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Those maxima you see in the total waveform do not represent anything the ear is necessary going to hear. An audible beat is a perceived variation in amplitude at a given frequency. It is not related to the peak values in the composite waveform. It is only coincidentally related in the simple case of two nearby sinewaves beating. If beats were a function of maxima in total waveforms, then how would it be possible for a person to correctly hear and count the beats in both the 4:2 and the 6:3 octaves as separate things?

I have no doubt that one can hear beats at 1320 when listening to your synthesized tones. The human ear is very sensitive. Even a very good sound system produces enough harmonic distortion to create the needed energy at 1320.

When we tune unisons, there is a "mode locking" that occurs when the pitch gets close since the strings are closely coupled. I think Oleg has a discussion of Viennese style tuning which illustrated this effect.

I think we need to separate the physics from the psychoacoustics. Remember, it is what we hear that's important. All of the physics is to help up get to a good sound from a very imperfect system.

A viennese style unison, BTW, is not really "mode locking"--I think it would be what Olek calls a "smiley unison;" one string is slightly sharp, and quickly gets pulled-in. I call this a "sustain unison." A viennese unison (i.e., a colour unison), like the Viennese themselves, is about internal conflict: there are three strings that are tuned in slightly increasing amounts ++,+,0 [in that order]. This unison struggles internally, which is what produces more colour, in the unison. This kind of unison has the most flexibility to move up/down with coincident partials of other note combinations. <----the Viennese piano builders have known about this for a long time; there was a period of time where their string lengths in the unison were purposefully cut at the bridge to increasing lengths, just like like that style of tuning.

The american DOA [dead on attack/arrival] unisons are great for a slightly bigger attack sound, but there is a sacrifice in the decay shape for doing so. "Attack unisons" are the least flexible in terms of the amount they are able to move to align with other coincident partials.

Yes, I agree that practically speaking, when tuning piano strings, I listen for and can hear partials beating in both the 4:2 and 6:3 octaves simultaneously.

As you say, the human ear is remarkably sensitive. As an amateur radio operator, I learned to use the Just Noticeable Difference ability to listen to Morse code operators that were transmitting close in frequency. It became easy to hear two or three conversations simultaneously and choose the one to which to listen.

Thanks for elucidating this point. The internal conflict in the Viennese unison fights with the mode lock, I think, giving more color to the sound. The unison locks and unlocks due to the tuning. I certainly agree with the comment on American style unisons. Everything sounds very dry without much singing quality.

Come on Robert, you're a DSP/electronics guy (as am I) - this is patently false. If it was true (that you can't hear amplitude variations in a composite waveform), then there would be no market for tremolo pedals!

For anyone with Audacity, generate a 440Hz sine wave and then apply the tremolo filter with a "wet" level of 10% and a rate of 2Hz. See if you can hear anything. If you can't Robert is correct, if you can, then all the pedal makers are correct.

Of course beats are different since they give a 100% cancellation and are essentially equivalent to tremolo with a "wet" level of 100%. But so say we can't hear periodic amplitude variation in complex waves is just not true - though it does sound different. If it was not possible to hear this, then it would be impossible to tune two sines at integer ratios. For anyone with an analog synthesiser, they will quickly show how easy this is. No distortion is needed to achieve this, either.

Paul.

Paul, we are talking about two different kinds of amplitude here. The tremello effect you describe is an amplitude variation that is applied to the whole waveform, so the RMS power goes up and down just like the peak to peak level goes up and down, and of course it is a variation that can be heard, even if that variation is only 10% of the total.

However the peak to peak envelope variations seen in mixing 440 Hz with 663 Hz are not this kind of audible amplitude variation. Those peaks represent the result of the phase relationship between the two waveforms changing. The total energy remains constant, as does the energy in each component. Another way to illustrate this effect is to compare the graphs of cos(t)+cos(2t) and cos(t)+sin(2t). The first graph has a peak level of 2.0, while the peak level of the second graph is 1.76. Yet both waveforms will sound equally loud because they contain the same amount of energy at each of the two frequencies involved. The only thing that changed was the phase relationship between the two components which caused the peak to peak value to change.

I think the problem is that it is not done on purpose , the "offocial" explanation is that as it will drift it is better to be that straight. And the loss of singing quality is corrected by using very forgiving hammers.

So that is mostly relate to a "way" of listening. But this is not only "Americacn way" mastering the string an pin so they stay where they are ask so much concentration the ear can get caught in a "listening mode" where the hardness is not noticed anymore.
The mistake in that concept is that if we do not know how to manage the thickening an the lengthening of the tone (phase lock) it does not install by himself by miracle, , if the unison are not "pushed" in a certain direction they can move later in any one, hence tone defects, whistlings , moanings, energy losses.

I worked today on a vertical that was tuned 3 years ago at last, and moved to my shop.

I hardly find really objectionable unisons, most of them are "locked" in "smiley" shape (I made a small video plucking the strings)
When I tapped the bridge pins the strings did raise in pitch, not the opposite. (I also made a video, before/after, while I don't know how good it is)
Hen, lightly tapping on the pins, the tone of the tap tells if the pin bottoms or no. The ones that do not transfer their little loss of firmness in the string (no false beats, just some parasitic noises and loss of clarity for the tone body)

LOL...well, the "official" explanation is that they start on the R, then tune the C with a "good sound," and then tune the L to the C with "a good sound." Most of them think they are tuning DOA, but that is far from what is really happening. Much translation and measurement is required to understand what is really happening.

