# 2 notes on a keyboard

1. Jun 20, 2011

### pkc111

A friend of mine says they can tell which 2 notes are being played together on a piano keyboard. How can this be if the 2 notes combine to form a single wave (sum of the 2) ?

2. Jun 20, 2011

### Born2bwire

The single wave is still a combination of the two frequencies of the keys. So if you were to look at the frequency content of the wave, it would look like the summation of the frequency content of the waves from each individual key. So as long as you have a way of processing and filtering in the frequency domain (which people can do) then your friend can live up to his or her claims. Still, we have our own set of psychoacoustic faults that make this difficult. For example, we have a hard time distinguishing sounds when they have a large difference in volumes. Sounds with frequencies close together would probably be harder to distinguish (although using the resulting beat may help in that).

3. Jun 20, 2011

Musicians can develop what I think is sometimes referred to as "trained musical ears".Their hearing becomes much richer because they can detect subtleties of sound not perceptible by untrained ears.

4. Jun 20, 2011

### pkc111

[PLAIN]http://calendar.arvo.org/9/3/12/images/fig01.gif [Broken]

I still dont really get it, although Im sure you are right.

From the picture above (rowA), to me the resultant wave does not seem to have the frequency of either wave (does it ?), but somehow both frequencies can be detected by the ear at once ?

Last edited by a moderator: May 5, 2017
5. Jun 20, 2011

### vanhees71

The superposition of two harmonic oscillations can be written as a product of harmonic oscillations. E.g.,

$$\sin \alpha + \cos \sin =2 \sin \left (\frac{\alpha+\beta}{2} \right) \cos \left (\frac{\alpha-\beta}{2} \right ).$$

For two frequencies not too different thus the first factor is a harmonic nearly of the frequency of the two tones, while the second one is of very low frequency which leads periodic variations in loudness (beats). That's why they depict these two signals in the second row. Of course the correct equation as a function of time is obtained by the one above by setting

$$\alpha=\omega_1 t=2 \pi f_1 t, \quad \beta=\omega_2 t =2 \pi f_2 t,$$

while the formulae printed above the figures cannot be right, because they don't depend on time at all!

6. Jun 20, 2011

### pkc111

Im still confused how an ear can make out the two component waves shown in (row A) from the wave received (rowA) ?

7. Jun 20, 2011

### Born2bwire

While the temporal (time) picture of the combined waveform looks confusing, the spectral (frequency) picture of the combined waveform will distinctly demonstrate the two or more unique frequencies contributing to the waveform. Our brains do a lot of signal processing to the sounds we hear. The brain looks at things like dispersion, phase shift (between the sound at either ear), and frequency content to reveal the myriad of information that we perceive automatically. Things like identifying pitch, position of sources and so forth are examples of such processing.

8. Jun 20, 2011

Staff Emeritus
The ear doesn't. The brain does.

9. Jun 20, 2011

### atyy

See Vanadium 50's very important point above.

Anyway, the ear performs a Fourier decomposition of the incoming sound into its component frequencies. Essentially, by having elements with different resonant frequencies (it's actually very complicated, and still being worked out http://www.cornell.edu/video/index.cfm?VideoID=698).

The notes on a piano are not pure tones, and are composed of many sinusoidal waves, and omitting the lowest frequencies from each note would still render them perceptually separate.

If you play pure tones that are a semitone apart (neighbouring black and white piano keys), you will not hear them as separate notes. You hear one note, oscillating in loudness. As the frequency difference approaches at least one third of an octave, you begin to hear two notes, each of constant loudness. http://www.sfu.ca/sonic-studio/handbook/Critical_Band.html

I believe there is still no definitive understanding of the critical bands in the central nervous system. One theory is http://www.ncbi.nlm.nih.gov/pubmed/9237756.

Last edited: Jun 20, 2011
10. Jun 21, 2011

### pkc111

OK I can accept that the brain may be a giant computer that can do a Fourier analysis on waves to analyse their components.

I asked a Science teacher today and they gave me another possibility that I wanted to put out there ie:
"There are hairs in the Cochlea which resonate at different frequencies (like antennae). The components of the summed wave are able to effect each antenna separately according to which frequencies match which hair."
This sort of made sense because I imagine radio receivers face the same problem as ears and brains. They would receive a "wave sum" of all the radio waves in the area and would therefore only have to be able to resonate at the frequency of one of the component waves .
Is this a correct analogy?

