(13.2*pi*x) is the one that will appear to be beating as the first coincident partial. But in the case of pure sinewaves there is no sin(13.2*pi*x) present. So it can't beat in the same way it does with piano notes. One more difference is that the waveform envelope that you want to call a beat is not actually sinusoidal. But it is sinusoidal in the case of graph sin(4.4*pi*x)+sin(4.41*pi*x). That should be a clue that something very different is going on. When I use Audacity to generate 440Hz + 661Hz, I hear a variation in loudness at about 0.5Hz, not 1Hz. I think the RMS figures are a red herring since what time period do you average over? A sine has zero average amplitude over the period of one cycle, but it certainly transfers power that is given by the RMS value. I know that non-linearity will introduce distortion (essentially you get a multiplication operator which does produce sidebands) and I know that sin(A)+sin(B) is not the same as true AM. What I hear when I listen to the 440Hz+661Hz is both frequencies varying in loudness - it's not the same sensation as beating partials, but it can be heard and you can definitely tune sinewaves that are integer multiples of each other - you just adjust for maximum "smoothness". It is not the same as beats, but it can be heard. Really the argument was that it is impossible to "tune" two sines - this is definitely not true. As for the other argument that you can't hear beats in a single string due to iH - an extreme example of this is a badly shaped bell where the partials are all over the map and you get all sorts of complex beats produced that are very audible. Paul.
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Here is a sound file of two pairs of two simultaneous sine waves - the first pair is 440/883Hz for 5.6 seconds, and the second pair is 440/880 for 5.6 seconds. One can CLEARLY hear the 3bps of the first pair and nothing in the second pair. It would be easy to tune using pure sine waves. This file is amenable to discrete FFT analysis using 2^18 samples at 48kHz. I measured the samples at 440.0000, 882.9999, and 880.0000 Hz respectively. The samples were recorded at -8db and the noise floor is -160db. There is no distortion or non-linearity in the recording above the noise floor in the frequency range of interest. Edit: This is a good test of your audio system. It may be that what I hear, and you will hear, IS non-linearity in the reproducing system, including your ears. It doesn't matter though. If you can hear it, you can tune it.
Last edited by prout; 07/26/14 03:20 PM.
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Try any other intervals, to see if you can still hear the 'beat'--they are not as audible with 5ths, or any other tuning interval as with the octave example. Here is an online tone generator for those with curiosity and want to play around for themselves; open up two windows and play around with the distances between the two tones: http://onlinetonegenerator.com
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Try any other intervals, to see if you can still hear the 'beat'--they are not as audible with 5ths, or any other tuning interval as with the octave example. Here is an online tone generator for those with curiosity and want to play around for themselves; open up two windows and play around with the distances between the two tones: http://onlinetonegenerator.com is a file of 440Hz beating against 663Hz, followed by 440Hz beating against 660Hz. The result is clearly audible - more so than in the 440-883-880 example above. In this example, you can hear the resultant at 223Hz and 220Hz as well. These exist in your brain. The math clearly shows that amplitude variations are possible (Not amplitude modulation as I stated in a previous post - my bad - as an old amateur radio operator I should know better!). The physics show no such variations. Yet, we still hear it. Hence my point about "truth', 'existence', and observation. You can choose to 'believe' what you hear or see, or you just accept it and use it to do your job. The recording parameters are the same as the previously posted recording. Edit: I also have a recording of F3A3 sine waves beating at 6.93bps and 2.93bps.
Last edited by prout; 07/26/14 12:01 PM.
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Here is a sound file of two pairs of two simultaneous sine waves - the first pair is 440/883Hz for 5.6 seconds, and the second pair is 440/880 for 5.6 seconds. One can CLEARLY hear the 3bps of the first pair and nothing in the second pair. It would be easy to tune using pure sine waves. This file is amendable to discrete FFT analysis using 2^18 samples at 48kHz. I measured the samples at 440.0000, 882.9999, and 880.0000 Hz respectively. The samples were recorded at -8db and the noise floor is -160db. There is no distortion or non-linearity in the recording above the noise floor in the frequency range of interest. Edit: This is a good test of your audio system. It may be that what I hear, and you will hear, IS non-linearity in the reproducing system, including your ears. It doesn't matter though. If you can hear it, you can tune it. There are beats, with almost all intervals, but I wonder if they are not simply due to our digital equipment . It should be done with an analog synthétiser to be sure, I believe. I am lost in your explanations an acronyms so this is just a guess. I usedd http://onlinetonegenerator.com binaural beats.
Last edited by Olek; 07/26/14 12:02 PM.
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I wish to add some kind and sensitive phrase but nothing comes to mind.!
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Here is a sound file of two pairs of two simultaneous sine waves - the first pair is 440/883Hz for 5.6 seconds, and the second pair is 440/880 for 5.6 seconds. One can CLEARLY hear the 3bps of the first pair and nothing in the second pair. It would be easy to tune using pure sine waves. This file is amendable to discrete FFT analysis using 2^18 samples at 48kHz. I measured the samples at 440.0000, 882.9999, and 880.0000 Hz respectively. The samples were recorded at -8db and the noise floor is -160db. There is no distortion or non-linearity in the recording above the noise floor in the frequency range of interest. Edit: This is a good test of your audio system. It may be that what I hear, and you will hear, IS non-linearity in the reproducing system, including your ears. It doesn't matter though. If you can hear it, you can tune it. There are beats, with almost all intervals, but I wonder if they are not simply due to our digital equipment . It should be done with an analog synthétiser to be sure, I believe. I am lost in your explanations an acronyms so this is just a guess. They could be, and most likely is the case. Nevertheless, the math shows that a variation in volume should occur, and that is what we hear, whether or not it exists! 
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I don't know if these sinusoidal waves are really sinusoidal.
But, to circunvent this problem, there is an experiment that I have posted in another thread. It uses an A 440 tuning fork, which has a very very weak 2nd partial and practically no fourth partial.
1) If you sound the fork and play F4 in the piano, there are no beats.
But if you play A4 and F4 in the piano, there are beats.
2) If you sound the fork and play F3, there are no beats.
But if you play A4 and F3 there are beats.
3) If you sound the fork and play F2 there are beats.
If tou play A4 and F2 there are beats.
The conclusion is obvious. The fork has a very very weak 2nd partial and the beats with the fifth partial of F3 are not audible. The fork has practically no fourth partial, so there are no beats with the fifth partial of F4.
But the fork has a strong first partial, which clearly beats with the fifth partial of F2.
The piano note A4 produce beats with all three notes F4, F3, and F2 because A4 has audible 1st, 2nd and 4th partials.
Beats are produced by coincident partials.
Every piano tuner knows that.
Beats at the fundamental/fundamental level are produced only by unisons.
Last edited by Gadzar; 07/26/14 02:47 PM.
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I don't know if these sinusoidal waves are really sinusoidal.
They are sinusoidal to the extent that there is no harmonic distortion in the waveform greater than 152db below the fundamental. This level of distortion is so ridiculously low as to border on the absurd, so I can say with confidence that the waves are sinusoidal. Any person who has analyzed the posted recording should achieve the same results within their device's measurement error limits. Edit: You should understand that my interest in tuning, and the physics-based math that predicts it, is based on measuring partials. I just don't find it useful or helpful or fruitful to argue about the absolute existence of something. It is as if your universe would come falling down about your ears if someone contradicts a belief you have held for a lifetime. Who cares. I use partial theory because I find it useful - BDB does not, and it appears he has not suffered a lack of clientele as a result.
Last edited by prout; 07/26/14 03:13 PM.
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