# Why does almost sounding right kind of sound wrong?May 19, 2007 2:09 PM   Subscribe

Guitar tuning through harmonics: Could you fine fine folks explain the physical/mathematical principles behind the increased "distortion" as two guitar strings being tuned through harmonics approach being tuned?

Distortion is almost certainly not the right word, but I'm guessing any guitarist who tunes through harmonics knows what I mean. My original impulse was to refer to it as "the little woobywooby sound..." You know, the overtone vibration that almost seems to resonate in another part of the ear. The sound/sensation that, upon synching up, tells you the strings are in tune.

A layperson guess based on some wikipedia reading: Is it due to overtones with relatively short wavelengths that are too close to cancel eachother out but still not quite harmonized (or whatever)?

Since I think this is also my "in" to starting to understand musical theory (and maybe even a bit of math), any additional explanations of general principles, interesting tidbits, or links to useful resources would be appreciate too. Keep in mind my education in math and musical theory is nigh nonexistent, but I love learning. Thanks!
posted by poweredbybeard to Science & Nature (24 answers total) 8 users marked this as a favorite

Best answer: The term you're looking for is "beating." Wikipedia has a good explanation of what's going on.
posted by danb at 2:14 PM on May 19, 2007

Best answer: When you hear 2 frequencies together that are not in tune, you will also hear a tone that is the difference between the two.
For example, if one A string is at 440 Hz and the other is 465 Hz, you will hear a 25Hz woobywooby.
This is just the result of adding the waveforms together.
posted by MtDewd at 2:16 PM on May 19, 2007

Also, a tuning guide was in this post recently.
posted by MtDewd at 2:22 PM on May 19, 2007

Response by poster: Thanks to both respondents so far. Quite helpful.

MtDewd - for clarification, if you don't mind: Why do you hear a tone that is the difference between the two? What's physically causing that? Was I at all close with overtones cancelling? Your last sentence (re adding waveforms) may be the answer, but I'm the type who needs to approach math through real things, and not real things through math.
posted by poweredbybeard at 2:28 PM on May 19, 2007

Best answer: It's nothing to do with overtones; you'll get beats with two pure sine waves.

You know when you're sitting at the traffic lights, and you have your turning indicator on, and so does the car in front? And lights on the car in front are flashing slightly faster or slower than your car's are?

And you know how for a few seconds, it'll look like they're almost exactly in time, and then they'll start slipping out? And a while later, they're almost exactly out of time, so his light is on when your light is off, and vice versa?

There's a repeating pattern that goes: in-sync, out-of-sync, in-sync, out-of-sync. And the pattern repeats much slower than the actual flashing of the lights. It takes maybe 20-30 seconds for the pattern to repeat, whereas the lights are actually flashing about once a second.

That's beats. The same thing happens with your guitar, only the frequencies involved are much higher.

When the strings are vibrating almost together, you get a strong noise, and when they're almost completely out of sync, you only get a quiet noise, because they cancel out. The repeating loud/soft pattern is picked up by your ears as an extra note, much lower than the actual notes being sounded. And, as you tune, the pitch of the extra note changes as the in-syncness of the two strings changes.
posted by chrismear at 2:39 PM on May 19, 2007 [4 favorites]

powered by beard--the beat frequency is caused by the syncronization of the two beats ever xHz where x is the Hz difference between the two strings. If you imagine their waveforms, due to the Hz discrepancy they won't superimpose perfectly, and when they periodically syncronize is where you hear the beat.
posted by Kifer85 at 2:50 PM on May 19, 2007

Response by poster: chrismear - lovely answer.

So, then, the effect is more pronounced when you're tuning through harmonics because both tones have relatively high frequencies?
posted by poweredbybeard at 2:50 PM on May 19, 2007

Not sure about that one. My hunch would be that a harmonic is a purer sound to the human ear than an open string is, so it's just easier to hear the quiet noise that the beats are making. But that's a complete hunch.
posted by chrismear at 2:57 PM on May 19, 2007

Best answer: Wow, good answers already. poweredbybeard and chrismear are both right. Here, I'm using scientific terms with common terms in parentheses for clarity.

