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Harmonics are integer multiples of a fundamental frequency

  1. Apr 14, 2005 #1
    I understand that harmonics are integer multiples of a fundamental frequency. Also, that the relative strengths of the harmonics are what make the same note on different instruments sound different.

    Why are these other frequencies made?

    How many integer multiples are there?

    Why do our ears read the 'fundamental frequency' and not any other? Is it simple because it's the lowest frequency it can hear?

    If a note is played and makes a 15 Hz frequency, below the hearing level, will the harmonic of 30Hz (within hearing range) be picked up by the human ear?
  2. jcsd
  3. Apr 14, 2005 #2
    You have harmonics because the musical instrument cannot be described by a linear differential equation. Which means that the movement of the mechanical parts , producing sound, cannot be expressed by a simple sine (cosine) function.
    Mechanical vibrations of the musical instrument exist with fixed frequencies (harmonics). That follows from a finite instrument size (neglect vibrations' decay). That means that the mechanical movement is also periodic. Now, to describe the movement at given frequency, we can use a set of functions, which have at least one common period.

    It is known that if we consider a set of sine and cosine functions with periods which are integer multiples of the base period, we can describe any function which are periodic with this period. Strictly speaking there are may be an infinite number
    of harmonics, but usually only some of them are strong enough to hear. And we may hear all of them, but the fundumental frequency is usually stronger. However it is possible to have an instrument with strong second, and weak first harmonic. It may be hard to do mechanically, but easy electronically.

    And human ear do not pick harmonics, (although we can be trained to distinguish them) neither we have built-in fourier spectrometer. We recognize air vibrations in a certain range. So we detect anything which is withing the detectable range and amplitude, person dependent.

    There are many reasons why the musical instrument is non-linear. For example, if you pull a string strong enough, you will break it- that is non-linear effect.
  4. Apr 16, 2005 #3


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    these two concepts are, for the most part, unrelated. an ideal string, not vibarating into any significant nonlinearity, can have modes of vibration that are not the fundamental mode. it's linear and it has harmonics.

    sometimes non-linearities can cause overtones to be detuned from their harmonic values, so there is some kind of relationship there, but non-linearities is not what solely causes harmonics. driving functions that are not a pure sinusoid is more often the cause.

    also human hearing (the ears are one component) do pick up harmonics, when our perception fuses all the harmonics (inc. the fundamental) into a single tone, then we perceive these harmonics as timbre. but sometimes we isolate and perceive a harmonic or two as separate tones if they are loud enough. Tuvan throat singers have been doing that for centuries.

    r b-j
    Last edited: Apr 16, 2005
  5. Apr 16, 2005 #4
    That's really good because not understanding makes answering your questions a bit more complicate; meaning I'd have to fry more braincells thinking of how to answer :wink:

    The fundamental frequency of an object is the minimum force that can be applied to set that object into vibration. Multiples of this frequency carry that minimum force and then some. So they set the object into it's simple harmonic vibration. The reason the pitch of the frequency is higher is because of the extra energy the wave carries in it which is do to the extra force.

    There can be an infinite number of these integers. However, they can never be below the fundamental. For example you can multiply the fundamental by three to give you the third harmonic of the fundamental frequency, but you could never multiply the fundamental by 1/3 because the fundamental, as I mentioned above, is the minimum frequency that will sympathize and set the object into simple harmonic motion.

    Our ears hear the fundamental frequency as well as the harmonics of that frequency as long as they are within the range of 20-20 000 Hz. Anything infra or ultrasonic is inaudible to humans. So our ears do hear the harmonics too. It is sometimes harder for some people than others. Like me, I can't stand it when someone strikes a tuning fork of any frequency because I can hear the frequency and when it isn't resonant. It is like nails running down a chalkboard for me. I have a hard time in my band rehearsal when every one is tuning their instruments because I can hear all the harmonics and the people who are not intune. Same goes for people who tune guitar or string instruments. I can hear all the frequecy changes...which sounds cool, until you actually experience it and then its not as cool as it sounds.

    The 15 Hz frequency is infrasonic meaning it isn't audible to humans. However the 2nd harmonic is. Remember how I stated earlier that humans can hear the harmonics of a fundamental frequency? This senario would be another example of that. The pitch of this frequency would be higher than that of fundamental because the frequency is higher. It has to do with the number of nodes and anti-nodes of that frequency as well as what kind of system your dealing with. Even with those, you would still be capable of hearing that frequency.
  6. Apr 16, 2005 #5


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    Just a nitpick -- the amount of force doesn't really have anything to do with it.

    Interestingly, I don't think the fundamental frequency doesn't even have to exist! I was playing with making pure tones before, and I tried to create two pure tones a fifth apart... but I didn't hear a chord! What I heard was a tone at the pitch of what would have been the fundamental if they were both harmonics!

    IOW, I tried to make a pure C and a pure G, and what I heard was an octave below that C.
  7. Apr 16, 2005 #6
    :surprised:WHOA!!:surprised: I've never heard that! How did you make that happen?! I wonder how there could be no fundamental frequency...that's gonna keep me up tonight :wink:.

    What does IOW stand for? I'm not really great at the abbreviations because I rarely use them. :blushing:

    Sorry about the force thing. I didn't even realize it until you said something. It was the wrong thing. Ah, it way late and I'm wicked tired...my pure love of science is what's keeping me up. :biggrin:
  8. Apr 17, 2005 #7
    Alright, I was unclear about the whole "15Hz and 30Hz harmonic" thing. Let me put it this way.

    When someone plays a 200Hz note on a guitar and then a 100Hz note, I know and can hear the 100Hz tone is lower (even though I am hearing a 200Hz harmonic from this one).

    Now, say these two notes were a 30Hz (note A) note and a 15Hz note (note B). The lowest frequency I could hear of either note and their harmonics would be 30Hz. Would I perceive these as the same note?

    Or, perhaps since all the note A harmonics would be higher than the note B harmonics, I would still perceive it as higher. Is this correct?
  9. Apr 17, 2005 #8
    I think you were cheeting on your brain. For example, we are able to imagine 3D world on a 2D movie screen. But to check this it is necessary to tape this sound and to put it on the oscilloscope.
  10. Apr 17, 2005 #9
  11. Apr 17, 2005 #10
    The problem is a speaker making that low of a sound. I could put a subwoofer in a closet, and that would help, but it still couldn't go below audible.
  12. Apr 17, 2005 #11


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    not only that, but if your 100 Hz tone was missing its fundamental harmonic (the one at 100 Hz) you would still hear it as one octave lower than the 200 Hz tone even though it has no frequency components lower than the 200 Hz tone. the difference is that one has components at 200, 300, 400, 500,... and the other has components at 200, 400, 600, 800,... . one tone has a (shortest possible) period of 1/100 second, the other a shortest possible period of 1/200, and for harmonic sounds, that's what your hearing will base the perceived pitch on.

    no, it's perceived on the basis of highest possible fundamental frequency.

    r b-j
  13. Apr 18, 2005 #12
    Ok: you might perceive the notes to be the same since you can not hear the fundamental frequency. But if you know the sound you are hearing that is 30 Hz and you know the fundamental for that frequency is 15 Hz then it becomes a moot point because you know that its the second harmonic for the frequency, even though you can't hear the first note. Everything after the 30 Hz frequency is something you would be able to hear and distinguish. You would just have to keep in mind that the 30 Hz is NOT the fundamental.

    Do you mean if the 15 and 30 Hz were two separet frequencies that have different harmonics? If that were the case then I think you are right.
  14. Apr 18, 2005 #13
    I think Shyboy might be onto something with the oscillioscope.
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