When you hear an electric guitar, it's almost always distorted. Even 'clean' electric guitar is usually somewhat distorted - a Fender Twin, the standard 'clean' guitar amp, has an order of magnitude more harmonic distortion than a typical home stereo. And 'crunch', 'metal', 'high-gain' tones carry even more distortion to them. Why do guitarists love distortion so much? It allows notes to be held and sustained, attack and decay to be manipulated, feedback to be induced, the touch sensitivity of the instrument controlled, and the production of the great variety of tones that the electric guitar has to offer. There are as many 'signature' tones as there are great guitarists, and distortion is a part of just about all of them.
Everyone who picks up a guitar learns the power chord almost immediately. Put a finger on the lowest string. Put another finger on the next string, 2 frets up. Play it loud! It's a big sound, especially with plenty of distortion. It's addictive. Compared to the sound of a single note, this root-fifth power chord is huge, monstrous! But why? Wikipedia will tell you that it generates a wide spectrum of harmonics, including a tone an octave below the fundamental. Big. Bassy. Loud. Most guitarists never think about the details of this. But the details are rewarding.
Any guitarist will tell you that distortion affects the tone of single notes, often drastically. But suppose you play 2 notes at a time? Distortion will generate sum and difference frequencies. This is a phenomenon known as intermodulation. The Wikipedia article linked to here, at this point in time, proudly proclaims that
Intermodulation should not be confused with general harmonic distortion (which does have widespread use in audio effects processing). Intermodulation specifically creates non-harmonic tones ("off-key" notes, in the audio case) due to unwanted mixing of closely spaced frequencies.'
WRONG WRONG WRONG WRONG WRONG!
This is a fairly common opinion. And it's wrong. All distortion will generate intermodulation. There's no escaping it! And the frequencies of intermodulated tones always have a linear relationship to the tones present in the original signal. Are they 'off key'? Not really. Even if not 'harmonic' they're definitely musically related. The confusion here is because distortion isn't just one thing. It's basically a function applied to the input signal, and this function can be characterized by a Taylor series. The harsh sounds mentioned in the Wikipedia entry refer to the intermodulation products produced by high-order terms of the Taylor series. Guitar effects and amplifiers are often designed to minimize these high-order terms. Although, as always, it's subjective. Some of us may like the mellow tones of Eric Clapton, while some of us like Atari Teenage Riot. It's art!
What makes a power chord really work? The simplest intermodulation products, the lowest terms in the Taylor series. The most prominent such terms in guitar distortion are the 2nd through 6th. The 'first' term is clean, undistorted signal, so we won't worry about it here. The fourth and fifth provide mostly high-frequency edge. So the low-end stuff that makes a power chord really MOVE is all about 2nd and 3rd order intermodulation products. A typical distortion tone for guitar has mostly odd-order distortion products - the 3rd will usually dominate. And now, it's time for math!
Suppose you are playing 2 tones at once, a 'diad' chord. Let them be pure sine waves for simplicity, let the lower-frequency tone have frequency X and the higher frequency tone have frequency Y. The lowest-frequency product from 2nd order intermodulation is going to be Y - X, while 3rd order intermodulation gives a tone of frequency 2X - Y.
Now let's consider a low power chord, low E and B above it. 80 Hz and 120 Hz. The 3rd order intermodulation product, usually the loudest, is 2*80Hz - 120Hz = 40 Hz, an octave below the chord's root! The 2nd order product is 120Hz - 80Hz = 40Hz as well. They reinforce each other, and this octave-down tone gives the power chord its grunt. Makes it work.
And now it's time for a more general theory. What about other musical intervals besides fifths? We can start by writing out the relative frequencies of a diatonic scale. Guitars do not play the diatonic scale, but an 'equal tempered' approximation in order to be tunable in all keys. Still, this is the best place for the math to start. Due to the phenomenon of beat tones, the diatonic intervals are what matters anyway. More explanation of this later.
| C | D | E | F | G | A | B | C' |
| 1 | 9/8 | 5/4 | 4/3 | 3/2 | 5/3 | 15/8 | 2 |
We can now use this scale to calculate 2nd & 3rd order low frequency intermodulation products for each interval found in the scale, as we did with the power chord (fifth).
| Interval type | 2nd intermod (Y-X) | 3rd intermod (2X-Y) |
| Octave (C' - C) | 1=C | none |
| Fifth (G - C) | 1/2='C | 1/2='C |
| Fourth (F - C) | 1/3=''F | 2/3='F |
And we see that the fourth is a lot like the power chord. Just like in Smoke on the Water! It's a little muddier than the power chord, because the low-frequency products don't coincide as nicely, but it's got its own sound and charm to it. Let's continue through the intervals.
