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Fine Ear Games! Thanks Chris.

I hear the secondary beats when I crank up the volume, but not when I play it softly, unlike the primary beats which retain the same relative subjective loudness irrespective of volume.

Of course what we hear when piano intervals beat is 99.9% primary beats between the nearly coincidental partials, "graphing calculator talk" notwithstanding.

Kees

Of course, stated without any explanation or evidence.

Oh please. The model of near-coincident partials leading to beats at specific pitches, has been described and explained umpteens of times, both on this forum and in other places. Overtone/harmonic/partial series (very illustrative videos by Mark C.), beat locator charts, ghosting, filtering, F2-A4 vs. F2-fork both beating at A4, etc. etc. etc. Both explanation and evidence have been offered. One can only lead a horse to water...

The same abundance of evidence and explanation cannot be said of the graphing calculator talk - least of all how this model could explain that
i) beats are heard at specific pitches (at least by some people),
ii) these specific pitches are those of the nearly coincident multiples/partials of the fundamentals, as can be (and has been) confirmed in spectrograms, filtered recordings, etc.

The graphs show that the addition of a fifth results in three distinct frequencies being generated, and if they are not exact, there are beats on each of them. They also disprove a central point of the idea that partials are what beat: that beats depend on there being partials.

What the graphs do not show, which is the weakness of the partial series, is where the partials are supposed to come from. As far as I know, except for organs, no instrument consistently generates partials independently of one another.

All of that is incidental to the claim that "what we hear when piano intervals beat is 99.9% primary beats between the nearly coincidental partials." The 99.9% just seems to have been pulled out of thin air.

But does the graph really show "three distinct frequencies", BDB?

I plotted
y = sin(10*2Pi*x) + sin(15*2Pi*x)
(I call it "Pi" here, because apparently the special ASCII character that I used previously, displays as "n" on your PC.)

This corresponds to the sum of a sine wave of 10Hz and one of 15Hz. I inspected this graph, measuring the x-values of the maxima and minima, to determine the distance between successive peaks and successive troughs.

Here are the graphs I used (I hope the links work); the one shows one period, the other about three periods. I tried to measure down to 0.001s, which should be accurate enough.
https://www.dropbox.com/s/6lf0dkbengf6423/10Hz%2015Hz%201period.jpg?dl=0
https://www.dropbox.com/s/vashhsycjqadf6n/10Hz%2015Hz%203periods.jpg?dl=0

The only clear periodicity I can see, is one of 5Hz (i.e. repeating every 0.200s), being the difference tone between the two sine frequencies. From what I can see, the other maxima and minima do not occur at regular intervals from one another, i.e. the distances between the peaks are not regular. If indeed there are "three distinct frequencies", should they not be identifiable as maxima and minima at regular distances? The distances should be the inverses of their respective "distinct" frequencies, in this case:
10 Hz ... 0.100s
15 Hz ... 0.067s
Presumably the third "frequency" you speak of, would be the lowest common multiple, i.e. 30 Hz ... 0.033s.

But the closest spacing between peaks is 0.053s (measured between the first and second trough, and mirrored between the second and third peak). This corresponds roughly to 18.9Hz. All other peaks (and mirrored in the troughs) are more widely spaced:
... first to second peak is 0.071s (14.1Hz) and
... third peak to first peak of the new period is 0.076s (13.2Hz).

And none of these spacings repeats itself continuously. They only occur once per 0.2s period. In fact, not even the 10Hz and 15Hz signal are clearly visible.

Could you please point out the "three distinct frequencies"?

Mark R.

Did you notice the 0.08 sec periodicity? That would correspond to 12.5 Hz, the average of the two frequencies, beating at 5 Hz.

No, I did not, because there is none.

There are, in total, three wave-subforms in the 200 ms window. The first and the third have an 80 ms "periodicity", and the one in the middle has a 40 ms "periodicity".

But all of this is really a moot point. To speak of periodicity that is actually sensed by the ears, we have to look at maxima and minima, not the x-axis intercepts. Why? Because ears don't sense zero sound pressure. They sense pulses, i.e. maximum deflection. And I have already looked at those in the graph. There is no 80 ms periodicity amongst the maxima and minima. [Edit: at least not as far as I can see.] Sorry.

[Edit 2: and even if we were to look at X-intercepts: what sort of "periodicity" does a wave form have that has segments of 80, 40, 80, 80, 40, 80 ms...?]

Ellipses marked at 0.08 sec intervals.

