Physics on harmonics...

Discussion in 'Strings [BG]' started by Achilles LS, Apr 9, 2003.

  1. What happens physically when you play a harmonic? I've wondered so much about this...

    Don't be afraid to get way too technical/scientifical - that's what I want to know about! :D
  2. I'm no pro by any means, but I'm pretty sure it's similar to playing a regular note except it doesn't have one main note being articulated. The undertones are the main part. That's as much as I know. :)
  3. moley


    Sep 5, 2002
    Hampshire, UK
    Overtones, not undertones :)

    I'm not sure of all the physics of it, but what happens is you're muting the actual fundamental (the open string, or whatever note you're fretting, with artificial harmonics) - and depending on where your finger is on the string, a different harmonic resonates.

    At half way along the string (i.e. 12th fret), the 2nd harmonic (octave) resonates, at 1/3 and 2/3 of the way along the string, the 3rd harmonic (octave + 5th) resonates, at 1/4 and 3/4 (not 2/4, because 2/4=1/2) of the way along the string, the 4th harmonic (2 octaves) resonates, etc. etc.

    To get the nth harmonic you have to touch the string at any of the following fractions of the string length:

    1/n, 2/n, 3/n ... (1-n)/n

    *Except* where any of those gives you a lower harmonic. For example, 4th harmonic would be 1/4, 2/4, 3/4 - but it's only 1/4 & 3/4 because 2/4 = 1/2, which gives you the 2nd harmonic. So, you get the lowest harmonic corresponding to that fraction of the string.
  4. Aaron


    Jun 2, 2001
    Bellingham, WA
    A friend of mine read a book about physics in music... I should probably ask her.
  5. :eek:

    You are a horrible person! :mad:

    :p just kidding
  6. Jeff Moote

    Jeff Moote Supporting Member

    Oct 11, 2001
    Beamsville, ON, Canada
    Heh, we're studying this in physics right now... and I knew a bit to begin with.

    What is happening is that the string is vibrating in not one eliptical path, but two or more. There are one or more places along the length where the string does not move at all, called nodes, and the places inbetween that do vibrate are called antinodes. The frequencies at which these antinodes vibrate are directly related to that of the open string (or fretted note) and are gennerally multiples of that.

    This is far more easily explained with a diagram, but I'm too lazy to find/make one. For more information, you can likely find it easily in a fairly introductory physics text.
  7. dmaki


    Apr 29, 2000
  8. all i suggest is finding notes on simple harmonic motion... you will find everything you ever wanted to know(about notes and harmonics, the other stuff can be found in oz).... its quite simple but i dont feel like trying to find my old class notes from physics... just read about simple harmonic motion and you will start to understand...
  9. i think that between montyp and moley u pretty much have it right. The waves on a string are stationary waves, usually, the first half goes from one end to the other and the second half is the other way, so you see an oval shape. As the wave reflects back on itself with a phase change of pi radians, you have a standing wave. With a second harmonic you have two wavelengths, as you can see two going either way.

    simple harmonic motion is the projection of circular motion onto a plane. Its acceleration is proportional to its displacement. can also be defined by the formula:

    a = -w^2x

    where "a" is acceleration, "-w^2" is a constant and "x" is displacement.

    (sorry to get all physicsy but i thought ud like to know and this is in my A levels this summer, so i get a bit of revision in!)
  10. wulf


    Apr 11, 2002
    Oxford, UK
    A note is made up of a series of harmonics - the fundamental blended with a range of overtones. These could be written X, 2X, 3X, 4X, etc.

    Therefore, with the open A string you have:


    When you play a harmonic you're actually filtering out the lower pitches from the series.

    Touch the string at the twelth fret and you are left with 110, 165 and 220Hz - 110Hz is an octave above the open A but the different mix will mean the timbre of the note is a little different.

    Touch it at the seventh fret and you get 165, 220, etc. 165Hz is the fundamental of 12th fret on the D string - an octave and a fifth above open A.

    As you go higher up, you filter out more and more of the lower series, which is why the tone of the harmonics becomes clearer and more bell-like, while the volume decreases (less pitches reinforcing each other).

  11. thrash_jazz


    Jan 11, 2002
    Ottawa, Ontario, Canada
    Artist: JAF Basses, Circle K Strings
    As mentioned, a note is composed of the fundamental (the pitch of the note you are playing) and an infinite amount of overtones.

    Moley, MontyP and Wulf have already mentioned how the harmonics are produced and why they sound the way they do, but I just wanted to add a few things (all of them IIRC):

    Overtones are largely responsible for the timbre of the sound. This is why a lot of synthesized instruments sound like crap - they don't reproduce the higher overtones properly.

