In order to get a 2:1 ratio we need to use a harmonic in the middle of the wave above, which would be directly above the twelfth fret on a guitar with correct intonation. This "divides" the wavelength in half and doubles its frequency, singling out an octave. It looks like this:
(Top of String) / | | | | | | \ (Bottom of String)
If you keep dividing the wave/string into exponents of 2 (ie 2:1, 4:1, 8:1, 16:1 ect. ect.) you always get a higher octave note. But if you divide it into other ratios interesting things start to happen. The first interesting thing happens as soon as you make a 3:1 ratio, in other words the harmonic on the 7th fret. The note you end up with is a perfect Fifth above the second Octave, which on a "C" string would be a "G". These to notes together make up a "C" Power Chord, or a "C5" chord. After the 4:1 octave, at the ratio of 5:1 (Around the fourth, or ninth frets), you get a Major Third a couple octaves up, which on a "C" string would be an "E". This makes the first three notes we have "C", "G", and "E", or in another form "CEG" which is the 1-3-5 major triad. If you keep moving up in this fashion, you end up getting more notes on a major scale, and then eventually chromaticism, and microtonality. (Yeah. blah, blah, blah. Let's keep going.) Now we know where major chords and scales come from, let's talk about modes. (If you don't know what modes are, look into that before moving on. I don't want to lose you.) Diatonic modes (On a major scale) can be discribed by their level of "Flatness" or "Sharpness", The "Sharpest" being Lydian (1 2 3 #4 5 6 7) and the "Flattest" being Locrian (1 b2 b3 4 b5 b6 b7). Let's see this in order: Lydian(1 2 3 #4 5 6 7) Ionian(1 2 3 4 5 6 7) Mixolydian(1 2 3 4 5 6 b7) Dorian(1 2 b3 4 5 6 b7) Aeolean(1 2 b3 4 5 b6 b7) Phrygian(1 b2 b3 4 5 b6 b7) Locrian(1 b2 b3 4 b5 b6 b7) All the "major" modes, "minor" modes, and the half diminished mode are all in perfect little groups. Also, every mode in this order is a fifth away from the next if they were all derived from the same major scale. Also notice the order the intervals are becoming flat: 4,7,3,6,2,5. The next logical interval would be 1, but if you flatten the 1 you get another Lydian mode, but a semi-tone down (1 fret). Alright lets talk about chord progressions for a second. You'll see alot of chord progressions that move in fourths (ie: I-IV, or V-I). Let's explore why. As you may have noticed, on our chart of the modes above, the most tense mode, Locrian, is the most flat. It is also the farthest in terms of Fifths. So thus, moving up in Fifths adds alot of tension. Therefore, if we move through the inverse interval of a Fifth, the Fourth (In "C" Major, instead of going up five intervals to the fifth note "G", you go down five intervals to the fourth note "F") we can move very smoothly through all the chords, in the order 1 4 7 3 6 2 5 1. If you remember the order the notes become flat on our mode chart, you also see the same pattern (4 7 3 6 2 5) As you can see, it's not just some randomly chosen pile notes and harmonies, it's all nature and math. Thanks for reading.
(Top of String) / | | \ \ | | / (Bottom of String)