When you have a harmonic, your waveform is closer to sinusoidal (sine-wave), so you have fewer higher order (harmonic) components interfering. The interference of those higher-order components might easily disguise the heterodyne (beating) frequency. In very simple terms, the more "sounds" you have, the harder it is to distinguish one.

Also, at higher frequencies, the beat frequency (generally quite a low frequency if you're close to in-tune) will be quite different from the fundamental frequencies. You can imagine that a 50 Hz tone is easier to hear with a 1000 Hz and a 1050 Hz tone than it is with a 100 Hz tone and a 150 Hz tone....
posted by JMOZ at 3:11 PM on May 19, 2007 [2 favorites]

Best answer: Oh, and fundamentally, the heterodyne (beating) frequency comes from alternating constructive and destructive interference. Total constructive interference occurs when the frequencies are aligned and the peaks of the waveform overlap. (You can think of this as two speakers with the same signal being louder than one.)

Total destructive interference occurs when the frequencies are anti-aligned (the peak of one overlaps the trough of the other). In this case, they cancel each other out. This is how active noise-cancelling headphones work, actually. if you want to demonstrate this, hook up 2 speakers right next to each other, but with one out-of-phase from the other. You can get pretty good cancellation, though it will only be in certain locations (where the waves are spatially in-phase).

The alternating between constructive and destructive interference will be sinusoidal (assuming, of course, sinusoidal sources) and the frequency, as mentioned, will be the difference between the two.

To complicate things more (i.e. feel free to ignore this section): if you don't have sinusoidal sources, you'll have a more complicated waveform, which will consist of the weighted sum of some number of sinusoidal sources of varying harmonic frequencies. Breaking this complicated waveform into the weighted sum of sinusoidals is called Fourier analysis. In this case, you are going from the time-domain (a complcated wave function of intensity vs. time) to the frequency domain (a weighted list of frequency components).
posted by JMOZ at 3:17 PM on May 19, 2007 [1 favorite]

Response by poster: Oh my god, I'm completely geeking out on this.

JMOZ, I think you might have already answered my next question, which was regarding what to call that additional tone (with a frequency = the difference between the two waves).

So, I would be correct, then, to refer to it as the "beat" or "heterodyne tone?" Other suggestions?
posted by poweredbybeard at 3:40 PM on May 19, 2007

Response by poster: Oh, and, from which points in the wave of that heterodyne tone (or whatever) would its frequency then be measured? From the points of total constructive interference? Destructive? Or is it the length between two points where there is the least interference?

Does that question even make sense?
posted by poweredbybeard at 3:49 PM on May 19, 2007

from which points in the wave of that heterodyne tone (or whatever) would its frequency then be measured?

You can measure the period using any point in the wave you like, as long as you use the same point every time. You'll get the same measurement regardless of where you measure from, since neither the frequency nor the shape of the wave are changing. You just use whatever point is most convenient to measure.

If you were timing a pendulum, you could start the timer when it's at its leftmost position, and then stop the timer when it's at it's leftmost position again. Or you could it stop and start the timer when it's at it's rightmost position. Or you could stop and start the timer when it's lowest position, swinging to the right. You'll always get the same result.

Of course, if you started the timer when the pendulum was at its rightmost, and then stopped the timer when the pendulum was at its leftmost, you'd get a different time, which wouldn't be the true period of the pendulum. You have to measure to and from the same point in the cycle.
posted by chrismear at 4:02 PM on May 19, 2007

Well, the frequency is the heterodyne, or beat frequency. I suppose it would be pretty reasonable to call it a beating or heterodyne tone.

To be technical, there's a single waveform, and the "tone" is actually a modulation of the sound. That is, if you have a waveform (again, a sum of all the Fourier components), then the signal is modulated at that frequency. (That is, the beating serves as an "envelope" function which modulates the amplitude.

If the beating frequency were sinusoidal (which would be the case if it were resulting from two sinusoidal tones), the wavelength (and frequency) would be the same whether measured peak-to-peak or trough-to-trough (both correspond to max. constructive interference; one is max positive and the other is max negative) or null-to-null, which would be max destructive.

