art, math, music, physics

Mathematical Music Theory 2: Derivation of the Western Scale System

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.

Mathematical Music Theory:

Part 1: The Overtone Series

Part 2: Derivation of the Western Scale System

Part 3: Combination Tones

Part 4: Intro to Post-tonal Methods

(Part 5:) The Stack of Pitch Class Sets


8 thoughts on “Mathematical Music Theory 2: Derivation of the Western Scale System”

  1. Hi, I’m playing around with your idea and if you had a moment I was wondering if you could clarify why you see moving the overtones down a single octave is acceptable but not extending the overtone series themselves to higher frequencies? Is it damping/audibility of the higher frequencies?
    i.e. (quote:”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 taking the 64Hz example for C, I’m essentially looking at this table (second column is Hz, and 3rd column is fairly close approximation of the musical note on equal tempered scale):

    series factor overtones note
    1.0000 64 C
    1.0000 128 C
    0.5000 192 G
    0.3333 256 C
    0.2500 320 ~E
    0.2000 384 G
    0.1667 448 ~
    0.1429 512 C
    0.1250 576 D
    0.1111 640 ~
    0.1000 704 ~
    0.0909 768 G
    0.0833 832 ~
    0.0769 896
    0.0714 960
    0.0667 1024
    0.0625 1088

    and the first overtone that really seems out of place is 448Hz. It lives somewhere uncomfortably between A and A#. Is there thinking/idea/guess as to why that overtone is unmusical vs. the other “musical” overtones? (or maybe 448 would be musical despite not being in the equal tempered scale)? Again, just playing around with this idea and curious what you might think about the anomalous overtones? If you have thoughts I’m curious. thanks in advance.

  2. I think this link may actually be related to the above discussion.

    Octave spiral with the first 16 elements of the harmonic series

  3. Thanks for the comment! Of course there are lots of different tonal systems you can derive using different methods that are similar to what I did. I was just following what Hindemith does in “The Craft of Musical Composition.” Historically speaking, this of course is not how our current system was developed. It is merely meant as an after-the-fact justification of why it is “natural” that it sounds good.

    My guess is that using the 7th overtone exactly where it is would be kind of cool (maybe a little “jazzy”) and probably wouldn’t sound bad if only using it in some overtone scale built off of the fundamental. You should check out the book I mentioned. Hindemith devotes an entire section (Chapter I section 9) to talking about how to handle the seventh overtone.

    I think the only reason for saying “move down an octave” rather than allowing any octave jumps is that you don’t get new notes by moving them up? I’d have to re-read what I wrote and think about this to be sure. But you test out notes formed in the first spot, then first two spots, then first three spots, etc, so it seems moving notes up an octave would only ever move them out of the allowable range.

  4. thanks for reply. I’ve had a lot of fun learning about this and your post pushed me in that direction. I found this piece of music (I think original) written in harmonic scale by a fellow named Dante Rosati. It’s different, but certainly musical – not just bad sounding like some constructed scales. The words that came to mind as I listened were “foreign but cohesive” and “hypnotic.” I think as I listen my ear adjusts to it progressively so I find myself more accommodating as it goes along. Anyhow,

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