| Interval type | 2nd intermod (Y-X) | 3rd intermod (2X-Y) |
| Major 3rd (E - C) | 1/4=''C | 3/4='G |
| Minor 6th (C' - E) | 3/4='G | 1/2='C |
| Minor 3rd (G - E) | 1/4=''C | 1=C |
| Major 6th (E' - G) | 1=C | 1/2='C |
Now we look at the major and minor 3rds, and their inversions the 6ths. None of these have nicely coinciding low-frequency intermod products, so none of them are likely to hit like the power chord, but hey what does? Look at that 2-octave down tone for the major 3rd! Even deeper than the power chord, must sound good, right? Sadly, no. It's too low and just makes 3rds on the low strings sound muddy, unless you're Dave Mustaine and use the right EQ to keep the low end clean. Most people don't use low 3rds, but you can! What's really interesting here, to me, is the 2nd intermod product. Whenever you play 2 notes of a major chord, the second intermod product will provide a third note to fill it out, shifted down there in the bass! Like a harmonizer only pure, pure analog.
Except how do you get just the 2nd product? Just about all guitar distortion makes both 2nd and 3rd, with 3rd usually dominating. You can build an FX pedal if you want. But a really cool pedal to experiment with is the Foxx Tone Machine. Pretty, beautiful box, I think it's even US made, but if $200 is too much for you, you can get the Danelectro French Toast which is the same circuit in a cheap Chinese made plastic box that won't be as durable and may pick up random noise and radio interference too! I paid $25 for mine at the local music store. Not that I'm a miser, but hey it was there. Flip the 'octave' switch on it, turn 'fuzz' all the way down, it'll give you mostly 2nd order intermod. As you turn up 'fuzz' it brings in more 3rd. You can totally experiment with this thing, it's great. Not to mention the Octavia, it'll ring out the 2nd intermod product nice and loud if you can find one. There's many other pedals that do this sound too - look for octave-fuzz in general. NOT the frequency-dividing octave pedals, but a true octave fuzz. Try using the neck pickup. Roll down the tone. Crank up the bass on the amp. You'll get it!
Finishing up our interval table:
| Interval type | 2nd intermod (Y-X) | 3rd intermod (2X-Y) |
| Major 2nd (D - C) | 1/8='''C | 7/8≈Bb(neutral 7) |
| Minor 7th (C' - D) | 7/8≈Bb(neutral 7) | 1/4=''C |
| Minor 2nd (F - E) | 1/12=''''F | 7/6≈Eb(neutral 3) |
| Major 7th (B - C) | 7/8≈Bb(neutral 7) | 1/8='''C |
| Tritone (B - F) | 13/24≈'Db | 19/24≈'Ab |
These mostly sound so harsh that they're pretty much useless. The 'neutral 7' and 'neutral 3' are notes that don't fall on an actual fret very well, but are found in some harmonica tunings, and have some use in vocal music. Still the only one I really like to play with is the tritone. It really fills out into a full diminished chord. Sounds nicer than it really should. At least, when you've got the 2nd intermod product nice and clear, it sounds amazing!
And now back to the question - why did I do all my math with diatonic intervals when a guitar uses equal tempered tuning that divides the octave into 12 equal parts? Well, all the intervals in equal tempered tuning are a bit off. This won't just move the intermodulation products, but will make them 'beat', just like when you're tuning and trying to make the beats go away. This is due to the higher harmonics interfering with each other. So you'll probably do a bit of bending to get the intervals as close to the pure diatonic ones as possible. It isn't easy to describe but it's not too hard to learn. Sometimes the damn frets are just not in the right places.
Assuming you find an octave fuzz and set it up as described earlier, a good way to start experimenting is to play an A-C# diad on the G and B strings, up on the 14th fret. Hear the low A come thrumming out. Now hammer on the 15th fret on the B string, making an A-D diad, and the low A rises up to a low D! Harmony! Do the full 14-14-14 to 16-14-15 hammeron thing and you've got the "We will rock you" break. Yeah, this IS what Brian May is doing to sound so big!
You can hear similar sounds only far more wild when Jimi kicks in his Octavia in Who Knows, at about 6:40 in. Or heck, pretty much throughout Mudhoney's classic album Superfuzz BigMuff.
Something that I like to do with my French Toast pedal set for maximum 2nd intermod - suppose my band is holding an A7. I'll just blast out this simple chord:
E-9
B-8
G-9
E, G, and C#'. The G-C# tritone gives an intermod product that rings out at 'E, filling in with the E. The E and G give a low ''C while the E and C#' give a low A. The total effect is an A7 with an added C, yes, it's an inversion of the Hendrix chord!
Or just go wild. There's still a lot of noises out there to make. And the power chord is only the beginning.
![]](images/rt_edge3.gif){kind=small}
![[XML]](images/rssblue.gif){kind=small}
![[P]](images/print3.gif){kind=small}
