Why Are There Twelve Notes Per Octave? (Part 1)

Target audience: Advanced high-school algebra.

If you're at all familiar with Western music, you likely know that the Western scale is comprised of twelve notes: C, C# or Db, D, D# or Eb, E, F, F# or Gb, G, G# or Ab, A, A# or Bb, and B. (Why do we start with C instead of A? No particular reason.) The natural question is, why? Why shouldn't the chromatic scale have 13 notes, or twenty-seven?

The simple answer is that it sounds better that way. To fully understand why, you need to understand a thing or two about acoustics, which I'll explain in a subsequent post. For now, it will suffice to know that musical notes are associated with numbers called "frequencies." The higher-pitched the note, the higher the frequency. Skip the reasons for now and take my word that when we're building a scale, we're trying to construct a list of notes which obey the following rules:

1. The frequencies corresponding to each note progress in a geometric series; that is to say, we get obtain frequency corresponding to a certain note by multiplying the frequency of the previous note by a certain number. In musical terms, this means that no key signature sounds any different than any other, which allows us to modulate between keys.
   
2. The ratio of the frequencies between the first note of the scale and one note past the end of the scale is 2. From a musical standpoint, we are requiring that the top and bottom of the scale be exactly an octave apart.
3. Given the first two constraints, we are trying to set up a scale with notes defined so that we can achieve very close approximations of the frequency ratios 3/2, 4/3, 5/4, and 6/5. In musical terms, we want there to be a perfect fifth, perfect fourth, and major and minor third in the scale, or at least something very close to them.
4. We want to keep the number of notes per scale reasonable. Clearly with a near-infinite number of notes per scale it would be easy to satisfy all the conditions; this would be awfully hard on a pianist's hands, however, not to mention what sheet music might look like.
   

These rules will be derived from first principals later; for now, let's focus on how to construct a scale with these rules rather than why one would want to.

Historically, most scales were designed to satisfy condition (3) at the expense of (1) and (2), which meant that while the intervals may have sounded better than their modern equivalents, the sort of treatment of key signatures used by Bach and later composers would be impossible. Incidentally, Marin Mersenne, of Mersenne prime fame, apparently decided that violating (4) was a better solution, and produced an organ with about a thousand keys on it. As you can imagine, it was not a commercial success.

Now for the math: if we want to construct a scale with n notes, equally spaced, leading to a perfect octave, and our base note has a frequency f, then the freqencies of the notes will be f, f a, f a², f a³, ..., f aⁿ = 2f, where a is the ratio of the frequencies. Dividing both sides of that last equation by f, we get aⁿ = 2, or in other words a is equal to the n-th root of 2.

I prepared an Excel spreadsheet which calculates the relative frequencies of the notes in a scale of n equidistant tones and finds the closest approximation to each of 3/2, 4/3, 5/4, and 6/5 in each scale. It's not fancy, but it gets the job done. If anyone out there is a more talented Excel-programmer, I'd be interested in seeing if there's a better way of finding the element in a range nearest to another element without resorting to VBA like I did.

Anyway, If you download the spreadsheet and open it up (you will have to enable macros, if asked), you can see the relative frequencies for an equal-tempered scale with n notes, where n is the column number. Scrolling down to rows 101 - 104, you will see four percentages associated with each scale; these are the relative error on the perfect fifth, perfect fourth, and major and minor thirds, respectively. Naturally they are very high for the one-note scale, and the two-note scale, while an improvement, isn't terrific. There are plenty of scales which manage to get one of the intervals within one percent of perfect pitch, but compensate by distorting other pitches. Now scroll over to column L, representing the 12-note scale. It's considerably better than anything to the left or right; the fifth and fourth are better off than anything in the immediate neighborhood, and the thirds are pretty good as well. If you really like your minor thirds there's always the 19-tone scale (column S), and devotees of the major third might want to check out column V's 22-note scale, but for all-around performance there's simply nothing better than good old 12. That is, at least until we get to the 34-note scale in column AH.

If you would like to play around with your own scales, I recommend ZynAddSubFX, which is available for Windows, Mac OS X, and Linux, and has a feature which allows you to directly enter scale data. Here's a screenshot of me using the 19-tone scale from the spreadsheet:



If you've enjoyed this, I recommend checking out Temperance by Stuart Isacoff. It's a much broader and deeper discussion about the history and nature of tuning systems, albeit without quite as much math as I've gone into here.