    This is also why tube amps and solid-state amps sound different. Tubes tend to accentuate even harmonics, causing that nice warm sound when pushed; with solid-state amps, the edgier-sounding odd harmonics are emphasized.

    If you are [very] mathematically inclined, you can determine the harmonic content of a sound wave by performing a Fourier Analysis.
  12. where did you learn that from wulf? im not certain that ur wrong but what you said has muddled somewhat.

    Are you saying that a string ca resonate at two different frequencies at once? This would mean that there would have to be waves of two different wavelengths travelling along the string. Surely this isnt possible? I think the wound gets louder as you approach the twelf fret as the amplitude of oscialltion gets bigger. When you cut the wave short is stops it from getting a higher amplitude as it has got a smaller wavelength.

    Maybe im wrong, but there is seems to me to be something fishy about that.
  13. wulf


    Apr 11, 2002
    Oxford, UK
    I'm just passing on the knowledge I've received - I won't pretend to be an expert.

    However, think about the following two things:

    1. How comes the harmonic above the 7th fret is an octave and a fifth above the open string (eg. E on the A string - NOT D as I stated in my previous post... doh)? All you've done is touch the string and suddenly you've got a different note produced!

    2. If you record a single note on your bass and look at the wave file, you'll find that it isn't the simple sine wave of a pure tone but a much more complex beastie. If you've got some software that can perform Fourier analysis (like Transcribe!) you'll find that it detects a range of frequencies making up the note.

    For an external reference, see:

    which talks about how the phenomenon of a note being made up of a sequence of a harmonic series of frequencies as a way of psycho-acoustically creating the illusion of deep bass from small speakers.

    There was an online analysis posted by someone from The Bottom Line mailing list, which I found very helpful on this topic, but unfortunately I can't find that at the moment.

  14. thrash_jazz


    Jan 11, 2002
    Ottawa, Ontario, Canada
    Artist: JAF Basses, Circle K Strings
    No, Wulf is correct... The string doesn't have two fundamental frequencies, but what he meant was that each harmonic will produce overtones of different pitches.

    All of these overtones combine to form what we hear.

    IIRC the smaller wavelengths of higher harmonics don't have anything to do with their lower amplitude - it's simply the fact that they are higher harmonics that makes them less audible.
  15. I won't pretend to know anything about physics or harmonics, but hopefully this may help. I think you get louder playing at higher frets simply because the string gets closer to the pickup when you're fretting up there.
  16. thrash_jazz


    Jan 11, 2002
    Ottawa, Ontario, Canada
    Artist: JAF Basses, Circle K Strings
    The shorter the string gets, the closer the peak of the wave will be to the pickups... so yes. :)
  17. wulf


    Apr 11, 2002
    Oxford, UK
    The higher you go in the harmonic series, the lower the amplitude of the individual frequency in general.

    However, the TBL article I can't find suggested that the author's bass actually had a second harmonic that was more prominent than the fundamental! However, you still hear it as a low E because of the other associated harmonics - you can hear something round about 120Hz (Low E = 41.3Hz * 3) whereas the next E up should jump from c. 80Hz to c 160Hz (x1, x2). That's what the article I did post a link to discusses.

    The main factor is that the more lower harmonics you remove, the less frequences are going at once. If you add the first four harmonics together, the amplitude is A + B + C + D at any one point. If you've got a time when they are all at their maximum values, that's going to be reasonably loud AND certainly louder than B + C + D (lowest harmonic damped out) at the same point.

    That's also one of the reasons why an acoustic guitar can drown out an electro-acoustic bass - if they're playing six strings to your one, you've got to admit that you're outnumbered ;)

    Nb. if you were to get a number of tone generators and set them each to a related frequency, you could probably get an approximation of a bass sound (or any other instrument) by adjusting the balance of volume between them (although for a more authentic situation you'd also have to capture the varying balance during the attack and decay of your composite note).... and that's the basis of FM synthesis!

  18. Jeff Moote

    Jeff Moote Supporting Member

    Oct 11, 2001
    Beamsville, ON, Canada
    I must thank wulf and thrash_jazz for sharing their knowledge. Compared to where I am, they've studied much more, and thus are a much bigger help.
  19. I/Waves/Notes Standing Waves.htm

    go there... i think youll find most anything you want to know... also check out the other simple harmonic motion notes at: I/Waves/Wavesnotes.htm

    hope this helps so you can read for yourself.... im apathetic towards helping people so i prefer to have them learn it for themselves it also proves to be more effective in remembering things and furthering the knowledge rather than absorbing anything you are told and dont question.... ohh nevermind...