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Let’s take those 12 tones, in any
one register,
and see what the possibilities are of their melodic combinations. Through
an awe-striking mathematical formula, it turns out that the maximum number
of possible melodic combinations of these 12 notes is the following astronomical
figure: one billion, three hundred and two million, sixty-one thousand,
three hundred and forty-four, without ever repeating any one note in any
one pattern. Incredible! But enormous as that seems, it’s still a finite
number; and you’d think that eventually even that number
of combinations
would be exhausted. However, musical science supplies even further possibilities,
so all is not lost. Remember that our number of possibilities so far has
been limited to music that consists of only one note at a time moving horizontally,
as melody. Just think how it all expands wildly when the vertical idea
of harmony, or chords, enters the picture. Obviously the number of possible
combinations of the 12 tones as chords is also one billion, three
hundred and two million, and so on. Now we have all the possible melodic
combinations plus that same number of chords to go with them; and
each one of those chords can conceivably go with each one of the possible
melodic combinations! We begin to smell infinity.
My mathematician friends tell me that the maximum possible number of vertical and horizontal combinations of 12 notes or less comes to a figure which is expressed in 106 digits. The actual round figure, for those of you who are curious about such things, is 127 followed by 103 zeros. You may have heard of the word googol, which was invented as much for convenience as for humor, to express in shorthand any number consisting of a digit followed by 100 zeros, much the same way as astronomers use the word light-year to avoid having to spend their time mumbling endless numerical expressions. Well, melodically and harmonically speaking, we already have at our finger tips 127 googols of possible combinations. And on top of that, consider the element of counterpoint—different melodic lines going on at once—which means that it is also possible to combine all those billion odd melodic combinations with each other! At this point the brain begins to reel. It’s like counting stars in the Milky Way. And we haven’t even taken into account the all-important element of rhythmic variety, which provides a whole new Milky Way of possibilities—to say nothing of the whole range of tempo, or speed; to say nothing of innumerable instrumental colors, vocal colors, and the like—all of which add galaxy upon galaxy to this musical universe, whose limits are by this time beyond calculation. It turns out, after all, that the realm of music is an infinity into which the composer’s mind goes wandering, looking for his material, his way of selecting and shaping it. And that’s the most important component of all—the individual composer, the mind and heart with something to say, something to communicate through this vast mathematical language. I’d like to try to show you how some composers have availed themselves of this infinite musical variety, because I think that through such an exploration we can also get a glimpse of the infinite variety of the creative human spirit.
Leonard Bernstein
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