PS The wave pattern also crosses the x-axis at 0.04 sec intervals, a characteristic of a 12.5 Hz wave frequency.

So that's why we all hear differently!!!!

It differs according to whether the periodicity is measured in decimal or fractions.

Finally!!

Yes, exactly. These are patterns set up by whole number frequency ratios lining up and falling apart again as the frequencies change.

The patterns actually take shape from progressive changes in the waves that comprise the envelope itself.. more of a visual illusion. I'd be hard pressed to guess how we would hear this - if we would at all. It requires special equipment.

I am interested in it.. have been curious for years. Maybe Dave Koenig has the equipment to test this? We do hear the absolute value (or envelope) of pressure waves, so it is unlikely that these mathematical patterns are audible, but who knows without testing?

Interesting stuff this. Here is a wave file of two sines, 10Hz and 15Hz interacting. One can actually hear the 5 hz beating (more like pulses) very clearly, and not at any pitch obviously. This picture of the independent analysis of the actual recorded sine waves shows the same detail as the graphing calculator. Notice that there is essentially zero harmonic distortion of the sine waves.

When you have two source frequencies f1 and f2 within a few Hz of each other, I think it is generally accepted that you will hear a frequency of (f1 + f2)/2 and beats at (f2-f1). In Mark's example of f1 at 10 Hz and f2 at 15 Hz the resultant frequency is 12.5 Hz and the beats are at 5 Hz.

In this example there is a maximum and a minimum on either side of the intercepts occurring at 0.04 second intervals (see graph four posts above).

More realistic examples would have higher values of fi and f2. For instance you could enter 440 Hz and 445 Hz into an online generator. Does anyone hear anything other than a tone and a beat?

Tunewerk, Prout & MarkR:

Indeed, this sum of 10 Hz and 15 Hz is an interesting wave. Like several of you, I generated it and plotted it up. The top graph shows 5 seconds of the wave. The middle graph shows its line spectrum with the two spikes at 10 and 15 Hz. My line spectrum looks a little different because I chose to plot the power linearly instead of in dB. The bottom graph shows a close-up that matches the graphs that Tunewerk and Prout have referred to.

I listened to the generated wave and heard a beating or thumping that seemed to have a repetition of 5 times per second which does not show up on the line spectrum.

Then I squared the wave to force it to have a periodic pattern that the line spectrum would detect. The graphs below show the result of the squaring and the line spectrum of the resultant wave which shows a spike at 5 Hz which is what I hear. Also, the middle graph shows the repeating form that has two peaks occurring every 0.2 second. These pairs of peaks are the thumps that I am hearing. The line spectrum also shows spikes at 20, 25 and 30 Hz which is probably a consequence of the double spikes that occur every 0.2 seconds.



I then generated the thumping sound of the original wave again and recorded it using Audacity. I read the recorded wave back into Matlab and analyzed it. The last graph below shows the result in the time domain. The trace shows how the clean wave shown above in the first graph gets quite muddied after going through speakers, microphones and A/D converters. The graph also shows numbers which count the audible "thumps". There are 25 thumps over 5 seconds which is consistent with the line spectrum of the squared wave and consistent with what I (and, hopefully, you) hear.

That the sum of two waves at frequencies 10 and 15 Hz could generate a perceived tone at 5Hz is an ultra simple example of what happens with the low bass notes on a piano. The C1 note has little if any strength at 32.7 Hz. However, there is considerable strength in the higher partials that are approximate multiples of 32.7 Hz and therefore are spaced apart at approximately 32.7 Hz. When these partials are added, a wave with a repeating component having a period roughly equal to 1/32.7 seconds (the period of the C1 fundamental) is generated and is heard even though it does not show up on the line spectrum.

The following graph shows the line spectrum, cumulative line spectrum and autocorrelation for the C1 note. The line spectrum shows that the fundamental is about 40 dB down relative to the dominant partial. The cumulative line spectrum shows that there is negligible power at the fundamental. The autocorrelation shows that the pitch of C1 is approximately 32.4 Hz which is consistent with what we hear.


Going to bed.

There are two axis crossings between some of those ellipses. That's one-and-a-half wave forms, and cannot be termed "periodicity" in my books.

Similarly, the axis crossings at 40 ms intervals are only sporadic, not regular. Hence, that's no "periodicity" either.

All:

My frequencies were only chosen for convenience. They were not intended as an audio example, but as a simple example of a perfect fifth, i.e. one note at frequency f, and another at frequency 1.5f.

