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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!

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.

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.

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.

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.

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)

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.

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.

Lol...ok, so "shimmer," acoustically speaking = amplitude modulation (AM)!

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

Regular beats are essentially AM with 100% modulation index (as the volume cycles between a peak and zero).

Paul.

Amplitude modulation is what you hear when using open unisons and listening to a RBI and one if the unisons is not pure. The resulting slow pulsation is the carrier wave. Another way open unison tuning can help you find drifted unisons.

Pyro, I'm not sure what you were referring g to with static phase, but if two tones of identical frequency are played together, the ear can hear the phase relationship between in phase and out of phase. Watch my video on Beats and listen as the two computer sine tones of identical frequency are started multiple times. Sometimes they start in phase, sometimes out. When they start out by 180 degrees, the volume is almost zero.

BTW, that discovery demonstrated to me the danger of persuing dead on unisons.

OK..so they are both AMs, just with different modulation indices. Is there any other universal vocabulary that differentiates between 100% and something like 10%?

Can you explain why I seem to sense them differently?!? The beats with a 100% modulation index seem to penetrate/vibrate more deeply into the body, where as the other, c.10%, or whatever, seem to be more perceived by my ears (i.e., with earplugs, a 10% modulation index much harder to detect). I perceive these differently, not really on the same scale.

I guess this my be a question for psychoacoustics?!?

Real amplitude modulation occurs when you cause the amplitude of a carrier wave (e.g., 440 Hz) to vary. If it varies from maximum to 0 it is considered 100% modulation, if it varies between maximum and 90% of maximum, it would be 10% modulation. Intelligibility and ability to penetrate noise of a modulated signal is enhanced by higher modulation. Over 100% modulation (the power of the modulator is greater than the power of the carrier) causes all sorts of nasty consequences, including state administered fines, if the carrier happens to be in the radio frequency spectrum.

I was referring the the static phase between partials at different frequencies. Obviously, if you have two sines at the same frequency and have them 180 degrees out of phase, you won't hear anything. But if you take a sine at 1f, another at 3f, one more at 5f (i.e. the components of a square wave) it makes no difference what the starting phases are to the final sound (even though the waves will look different). If the phases vary, you will hear that (it sounds like FM if you do it at audio frequency).

Hope this is clearer now

Paul.

Then it is your turn to explain either what they are, or what beats are, or both.

Beats are periodic fluctuation in the volume level of two, (or more), sound waves that are near enough to each other in frequency to give rise to the auditory perception of them. If they occur between partials of two separate vibratory sources-the volume of the lower partials is unchanged by the interaction.

BDB, why the evasiveness? Do you have a well-thought-out theory of what is going on or not? I can't read your mind, if you should be understood, you would need to write.

Beats--from my standpoint--are auditory pulses, that I can both feel and hear. This readily occurs at coincident partials (aka unison), but only occurs under very special 'controlled' situations with sine tones of intervals larger than a unison (i.e., they must be perfectly lined-up; which is rarely the case with the piano, and certainly never with the case as it would be while the pitches are being adjusted) even so, it is at a completely different level of intensity. Intervals other than the union create a different kind of AM that are almost undetectable with earplugs in (i.e., they are more heard and less felt). I equate coincident partial "beats" to the sound/feel produced by a tactile transducer, whereas the other phase/resultant-sounds of other intervals comes across more like electrostatic speakers--they are pretty and clear, but lack body and feel.

It appears that the field of acoustics may not differentiate between these qualities, but because there is a difference in perception, I tend to feel inclined to point out this is necessary in terms of piano tuning. Specifically, because 100% AM indices pull-in other unisons that are close enough to each other, whereas the other AM qualities behave differently (i.e., the frequency elasticity--they don't gravitate to each other in the same way). That is why I make the reference to gravity and mass--though I do understand that this is not acoustically the case. For whatever the reason is, be it coupling at the soundboard, air, or whatever, 10% and 100% AM indices behave differently in the piano in terms of how they pull on other notes, as well as how they are perceived.

With all that said, I still have no idea what your theory is about. Would you please offer more detail. It sounds like there are plenty of science/acoustic people here to help sort things out.

is not air allowing coupling of waves N

This screenshot is a 10kHz sine wave modulated to 100% by a 10Hz Amplitude Modulator. Play this wave file to hear the beats.



If you do a spectral analysis the file, you will see that there are three spikes - 9,990Hz - 10,000Hz - 10,010Hz. This is expected, since Amplitude Modulation creates both upper and lower sidebands separated from the carrier by the frequency of the modulation.