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With your feet in the air, and your head on the ground . . .


{Wednesday, June 14, 2006}

I bought a nice guitar as a birthday present to myself, and I've been having a great time beginning to learn to play. I can just about play the four power-chord intro to the White Stripes "Fell In Love With A Girl", now. :-p On the whole, my chord repertoire now includes A, Am, B, C, D, Dm, E, Em, and G . . . and in my next on-line lesson I'm going to learn the blues scale, which I'm psyched about.

Anyways, Jess being a pretty expert musician (she's played piano, french horn, cello, bass, and she sings) I asked her what the deal was, exactly, with chords; what makes some consonant and some dissonant, physically. She told me not to worry about it, and that it would just get in the way - that I should just focus on learning to play. This illustrates a deep difference between us that we have trouble with, sometimes: I value understanding itself, regardless of whether it's practical or useful; she doesn't, really. Or maybe, to be fair, she just values and prioritizes productivity and efficacy more than the merely theoretical or abstract.

In any case, unsatisfied, I went out on my own into the wilds of the www to try to find how how physical wave "harmonics" relate to musical scales, chords, etc.

Here's a good beginning: Deriving The Musical Scale

In short:

If the (physical) frequency of a note is x (like, 440 Hz is an "A"), doubling the frequency (say, by vibrating a string exactly half as long) produces the same (letter) note, an octave higher.

If "x" is the frequency of a "C", then:

2x = "C"
4x = "C"
8x = "C"


The chromatic scale used in most modern western music is "twelve-tone equal temperament", meaning that an octave is divided into a series of 12 equal "steps" - which is to say, equal frequency ratios.

So regardless of what octave you're in, or what notes you're playing, x * y^12 = 2x, where y is the frequency ratio that defines the steps (if you go up 12 steps, you're up an octave, meaning the frequency has doubled.) We can actually solve for y in this equation, and it comes out to about 1.059, though that's not really too important.

What's cool, though, is that chords "sound nice" because they minimize dissonance.

The ratio of a "major third" is y^4 (~5/4), and that of a "minor third" y^3 (~6:5) . So a "major" chord has fundamental frequencies approximately equal to:

x, 5/4x, (6/5)(5/4)x

or more simply:

x, 5/4 x, 3/2 x

Each of these notes has overtones, since when you pluck a string with length L, the fundamental frequency is f = 1/L, but you also create waves of lengths L/2, L/3, L/4, L/5, etc: these have frequencies 2f, 3f, 4f, 5f, etc.

The more the overtones of a set of notes tend to match, the more "harmonic" (and the less "dissonant") the set sounds. For the major chord, we get:

x | 5/4x | 3/2x
2x | 5/2x | 3x
3x | 15/4x | 9/2x
4x | 5x | 6x
5x | 25/4x | 15/2x
6x | 15/2x | 9x
7x | 35/4x | 21/2x
8x | 10x | 12x
9x | 45/4x | 27/2x
10x | 25/2x | 15x

But keep in mind that any pair of frequencies {f,2f} are heard as the "same" note - so, e.g., 15/4 x is the same note as 15/2 x, just an octave lower. With that in mind, you can see that only the 6th overtones of the fundamentals (7x, 35/4x, 21/2x) and the 8th overtones of the second and third notes lack harmony with the other notes in the chord, within the first nine overtones; all the other overtones comprise only 6 notes, in a range of different octaves. The 11x row admittedly gets a little messy, but the 12x row is again perfectly harmonious.

The overtone structure of a minor chord (a minor third first, and a major third second) is left as an exercise for the reader. :-)

One other interesting note: the "perfect fifth" interval corresponds to a ratio of 3:2, which is the smallest integer ratio next to the 2:1 octave ratio. What does playing a just a note and its perfect fifth give you? A "power chord":

x | 3/2x
2x | 3x
3x | 9/2x
4x | 6x
5x | 15/2x
6x | 9x
7x | 21/2x
8x | 12x
9x | 27/2x
10x | 15x

This is dominated by just two notes: the fundamentals and their octaves comprise 5/6 of the first three overtones, and aside from the 6th and 8th (again), there are only 4 notes represented in all the overtones through 10x. This is why power chords sound so "pure" and, well, powerful.

Which in part, to bring us full circle, explains why the White Stripes rock.

posted by Miles 2:11 PM

Hey Miles!
Just dropped by your blog on a whim tonight... Mine has been picking dust lately too. anyhow, you know what your blog reminded me of? That little Disney cartoon with Donald Duck when he demonstrates the usefulness of math and does a little harmonics demonstration.

you're not donald duck. but that's okay ;)
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