It is extremely important that you believe that the overtone series is inherent to music in a natural way before proceeding, because now what I want to do is make an argument that the reason the 12 note chromatic scale that is so fundamental to Western art music and sounds like the “natural” way to divide up the infinitely many possible pitch choices is that the scale is derivable from nature.
I think most musicologists would probably say the scale that sounds natural to you is based on the culture you grew up in and there is no objective reason to favor one over another. Of course, just like what language sounds natural to you depends on what you grew up with, what musical language sounds natural will also depend on culture. But except for a few very rare exceptions (I actually don’t know of any) all scale systems that have had enough cultural significance to survive history actually can be derived in a similar way to what I’m going to do.
In Paul Hindemith’s book The Craft of Musical Composition, he spends almost a fourth of the book doing this derivation. Note that he wrote this in a time when most academic composers were extreme relativists and wanted to throw all Western conventions out the window (including the use of scales and well-defined notes).
My guess is that he wanted to make an argument that our scale system was not some subjective arbitrary system, but is objectively superior to a choice of scale system that is not derivable from the overtone series (N.B. this is not the same as saying the Western scale choices are superior).
Now that that rant is out of the way and I’ve alienated all readers we’ll move on. Actually, there isn’t much to derive if you fully understand the overtone series from last post. Let’s go back to C being the fundamental, because we need to pick some arbitrary starting point from which to derive the rest of the notes.
From the fundamental to the first tone of the overtone series, we get exactly an octave, so it makes sense to talk about moving a note down an octave (i.e. this is an allowable interval in our scale derivation). So really we just take the overtone series and move notes down by an octave until they are in the range between the fundamental tone and the first overtone.
Recall we got C, C, G, C, E. So at the fifth partial we only get two new notes which when moved down octaves give us C-E-G (this is a C major chord and now we see how the major chord can be “derived from nature”).
An interesting historical tidbit is that the Pythagoreans construct the rest of the notes only using this many tones of the overtone series. Now that G and E are well-defined notes, you just start their overtone series and go as far as the fifth partial to get a few more notes. Then start the overtone series on those notes until you’ve gotten 12 notes and repeating the process just produces ones you already have.
This is actually what was done back then, and if we listened to music tuned in this way it would sound horribly out of tune to our ears. In Bach’s time a switch from this “equal temperament” to the well-tempered system happened.
We’ll follow Hindemith’s construction. Instead of only adding in notes you get from the overtone series, you go backwards too. You take an allowable note and you consider fundamentals for which that note could occur in the overtone series and you only add in notes that occur in the right octave (between the two C’s). This just amounts to dividing the frequency of an existing tone by the number of overtones we’ve gone up.
That sounds confusing but here’s how it works. Take 64 Hz C. The second note in the series is 128 Hz C. Testing out C in both the first and second spots of the overtone series produces no new notes within the octave.
Thus we move to the third note 192 Hz G. We test out G being in the first, second, or third spot of the overtone series and see what new notes occur in the first, second, or third spot. We rule out G being the fundamental because all new notes would be outside the octave. If it is the second note of an overtone series, then the fundamental is G an octave lower (outside the allowable range) and the third note would be C already in the scale.
The last thing to try before moving on is testing C as the third note of an overtone series. The fundamental would be 256/3=85.33 Hz, which is our modern day F. A new note! Then you keep going. You test each new note as the first, second, third, or fourth overtone of some overtone series and see which notes land in the range 64-128 Hz (this just amounts to dividing by 1, 2, 3, or 4 respectively as mentioned). You keep doing this and you’ll get our modern 12 note chromatic scale.