Had I known that the example would also be listened to, I would have chosen 200 and 300 Hz rather, as those fall nicely in the temperament range. But in terms of graphing the resulting waveform, there is really no difference. I just wanted to show the shape of the waveform, and ask BDB where those "three distinct frequencies" are.

(Still awaiting his answer in great anticipation... )

[Edit: I just graphed y=sin(200*2Pi*x)+sin(300*2Pi*x), and the resulting waveform is exactly the same, repeating every 10 ms, i.e. at 100 Hz, which is the difference tone. So, except for stretching or compressing the x-axis, the waveform remains exactly the same.]

Mark, you can consider your example as one of two frequencies beating as well as one of a fifth. In the case of beating you get an average frequency and a beat frequency. The reason that there is an extra axis crossing within some of the 0.04 second intervals is a result of the way the two waves superimpose themselves on, or interfere with, each other. The reason that you cannot see the 10 Hz and 15 Hz components is that they have averaged themselves out into the complex 12.5 Hz wave that your graph illustrates.

Given the complexities that arise from such simple abstract models, highly informative as they are, I think one has to deal with the far greater complexities of piano sound by ear, or by Fourier analysis.

Dave, very interesting posts.. especially the included line spectra and information about the source of illusory lower frequencies on the piano. Thank you..

I have a more complete post to contribute to this discussion now, after a lot of thought. For those willing to follow, I will make the ideas as clear as possible. I believe this is a strong reductionist argument that answers this question with near certainty, without lab-grade equipment.

I add this on to address the part of the question still not quite answered, as well as to share some exploration..

The practice of examining complex wave functions is still a developing science. It is difficult to understand and visualize these things. Fourier analysis works off of the theory that any complex wave can be reassembled using a series of constituent pure sines. This is sometimes true to the components of the wave, but sometimes not. In other words, there is always a way a periodic function can be approximated using an infinite sum of constituent sines, but the original wave may have not actually vibrated that way in nature at all.

Even a simple string does not vibrate in a series of separate components. It vibrates as one integral, complex wave, including transverse, longitudinal and phantom frequencies. Why does it vibrate this way? The deep mystery of matter itself. Fourier transforms are a useful tool of analysis, but do not always let us understand waveforms.

I mention this to begin because there is a difference between many things mentioned above.. from mathematical wave forms, analysis and actual vibration of real material.

First, examining trigonometric identities, the law for addition of cosines gives a product formula that is an alternative way for describing a complex wave.

(1) cos(f[1]) + cos(f[2]) = 2 • cos((f[1] + f[2])/2) • cos((f[1] - f[2])/2)

Ian already mentioned this, in speaking about the two components of a beating waveform. This identity provides equivalents for parts of the wave we experience, and is true mathematically by the angle addition theorems.

(2) 2 • cos((f[1] + f[2])/2) - average frequency

(3) 2 • cos((f[1] - f[2])/2) - beat envelope frequency

However, even though we sense these waves, do they actually exist? Not in terms of when a complex wave form is generated by two pure sines, which is what is being discussed here.

These expressions describe the product of two different waves (2,3) that form the same final complex wave as the original waves in the addition (1). However, we hear things through wave addition.. and all compression waves in the physical universe behave through wave addition. What exists is in reality is only the trace, or amplitude modulation, of the original waves that follows the path prescribed by the product waves on the right side of the expression. Wave products are only a mathematical construct based on theory, and actually act as scalars to each other in the time domain - an equivalency, but a mathematical equivalency.

Another way to look at this to understand it deeper, is the Fourier transform. The approximation of real waves is based on the sum of a series. It is essentially just a longer, additive and progressive sum of the simple sine addition function (1) above.

(4)

When Dave squared the 10/15Hz complex sine and ran a line spectrum on his machine, it detected all 4 of these component waves: 5Hz, 20Hz, 25Hz, and 30Hz. Of course, all frequencies show as doubled because the spectrum is squared, so the negative phase is flipped over the x-axis. The actual component frequencies should read: 2.5Hz, 10Hz, 12.5Hz and 15Hz. The 2.5Hz sine is detected by the ear as the 5Hz pulsing beat rate.

Squaring the wave made these other frequencies more periodic and detectable, when before they were not - but nevertheless, they would not exist as true wave components in the real world, only 'traces' that we could sense.