Thanks for your ideas.

11. Jun 21, 2011

### atyy

Yes.

But please see http://www.sfu.ca/sonic-studio/handbook/Critical_Band.html. The discussion in the last 3 paragraphs of http://www.jneurosci.org/content/28/18/4767.long should also be helpful.

Last edited by a moderator: Apr 26, 2017
12. Jun 21, 2011

### RedX

I thought an antenna picks up waves of all frequencies, but it's up to you to have a filter with an LCR or bandpass filter?

Is there an acoustic equivalent of the lumped elements for electromagnetism: inductor, capacitor, and resonance?

13. Jun 22, 2011

### chrisbaird

Yes, an antenna picks up waves at many frequencies, but it has a natural resonance depending on its length and picks up waves best near its resonant frequency. So you still need electronics to isolate your frequency of interest, but designing the antenna with the frequency in mind boosts performance.

Yes, the hairs in the Cochlea are like antennas in this way. One hair can pick up vibrations at many frequencies, but picks up best near its resonance frequencies. With hairs of different lengths, and thus different resonant frequencies, we literally hear many frequencies at once using different parts of our ear. The brain gets sound in frequency representation, not in time representation. The original question is similar to the question, "Can our eyes see the colors blue and red at the same time?" Yes, because there are different parts of the eye that are tuned to these wavelengths. There are red cone cells, blue cone cells, and green cone cells.

14. Jun 22, 2011

### pkc111

Thanks everyone for your explanations, they are much appreciated.

I guess the question for me now is:

"Why does any object resonate at its resonance frequency when the wave sum that arrives is not at the resonant frequency of the object, but is rather only the result of the resonant frequency wave added to many others to create the odd sort of shape wave eg shown at the end of row A above. ?"

Cheers

15. Jun 22, 2011

### atyy

Most objects have many resonance frequencies. An object that has one resonance frequency is the simple pendulum.

Basically, the object itself must has a "natural frequency, at which it oscillates if left on its own after an initial perturbation. If an external force drives the object with a driving frequency that matches the natural frequency, the external force will evoke large oscillations. If the external force drives the object with a driving frequency that is far from the natural frequency, the external force will evoke small oscillations. So when there are two external forces of equal strength with different driving frequencies, the evoked response will contain both frequencies, but its amplitude will be much larger for the frequency that is closer to its natural frequency.

http://farside.ph.utexas.edu/teaching/315/Waves/node12.html

Last edited: Jun 22, 2011
16. Jun 23, 2011

### pkc111

Thanks atyy:
What are the "two forces of equal strength" ? The resultant wave of 2 waves summing only has 1 frequency right ?
How can "the evoked response will contain both frequencies " ? An object can only vibrate with a response of 1 frequency right ?
Im obviously missing something sorry...

17. Jun 23, 2011

### RedX

An infinitismal antenna should pick up EM waves of any frequency. If you have an antenna longer than the wavelength of the wave hitting it, then I think you would have to worry if the wave hits it obliquely, i.e., the wave number has a component in the direction of the antenna, since this would cause destructive interference since different parts of the antenna would differ in phase.

I think where resonance comes in would be where the transmission line connects the antenna to the receiver. So if your antenna has a length of half the wavelength of the chosen frequency, then the impedance is set at around 73 Ohms, which would require the transmission line to have the same impedance. So a wave of a different frequency hitting the antenna would have a different impedance, while the line was set to 73 Ohms, so you get reflection of the power by the receiver (where does this power go, back out the antenna?).

Would a hair be like a quarter-wave antenna, since there is only one follicle sticking out? What would be the grounding plane?

18. Jun 23, 2011

### RedX

The object should vibrate with all frequencies hitting it. But for frequencies near resonance, it'll vibrate even more, and for frequencies away from resonance, it'll vibrate very little, so that your mind probably ignores the very little or might not even be able to detect it.

The odd shape in row A is just the addition of waves, and not resonance. It looks like it demonstrates the phenomena of beating, not resonance.

19. Jun 23, 2011

### pkc111

Let me put it another way to show you my confusion:

If a C and an F are played on a piano, the resultant wave sum which reaches the ear has a frequency different to both C and F (right ?)

Why would detectors of C and F (elements with resonant frequencies at these pitches eg hairs) vibrate especially in preference to other elements which have a resonance frequency closer to that of the wave sum ?

20. Jun 23, 2011