© Volker Schubert 2023-03-27,
translated from the German article by Google
The major scale appears to us to be the most natural musical system in the world. Any child can already sing them using short names like "c d e f g a b c" or "do re mi fa sol la si do". The major scale has been so dominant in Western music since the Baroque that it seems to have a secret quality making it unique. If you take a look back at the history of music, however, you realize that it only developed slowly and along winding paths to its final form. In the following we want to consider the decisive contributions to the development of the major scale and evaluate their respective impacts.
The basic component of every scale are intervals, i.e. the proportions between two pitches. The identification of an interval does not depend on the real pitches, i.e. the frequencies of the two tones, but only on the mathematical ratio of the frequencies. The pair of pitches 400 Hz and 600 Hz form the same interval as the pair of pitches 200 Hz and 300 Hz. The frequency ratio is 2:3 for both pairs, so both are a fifth – or actually, represent the fifth. For an introduction to the physical nature of intervals see, for example, the video series Schubert01.
We will switch back and forth freely between frequency ratios (proportions) and their fractions, for example we will treat a proportion x : y as a number x/y in formulas. If one considers the frequency ratios of three tones x, y, z, then of course x : z = (x : y) · (y : z) holds, i.e. the juxtaposition of two pairs of tones corresponds to the multiplication of the frequency ratios. Traditionally, however, concepts and expressions of addition are used in music for the joining of intervals, like statements such as "a fifth plus a fourth results in an octave". Therefore, for the avoidance of doubt, we will distinguish between an interval and its frequency ratio, although an interval is entirely defined by its frequency ratio.
Let us denote the interval belonging to a frequency ratio x by x'. The fact that we multiply frequency ratios but add the associated intervals can be expressed as a formula like this: The following applies to all frequency ratios x and y
(x · y)' = x' + y'
In fact, the transition from x to x' is the logarithmic function, which we will never use. The octave is often not perceived as an interval at all, but as a unison. Therefore, if two tones or intervals X, Y differ by one octave, we also write X ≡ Y.
As early as the Stone Age, people made instruments that could produce different pitches. Measures for lengths and weights emerged with the first advanced civilizations. Although pitches could not be measured directly, certain quantities of the generating instrument could be measured, such as the position of the holes in a flute or the length of strings. It was certainly recognized early on that the tone of half the string length goes well with the tone of the whole string.
In ancient Mesopotamia amazing musical knowledge was available already around 3000 BC. However, what the used scales looked like, is again the subject of controversial discussion, after it had been assumed for a long time that these were our diatonic scales, of which much will be said later. Crickmore01, Dumbrill01, Rahn01
No later than 500 BC, the Greeks examined more complex ratios of string lengths or weights for their euphony, for example by the school of Pythagoras.