A "good sound" is one that swells melodically/fully once and is pulled in solidly before it has the opportunity to ever create a second wave. This is why I denote the one string sharp method as a Sustain unison (i.e., it pulls-out/increases the sustain). The Viennese unison, with its two strings sharp in increasing amounts, creates a broader/fuzzy attack tone, but it is much more flexible in pitch. It is also more colourful: the back-and-forth internal struggle to lock onto the mode doesn't happen in one predictable direction like the sustain unison.

Happy union tunings to all: <---this is where the real magic happens!

I still don't agree with you Robert. There's no difference in RMS power between the case of 440+441Hz (which most definitely beats) and the case of 440+663Hz. The average, integrated from 0 to some time t is the same, but the instantaneous power level does vary as per the amplitude variation (which is due to the constantly changing phase between the two sines).

In the tremolo case, the average power of some period of time is also constant, despite the amplitude modulation. What do you hear in the case of 440+661Hz? If it is an artifact of the non-lineararities or our hearing system, it doesn't mean that it's not perceptible? A simple bit of DSP would also be able to recover the envelope modulation too so I'm sure a machine would be able to "tune" two sines using only this, rather than directly computing the pitches.

As for hearing beats in 4:2 and 6:3 octaves, there's more than just those two sets of beats to hear, though they are the predominant ones and a tuner applies a mental bandpass filter to focus in on the partials of interest. Isn't the goal of "maximum consonance" effectively trying to reduce the overall amplitude modulation (coming from all the beating partials) to a minimum? Maybe it's easier for a human to do this since it's possible to hear all the partials simultaneously (though a machine could also be programmed to do this as well).

Paul.

I would say yes. This is the minimum entropy state so desired by all of us. To get a machine, as opposed to a human, to do this requires, as I am attempting, and ETDs that determine the actual partials, not just the iH, of every note, to compute a minimum entropy solution by effectively 'listening' to every note of the piano simultaneously.

Prout, in order to do that, you are going to need to compensate for the amount/distance that the notes are able to be pulled in different directions, in each part of the piano, in order to create "maximum consonance." The current software seems to look at a theoretical single sine waves in terms of coincident beat rate amount and how these amounts should be equally smoothed out over the range of the piano. However, that is not our only sense of what makes the piano sound in-tune. The piano's ability to pull itself in tune in different directions is probably more important--and doing so in an "equal" manner.

In general, I appreciate the Verituner's clean equally beating approach to everything, but that doesn't factor in how an aural tuner is able to maximise resonance in the process. The next evolution in ETD software refinements, IMHO, needs to address equal resonance as well as equal pull.

For example, I can tune an ET M3 in such as way [with a nice 12th/15th, and a wide colour unison] that it gets pulled pure after a few beats. <----this ability/control and resonance is completely missing from current software design (i.e., unless I'm not using it correctly).

BTW here is that video I took while making the tone cleaner.
used a cell phone, so the tone is not ideal but it does not saturates so much.
But there is a fan in the back of the shop, I think it creates some impression of false beats in some cases. I only have one string on those, that have a slow - discrete -false beat right string of A5 I think)
The result of resetting the pins is clearly hears on the few notes near the break. once done on all the section, I had a way more recent instrument

[video:youtube]EC3VcJvhtic[/video]

May be you can hear the different tone when tapping depending of the pins, and some strings moving.

That is refreshing to read that there is some explanation on why "listening" can differ. Thanks.

Well this topic has digressed. I don't see many posters weighing in on which of these two components; hammer or soundboard, is more significant in the energy balance of the partials contained in a piano waveform. I think the hammer is half of the whole shebang.

I do think the discussion of unison types needs to be defined by where in the compass "styles" of unison tuning may apply. In the wound notes almost all of the unisons will have some mis-matched partial pairs. The strings just aren't perfect enough to enable DOA unisons. Coupling is very strong in the long strings and becomes less powerful as you ascend the compass. So you can't depend on coupling to "pull" things together in the treble.

Ed , certainly lighter strings may have less effect on all that mass of the bridge and soundboard.
I think what I listen to is the result of the coupling in terms of spectra evolving in time.

Once differentiated from beats , false or real we can chase for the sustain and +- try to place it in a "hot spot" where there is participation from some parts of the piano elsewhere, the most the best I suppose.

I think you are describing yet an "evolved" instruction to build unison (generally used with strip muting).

The one I often have read is "make the 2 or 3 strings sound together as if there is only one" ( citing Ron Koval there, among others)

Go in the woods with that in mind , and cry because of those "false beats" that occur after a second or 2 !
Talk of a crude definition, I believe things have evolved in 10 years, hopefully.

Thanks for trying to pull the discussion back to the topic. My feeling, from the comments made early in the thread, is that the hammer design (felt type, hardness, shape, mass) will produce, for a given strike velocity and strike point, a fixed set of partials, and that voicing can raise or lower the 'energy' of all the partials such that they maintain the same relative intensities. The soundboard then modifies the spectra to produce the final relative partial levels.

Well, because of the non-linear spring rate of the felt, the spectra balance can change with dynamics more with one type of hammer felt than another.