Origins: Tuning Up

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White, black, white, black, white, white; black, white, black, white, black, white, white. This is the irregular series that the piano notes present across an octave from C to C. The series repeats itself from end to end of the keyboard. But why have two different colors? Or why not have them alternating regularly? Why is the whole palette of notes arranged in octaves? And why are half a dozen different notes, some high, some low, all called C, as if they were the same?

To find the answers we must go back to ancient Greece and Babylon. The starting-point is the fact that the octave makes the most convenient frame in which to organize musical space, because two notes an octave apart blend together particularly well and, in a way, almost do sound the same. A scale starting from one of them, however many notes you put in or leave out, feels complete when it reaches the other. There is a physical reason for this. Each note travels on a sound wave with a particular frequency—the higher the frequency, the higher the note—and when two notes are an octave apart their frequencies stand in the simple ratio of 2:1. This means that there is a regular coincidence of wave-peaks, as each peak of the slower wave matches every second peak of the faster one. The more constantly this agreement occurs, the better the sounds seem to blend.

The black and white notes of the piano divide up each octave into 12 equal steps, which we call semitones. Or if we just take the white notes (which look as if they are trying to hog the space), they divide it into seven unequal steps: five whole tones and two semitones, asymmetrically arranged. Now, there is no necessity to divide up the octave in this way, or at all. If you are playing a violin, or just crooning, you can slide up and down without making any fixed steps or stopping-places. Or you could choose to use a scale divided into six whole tones; Claude Debussy heard this in the gamelan music of Java, and it inspired him to use whole-tone scales in his own music. The possibilities are infinite.

However, most Western music, like most world music, builds on two favorite intervals within the octave: the fifth (an interval such as C-G, covering seven semitones or 3 1/2 tones) and the fourth (such as C-F, five semitones or 2 1/2 tones). They please the ear because, like the octave, they have simple frequency ratios, 3:2 and 4:3 respectively. Notes separated by these intervals sound good played either in succession or simultaneously; they literally give you good vibes. The fifth and the fourth added together make an octave, as mathematically 3:2 multiplied by 4:3 makes 12:6, or 2:1. For melodic purposes it is convenient and natural to divide the octave into these two unequal sections, 3 1/2 and 2 1/2 tones, and this explains the asymmetry in the layout of our white and black notes.

But what led us to use tones and semitones as our smaller units? Do they have a basis in physics? 007Well, only rather indirectly. The tone is not a consonant interval in itself, but it is derived from those two consonant intervals, the fifth and the fourth, being the difference between them. When an ancient musician tuned the strings of his lyre, he built up a scale by means of perfect fifths and fourths, which he could judge quite accurately by ear. First he would tune two strings an octave apart, say B-B. Then he would tune another string to give the note a fifth below the top B, namely E. Then another, going up a fourth from the E: A. He has now created a tone interval between this A and the upper B. Continuing with an alternation of downward fifths and upward fourths, he gets A, D, G, C and F. He now has a complete diatonic scale, BCDEFGAB. He has filled in the empty fifth that formed the upper section of his octave with three whole-tone steps built down from the top, leaving a small remainder between E and F—actually a little less than half of one of his tones. Similarly in the lower section of his octave he has filled in the empty fourth with two tones and a smaller remainder (B-C).

We know that Babylonian musicians, and probably Sumerians before them, did construct scales in this way. They had an elaborate set of technical terms for the different strings of the lyre, for intervals between particular pairs of strings, and for differently tuned scales. We have a cuneiform tablet from Ur, dating to the 18th century B.C., that explains how to work through a complete cycle of tunings by retuning one string at a time.

It was the Greeks, so far as we know, who first discovered that the major concords—the octave, fifth, and fourth—were based on mathematical relationships, on the ratios 1:2, 2:3 and 3:4. They could not measure wave frequencies, but they found out, for instance, that the concordant intervals could be produced by stopping a vibrating string at a half, two-thirds or three-quarters of its length. The discovery is attributed to Pythagoras (c. 560–500 B.C.). However, he was probably more of a guru than a scientist, and his later followers credited him with many of their own ideas. He is supposed to have heard hammers in a smithy sounding the concords and, on weighing them, to have found that the weights stood in those simple ratios. But from a scientific point of view this is nonsense. According to a more credible account, Pythagoras’s follower Hippasos of Metapontum produced the concords from metal gongs of equal diameter but proportionate thicknesses. That would have worked.

Later Greek theorists such as Archytas, Eratosthenes and Ptolemy calculated ratios for many other musical intervals. For example, the tone, defined as the difference between the fourth and fifth, was 9:8. (9:8 × 4:3 = 36:24 = 3:2.) The so-called leimma, the remainder left by two tones biting into a fourth, or by three biting into a fifth, was correctly computed to be 256:243. But one of the greatest and most influential of Greek musicologists, Aristoxenos of Tarentum (c. 370–300 B.C.), rejected this mathematical approach. He considered that musical intervals were to be measured by the ear. He held that the leimma was just half a tone, so that the fourth consisted of 2 1/2 tones and the fifth of 3 1/2.

He was wrong about that, if the tone is constructed and defined as above. Yet his analysis corresponds to what we see on our pianos: 2 1/2 tones from C to F, 3 1/2 from F to the next C. So are our pianos wrong? In a sense, yes; but they are wrong for a reason. Since the 16th century the need for musicians to modulate easily between different keys has brought about the general adoption of equal temperament in which the octave is divided into 12 equal semitones. Five of them are treated as making up a fourth and seven as making a fifth. But these tempered fourths are actually a fraction larger than the “perfect” 3:4 fourth, and the fifths a fraction smaller than the perfect 2:3 fifth. The tone (which equals two semitones) does represent the difference between the tempered fourth and the fifth, but it is a trifle smaller than the Greeks’ tone, which was the difference between the perfect fourth and fifth. With these compromises we have made Aristoxenos’s reckoning come right.

White, black, white, black, white, white; black, white, black, white, black, white, white. This is the irregular series that the piano notes present across an octave from C to C. The series repeats itself from end to end of the keyboard. But why have two different colors? Or why not have them alternating regularly? Why is the whole palette of notes arranged in octaves? And why are half a dozen different notes, some high, some low, all called C, as if they were the same? To find the answers we must go back to ancient Greece and Babylon. The […]