© Volker Schubert , 2024-07-05

"do, re, mi, fa, sol, la, ti, do“ - Every child can sing it. It seems natural. It is omnipresent. Western music is unimaginable without it. The major scale has been so dominant since the Baroque period that it seems to have a secret quality that makes it unique. In fact, however, it took a long time to evolve. On closer inspection, one is even amazed at the winding paths and detours that the development took. We want to follow the long path to the major scale and try to understand why the path was so difficult.

First, a few terms need to be clarified. The pitch of a note is determined by its *fundamental frequency*.
Two notes can be related to one another by forming the ratio of the two fundamental frequencies. Instead of
frequency ratios, musicians prefer to talk about *intervals*. Simple frequency ratios such as 2:1 and
3:2 are given names, such as "octave" and "fifth" here. Obviously, the actual pitches are not important for
identifying an interval, but only the mathematical frequency ratio. For example, the tone pair 400 Hz and 600 Hz
form the same interval as the tone pair 200 Hz and 300 Hz. The frequency ratio for both pairs is 2:3, so both
are a fifth – or actually, *the* fifth. For an introduction to the physical nature of intervals, see, for
example, the video series Schubert01.

We will not make a strict distinction between a frequency ratio and its fraction; in formulas, fractions are
often more practical. Traditionally, in music, an additive language for stacking of intervals is used, such as
"*a fifth plus a fourth equals an octave*", but actually, the
corresponding frequency ratios are multiplied, as in this case 3:2 · 4:3 = 2:1. To avoid any misunderstandings,
we will distinguish between intervals and calculations with frequency ratios, even if an interval is completely
defined by its frequency ratio. For a frequency ratio p, let p' denote the corresponding interval.

An ascending sequence of intervals is usually called a "scale"; in our usage, a scale is always independent from absolute frequencies.

(Imagination of Bing)

Even in the Stone Age, people obviously enjoyed producing sounds. Flutes have been found that were made more than 35,000 years ago. Over time, people discovered different ways of producing sounds, such as plucking strings or striking bodies, and in the process naturally also figured out how to produce different pitches. In addition to the material of the instrument, different sizes of the sound generators were also obviously important, such as the position of the holes in a flute, the length and thickness of strings, and the weight of a body. An early researcher then discovered at some point that the sound of one string matched the sound of a string half as long, for example by pressing down on the middle of the string.

By 3000 BC, ancient Mesopotamia already had astonishing musical knowledge, for example about the fifth and the tuning of stringed instruments. However, there are too few sources to gain a complete picture. By 500 BC at the latest, other intervals were being investigated for their euphony in Greek culture, for example by the school of Pythagoras . There was no direct access to frequency ratios at that time, and so measurable properties that were obviously directly related to pitch, such as the length of strings, were compared. The monochord was an important one- or two-stringed tool for precisely investigating the connection between pitch ratios and sound perceptions. In order not to complicate the discussion unnecessarily, we will pretend in the following that the Greeks had spoken directly about frequency ratios, and not about pitch ratios of strings.