Getting to the heart of the subject though, I felt Chris Leslie did the best job of laying out what the real question was in the nature of his graphs. He laid out the interference patterns between two sines at an interval of a pure 5th and a tempered 5th that show an alignment and a divergence.

The real question is not whether beating occurs between two pure sines. We know it does as they depart in frequency. The question is, are the interference patterns that take shape from the complex, evolving form of the wavelets in two pure sines at the fundamental causing any kind of detectable beating?

Here is a wave at a much lower frequency, but showing the same effect, composed of two sines that are at a frequency ratio of the octave (2/1).

Fig. 1:

And here is the following wavelet pattern that develops from that integer ratio departing by one unit cycle.

Fig. 2:

There's actually two 'beat envelopes' in this wave, or two sines being traced out by the departure of the two parent waveforms. Are those beats actually there? No, they are an interference pattern. But can we sense them?

If you look, you'll see that the interference pattern is actually created by a visual illusion in this case, by the shape of the wavelets progressively changing. There is also a pattern to the sines that trace these envelopes.

So the question remains, given that it has been defined what separates theory from how we sense things in reality, and given that it has been defined how these wave patterns exist in theory, can they be sensed in reality? Here's where it gets more interesting in tying together some real world data.

Below is a series of spectrogram images from data taken just yesterday. I think this is the best way to see sound information for general purposes because of how much it data it contains: time on the x, Fourier transform on the y, and intensity as we hear it on the z (through color coding of dB).

Fig. 3:

On the left is about 15 seconds of sample from one string of A4 in a Steinway L. You can clearly see the fundamental at 440Hz and all the partials neatly spaced as we would expect on a typical piano string, following the form, n•f[0]. There's a strong fundamental and 2nd/4th partials with a relatively expected decrease in the amplitude of the rest of the series. Of course, there's inharmonicity, but it's too small to be visually detectable on this scale. False beating is visible in partials 4, 6 and 7.

On the right, is a tuning fork. Same frequency, at A440. I hope many of you are as surprised as I was to see the spectrum of a fork turn out to be so similar to that of a vibrating string. It's clear to see a regular harmonic spectrum following the same linear distribution, with a strong fundamental and 2nd partial, but also a strong 6th partial (E7). The high frequency energy is interesting.. the fluctuating nature of it.. as well as the phantom partials.

Fig. 4:

Above is a series of related images taken from samples of an open-ended pipe, natural vocals, and whistling.

There's a very clear and detectable linear partial series in the resonating pipe, although I was more interested in the lower frequency region. It's interesting to see how the partials are less frequency specific, which mirrors what we hear. The break at about 11 sec. is from a break in the sound itself.

Vocals show a very linear partial series as well. The departure of the higher harmonics as the fundamental dipped down was interesting, but could be explained by a change in the vocal cavity. The whistling sample shows a simple spectrum, basically just a 1st and 2nd partial. The change in tone is me moving down by semitones, then up again at the end.

Fig. 5:

This final series of images gets to the point. Almost every type of vibrating membrane shows a linear partial series. Of course, every membrane has its own specific inharmonicity, but that is not significant or relevant to this discussion.. only the existence of partials themselves.

The first image above was taken from a pure sine at 440Hz running through a regular paper cone speaker. At the function generator, only a pure sine was present. Out from the speaker, multiple partials are present. As a matter of fact, it looks hauntingly familiar, doesn't it? Most similar to the piano spectrum, minus decay time.

The second image was taken from two sines running through the same system and speaker. Again, checked as pure sines electronically. In this case, one was dialed in at 440Hz, the other at 441Hz. You can see the 1bps pattern clearly in all partials from both series. Seeing a beat in the whole tone is what we experience when we tune unisons.. two vibrating strings with nearly the same iH, moving into or out of alignment with one another.

Finally, the third image shows one sine at 440Hz and the other starting at 550Hz, but moving to 550.6Hz at about 7.7 sec.; 0.6Hz sharp of a pure M3rd above. I took several samples with several different interval sizes and they all exhibited the same behaviour. The critical element, once you follow the rest of the connections? A beat rate.. but not coming from the fundamental.

Whether the beat envelope of two sines converging and departing from whole number ratios is audible or not is still up for technical discussion, but it's safe to say we haven't been hearing it. A definitive test would require lab-grade equipment with zero harmonic distortion.

My personal opinion is that technicians who are strong proponents of the 'natural' beat, or a beat supposedly coming from the fundamental frequency, are hearing an effect that Dave described earlier very well..



I hope all reading had as much fun on this journey as I did!