Is the division of the octave into 12 tones based on anything other than convention?
Question: I was explaining the basics of music theory to a friend (hopefully still my friend!) the other day and was stumped on one question: Is the division of the octave into 12 tones based on anything other than convention? or to put it another way, did "someone" just decide that the ear couldn't usefully hear more than 12 subdivisions of the octave? or to put it yet another way, was this some kind of arbitrary cut-off of the overtone series? Note that I'm NOT talking about tempering the scale, but about the idea of "12 tones" that I assume existed before tempering -- otherwise, what was there to temper?****Thanks -- and by the way, you're about the best theory guy on the web. Your answers are always "sound"! -J.M.

Answer: Good work with that line about being "sound."

I hope you have a few moments; this subject invites a lot of talk.

The division into 12 tones is not entirely convention, though all art has conventional aspects. Or perhaps I should say the 12-tone division is not arbitrary; there's a reason for it.

Acoustically the octave does not divide perfectly into any particular number. There have been keyboards with 17 tones to the octave, 19 tones, 31 tones, 36 tones, etc. But in a practical sense it does divide naturally in 12. You need to look back at the basis of the musical scales to see why.

The basic step is of course the whole step, and you are correct to assume that this has to do with overtones. That sounds theoretical, but it's a principle that will arise naturally for anyone tuning an instrument: it's natural to tune strings (or pipes; the principle is the same) using some interval that sounds solid and easy to pin down, like the fifth or fourth. If you tune three strings a fifth apart the outer two will be separated by a whole step plus an octave: corresponding to the 8th and 9th partials (this partial business is explained more fully in the appendix of Exploring Theory with Practica Musica; a string or air column vibrates in many ways at once, each of which produces a "partial" or "overtone;" you can hear when a fifth is in tune because certain partials line up.) Or tune by fourths and you'll still get that kind of step between the outer two strings, this time with the whole step below the octave.

So the whole step is entirely organic, and it's easy to see why it is found all over the world in traditional music from different cultures. But if you have only whole steps you'll miss hitting some of the major tourist stops in the octave: the fifth and the fourth as measured from your starting note. Those two intervals are acoustically very strong (see the partials discussion) and they are instinctive goal points. To reach them you have to add half steps: the difference between two whole steps and a fourth, or between three whole steps and a fifth. For example, to reach the fourth you'll add one half step to two wholes. Another whole step puts you on the fifth, so putting in that half step gave us a way to get to both of the guideposts. Then to get to the octave, a very natural place to begin anew, you'll add another half step to the remaining two whole steps, which is another fourth and completes the octave. That desire to include the 5th and complete the octave gives us a sound reason to use 7 claves (i.e. the keys of a piano or organ); that's how many claves it takes to fill in an octave with whole steps plus whatever half steps are needed to get to the fifth and/or fourth. So we also find, all over the world, scales in which there is a mixture of large and small steps, and these tend to be tuned so that three large steps plus a smaller one makes a fifth, etc. There are variations, which mostly have to do with whether your music is primarily melodic or also considers harmony.

We can carve ourselves a keyboard with 7 claves and tune them so that they play notes a whole step or half step apart, then start the pattern again at the eighth key (the octave). And because our five whole steps and two half steps add up so neatly to one octave - which anyone can hear as the same starting pitch but higher - we can keep going and repeat the pattern. That's a scale. Could be major, natural minor, Dorian, etc., or if we leave out a couple of notes it could be one of the world's several pentatonic scales.

Now, however, I might want to sing again the tune I just learned, but starting a note higher. That requires that I add keys to my keyboard. Maybe my tune starting from the higher note has a half step in a place where that simple keyboard has only a whole step available. So I must carve myself a new clave, tune its string or pipe so it's in between two existing notes, a half step up from the lower one or a half step down from the upper one. Maybe I'll paint it a different color. Eventually I'm going to have to do that in all the places where my original keyboard had only whole steps, which means adding 5 new keys to the 7 old ones. Total = 12. In fact I'd get the same result if my original keyboard was pentatonic; the desire to start a tune on any note would eventually require me to fill in the gaps.

Then there's the temperament part: once you've discovered 12 is a practical number of keys then you'll need to compromise on the tuning of the intervals they form, because recalcitrant nature just won't come out even in 12 steps if you're trying to make these intervals agree with the partial series for smooth harmony. But that seems to be something you already know about.

The above is an attempt to show inevitability in the 12-clave design. Historically we know how the design actually evolved, and it is by a similar principle but with more complex details: originally there were no major or minor scales but a system of hexachords: overlapping sets of 6 notes. These were built to encompass the fourth and fifth and correspond to "Do, Re, Mi, Fa, Sol, La." Each hexachord has only one half step: the one required to reach the fourth at Fa. The identical pattern can be sung starting on C or the fifth above C, G, and the result is a range of notes C, D, E, F, G, A, B. But singing the hexachord beginning on the fifth below C, that is, F, required adding a special lowered form of B, the first black key. Then the desire to play the hexachord on any starting pitch leads to filling in the other whole steps and again we reach twelve.

I am always interested to find universality in musical materials. As I've said in answer to another question there are simple mathematical principles involved here that make it likely that somewhere in another galaxy there are other civilizations that also have in their history 12-note scales and triadic harmony. And they will also have Indian-style scales and other interesting variations that correspond to the amount of musical variation here on earth, but I think we would recognize and understand what they were playing. They would have some sort of Mozart of their own, and an alien version of Dave Brubeck, but the themes would be different. This is my main motive for interstellar travel.

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