If you do the analysis rigorously, you should find that if the initial frequencies are well-definied (that is, a periodic oscillation which can be defined as the sum of Fourier components), that I believe you should find that the frequency is the same regardless, and frequency measurement is simply a matter of choosing the same point on each repetition of the waveform.

If any of this is unclear, let me know and I'll try and put together some diagrams.
posted by JMOZ at 4:09 PM on May 19, 2007

Response by poster: It's mostly clear within the range of my understanding, which ends in the vicinity of modulation (time to look that up, I guess).

I'm not sure what you mean by saying "there's a single waveform" though. Aren't there at least two in my example, ie. the tones of the two strings being tuned?

(And again, thanks to all for the answers)
posted by poweredbybeard at 4:16 PM on May 19, 2007

If you record something in, say, Windows Sound Recorder, there's a single waveform. That waveform might be complicated and is generally the sum of a variety of waveforms. (In the simplest case, two sinusoidals of different frequencies). If you think about how a (one-way) speaker works, in can only move back-and-forth. The total waveform corresponds more-or-less directly to that motion.
posted by JMOZ at 4:21 PM on May 19, 2007

Hold your guitar up in front of the TV and watch the strings vibrate from side on...you can SEE the sine waves!
posted by fire&wings at 4:22 PM on May 19, 2007

I put together a PowerPoint with a few waveforms (made in the excellent free program, Audacity) to illustrate the concepts.

It can be downloaded here (.ppt), but this account will expire in a few months, so if anyone things it should be preserved for posterity, feel free to put it on a better server.
posted by JMOZ at 4:53 PM on May 19, 2007

Best answer: Pardon my inability to make a properly formed HTML link. Try again: Here. (.ppt)
posted by JMOZ at 4:54 PM on May 19, 2007 [2 favorites]

You should know, by the way, that you should never tune your guitar with harmonics. You'll ruin your ear. Tuning with harmonics is a way to tune justly, but the guitar is not a justly intonated instrument. It is a kludgy hack that is meant to approximate a well-tempered instrument, not a just tuning.

If you tune it with harmonics, it'll sound more sonorous when you hit all 6 open strings at once, but as you finger chords higher and higher up the neck, it will sound more and more awful as you get further and further away from well tempered tuning.
posted by ikkyu2 at 7:27 PM on May 19, 2007

Response by poster: Thanks to everyone (especially JMOZ, for going to the trouble of making a PPT!). Immensely helpful, and quite fascinating.

posted by poweredbybeard at 8:34 PM on May 19, 2007

I got a new tuner from D'Addario that works like a mini strobe tuner. It looks like a big pick (plectrum).

The Planet Waves S.O.S. Guitar Tuner pulses two out-of-phase LED light beams directly onto the vibrating string. When the string is out of tune, the two lights will visually dance on the string being tuned. As you approach in-tune status, the movement slows down and eventually stops when you are perfectly in tune. This visual tuning system enables precision tuning in silent or noisy environments, without the need for audible sound or sensing of any kind.
posted by chuckdarwin at 1:25 AM on May 20, 2007

Sorry for not responding- I was out all night.

If you didn't get this from the powerpoint, here's an Excel chart.

This is just taking 2 sine waves- 5Hz and 6Hz and plotting them out, and simply adding them together, over 2 seconds. This is like what you might see on an oscilloscope or wav recorder.

In 2 seconds, the 5Hz has 10 identical waveforms, the 6Hz has 12 identical waveforms, and the sum(yellow) has 2 identical waveforms (making it a 1Hz signal).

This is way, way more simplified than what happens in real life, but your eardrum receives the sum. I think your inner ear has sensors attuned to individual frequencies, so your brain hears the component frequencies and also the beat frequency.
And then it gets extremely complicated once the brain gets ahold of it.
posted by MtDewd at 9:20 AM on May 20, 2007

lol woobywooby
posted by billtron at 8:39 AM on May 22, 2007

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