Nifty fractal

Why Is the Musical Scale Seven Unevenly Spaced Tones?

The white keys on a piano are the standard Western scale: do re mi fa so la ti (do), numbered 1 through 7 on the diagram below. One would expect that the seven notes would be equally spaced, but that turns out not to be the case. Each note on a piano is equally spaced, but there are twelve of them, not seven. The spacing between adjacent keys is called a half-step. You can see there are two half-steps between 1 (do) and 2 (re), but there is only a half-step between 4 (mi) and 5 (fa). Likewise between 7 (ti) and the start of 1 (do) from the next octave. So why does the scale only use seven notes and why are those seven not evenly spaced within the octave?

1 2 3 4 5 6 7 1 1x 2x 3x 4x 1 2 3 4 5 6 7 5x 6x 7x
Figure 1: The modern scale, labeled on piano keys. Frequency overtones are marked below the note. (7x is centered on the black key, 6½.)

At some point in the distant past, someone strung a piece of catgut on a wooden frame and made a stringed instrument. Plucking the string caused it to vibrate with a certain frequency. They discovered that if they pressed their finger in certain spots on the string while plucking it they would get a different note. The string can only vibrate in whole number multiples of the base frequency: 1, 2, 3, etc. If you place your finger halfway down the string it forces the string to vibrate at 2x, which makes it sound twice as high. We hear this 1:2 frequency ratio as an octave, which sets the ending frequency of our scale. Placing your finger ⅓ of the way forces the string to vibrate at 3x. We hear this 2:3 ratio (2x the original frequency compared with 3x the original frequency) as a fifth. Placing your finger ¼ of the way forces the string to vibrate at 4x, which is now two octaves above the fundamental, and we hear this 3:4 ratio as a fourth. Continuing, we hear the 5:4 ratio as a third. Higher overtones tend to end up about a whole step or half step, which effectively limits the divisions to multiples of a half-step.

1x 2x 3x 4x 1:2 2:3 3:4
Figure 2: Fundamental frequency and overtones on a string. Finger placement is marked with an arrow.

Since we have 12 half-steps in an octave, the obvious way to divide this up is with six whole steps (1, 2, 3, 4½, 5½, 6½), which is known as the hexatonic scale. Unfortunately, this scale does not contain the two fundamental intervals, the fourth and the fifth. Worse, it contains pitches close to, but not actually, a fourth or a fifth, which tends to sound out of tune. However, I doubt the original musicians of antiquity created their scale so rationally. Even if they were aware that the octave could be divided into 12 half-tones, they probably went by what sounds good, and fourths and fifths sound good.

So we want a division of the octave the includes at least the fifth, but since the fifth is seven half-steps up, this means that we cannot divide our scale evenly. If we use five tones (a pentatonic scale), then 1, 2, 3, 5, 6 divides fairly evenly, with an extra half-step in the middle (where 4 would be) and at the end. However, we could put the half-steps in a variety of places, for instance, 1, 2½, 4, 5, 6 (which puts them at the ends). The pentatonic scale found ubiquosly throughout the world, although the placement of the extra half-steps varies widely. Five-note scales are used in folk music throughout the world, particularly in older music, as the extra two notes of the modern scale were gradually added, for instance, as the seven-note scales of the Gregorian modes spread through Europe through the Church.

1 2 2 3 3 4 4 5 5
Figure 3: Most common choices of tones for the pentatonic scale

The two 3/2 steps are uncomfortably large, and the ancient Greeks split them into two more notes, giving the seven note, heptatonic scale, which forms the basis of Western music. This leaves two notes with only a half-step gap between them, and of course one can place the gap in multiple places. Since the fourth is such a fundamental interval, placing one of the shorter gaps after the 3 is logical, leaving the last to be placed after 7, which it is as far apart from the other short gap as possible. The different placements of the half-steps are known as modes. The placement described above is the Ionian mode, more well-known as the major scale described in the beginning.

However, one can generate a heptatonic scale a different way: if you keep adding a fifth you will generate the seven notes of the Ionian mode. Since a fifth is seven half-steps, then 1 + fifth = 5; 5 + fifth = 2; 2 + fifth = 6, etc. It is unclear to me if the ancient Greeks did it this way, since they obviously did not have access to modern pianos, which make it easy. It does demonstrate that the Ionian mode arises out of simple permutations of the primary physical musical interval, the fifth, so it makes sense that people would converge on it over time, regardless of how they arrived there.

Since fundamentally the ocative can be divided into twelve (almost) equal spacings, those notes have been added into musical instruments. In the fifth century BC, Aristoxenus wrote about equal-tempered (that is, evenly spaced) tones with twelve pitches. In the same century, the Marqis of Yi (which was a former Chinese state in the Warring States period) was buried with a set of bells with an octave of twelve pitches. In the 1500s, both Chinese and Europeans calculated the mathematical ratios. However, while the extra notes of the twelve-tone scale are often used to spice up modern classical music, using the full twelve-tone scale is rare, because it sounds unsettling. The unsettled feeling comes because it incorporates notes from several heptatonic modes: for instance, it has both the minor (Aeoloean mode) third (the 2½) and the major (Ionian mode) third (the 3), so the twelve-tone scale is both major and minor.

So now we have our answer: musicians have converged on the seven-tone, heptatonic scale because it most evenly spaces the notes while still including the fifth.

Additional information

“Pentatonic Scale”. Wikipedia, accessed Feb 25, 2024.
“12 Equal Temperament”. Wikipedia, accessed Feb 25, 2024.
Frequency and Music Intervals”., accessed Feb 25, 2024.