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.

BEGINNINGS AND INTERVALS

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.

Two-string monochord with movable bridges (source Monochord01 )

The philosophers in ancient Greece, especially the Pythagoreans, wanted a harmonious world order, which was preferably described with simple numerical ratios such as 3:2, 4:3, etc. This also went well with intervals. It is therefore not surprising that music theory in ancient Greece was initially a rather mathematical matter.

There was no question that the interval (3:2)' – what we call a fifth today – was the purest interval after the octave: on the one hand, 3:2 was the simplest imaginable number ratio after 2:1, on the other hand, the purity of the interval was also confirmed by the listening impression. Compared to the fifth, the fourth had the slightly more complicated ratio of 4:3. Also, the fourth could be seen as a mirror image of the fifth, because a fifth down is the same as a fourth up - ignoring octave differences. The fifth and fourth are said to be complementary intervals. For this reason, the fourth was also considered pure.

Let us abbreviate the octave ratio 2:1 with α, the octave α' with the big alpha Α, the fifth ratio 3:2 with β, and the fifth β' with the big beta Β. Then 2/β denotes the fourth ratio 4:3, and Α−Β the fourth, because an octave up and then a fifth down results in the fourth.

The octave, fifth, and fourth were generally accepted as elementary and pure intervals. After that, spirits parted.

Antiquity - layering of fifths

Music in ancient Greece essentially meant playing or singing a melody without any kind of polyphony, if singing an octave apart is not considered polyphony. But how did one get from the building blocks of fifths and fourths to a scale that was fine enough to form interesting melodies?

The first approach was a formal construct known today as the layering of fifths. It was probably developed long before the Greeks. You put fifths on fifths one after the other, and you get a rapidly increasing sequence of intervals or pitches:

B , 2B , 3B , …

But this strong increase is not desired. If the newly constructed interval exceeds an octave, one goes down another octave,i.e. one replaces 2Β by 2Β−Α . This ensures that the intervals never become larger than an octave. This procedure is called octave reduction . It is justified by the fact that people perceive the octave almost as unison, but at least not as really different.

Octave reduction of 2Β

If you apply octave reduction when layering fifths, to get back to the first octave you have to go back more octaves the further you go:

Layering of fifths with octave reduction

Note that the sequence is not sorted by increasing size, but jumps up and down. The layering of fifths is itself an ascending sequence, but of course this no longer applies after octave reduction.

The formulas can be simplified somewhat with the abbreviations Τ = 2Β−Α and τ = ββ/2 = 9:8 (for the associated frequency ratio), e.g. by transformations like 3Β−Α = Β+2Β−Α = Β+Τ. The octave-reduced sequence of layered fifths is thus written somewhat more simply as

Is that a good construction of a scale? Although the fifth is very harmonic, the layering of fifths with octave reduction provides anything but particularly simple mathematical proportions, for example ττ = 81:64. The local beauty of the layering of fifths actually belies the unsightly intervals in the resulting scale. This is in striking contradiction to the principles of the Pythagoreans, who were very interested in harmonious proportions. This can be seen as a further indication that the layering of fifths had probably already been developed in older cultures.

The layering of fifths has another fascinating property: The frequency ratio of 6Τ, the 13th tone in the sequence, is about 2.03. So 6Τ roughly corresponds to the octave Α, which is a happy coincidence and by no means a law of nature. This results in 12 pitches or intervals that are smaller than the octave, and the 13th pitch is almost the octave itself. In principle, the sequence could be continued. But you are so close to the octave and with the twelve tones you already have such a fine division of the octave that you like to break off the sequence here. That should create enough melodies.

Antiquity - whole tones and diatonic

The twelve tones were probably even a bit much for practice. So one concentrated on the beginning of the layering of fifths and stopped adding further fifths as soon as the tones were close enough together after octave reduction. From the reduced layering of fifths 0 , Β , Τ , Β+Τ , 2Τ , Β+2Τ , 3Τ , a 7-step scale is obtained by sorting, which is represented with the additional octave as follows:

The pitch steps are: 5 times the whole step Τ, in the middle one Β−3Τ, and at the top Α−(Β+2Τ ). The latter two steps are equal, as can be seen by replacing Τ by 2Β−Α :

Its size is about half of Τ , where the computational proof will be spared here.

Layering of fifths with octave reduction of length 7

In any case, the steps in the above scale are never greater than  Τ . The one shorter layer of fifths, on the other hand, lacks the 3Τ coming from 6Β, which lies between 2Τ and Β. Thus the scale of the one-shortest layered fifth has a pitch of about 1.5Τ, i.e. significantly more than Τ.

But they were satisfied with Τ as the maximum distance. The associated scale appeared to be a good compromise between the desire for as few steps as possible (7 steps if you don't count the octave) and the desire for the smallest possible distance between the tones (distance Τ or less). Also, Τ was special because the octave could be divided into parts of size Τ. The interval Τ, introduced as 2Β − Α, thus became a basic unit that indicated a sufficiently dense range of pitches. At that time it was called "Tonos", later Latin "Tonus", today whole step or whole tone distance.

Since the two small steps are roughly half the size of the whole step, we call them a half step — to be precise, a Pythagorean half step . We abbreviate the Pythagorean half step with the large Eta Η and its ratio 2⁸:3⁵ with the small Eta η (from Greek “ημι” = “(h)emi” = half). It is therefore Τ ≈ 2Η , or τ ≈ ηη.

The steps of the above 8-step sequence with maximum distance Τ can be written as

T , T , T , H , T , T , H .

The scale consists of 5 whole steps and 2 half steps. Extending beyond the octave results in an endless ladder with the following pattern of pitch steps:

Between two half steps Η there are alternately 2 or 3 whole steps. This pattern without a distinguished beginning we call the Pythagorean diatonic step scheme . The diatonic step scheme is exactly what is represented by the white keys on a contemporary keyboard. Although it is linked to the name "Pythagoras", its origins are probably much older. Each 7-step section of this pattern is now called a Pythagorean diatonic scale. By choosing a starting point from the 7 possibilities, you get a specific diatonic scale from the step scheme. Until the Middle Ages, no particular starting point or resting point was preferred, which was reflected in the large number of "octave genres".

A diatonic scale divides the octave into 7 small steps, T or Η . Conversely, 7 steps in the diatonic step scheme together always result in the octave. One can ask whether fewer steps together always result in a fixed interval. One quickly sees that 4 steps together usually result in a fifth, because Β = 3T + Η. Only at exactly one point does it happen that the 4 steps contain 2 half tone steps, namely in the section Η, T, T, Η. The same applies to 3 steps; these together almost always make the fourth. With 2 steps, on the other hand, you always get either T+T or T+Η. But this means: While 4 steps, and 3 steps respectively, form harmonic intervals (with one exception), 2 steps together always form formally inharmonic intervals.

Antiquity – diatonic and more?

The old approach to obtain a scale, the layering of fifths with octave reduction, was a simple but unmusical approach. Fifths and fourths dominated, all other intervals within the scale were unimportant.

The Greeks' new approach to deriving a scale was to break down the fourth into three small steps and combine the resulting 'tetrachords'. With two tetrachords and a whole step, you then also get a 7-step scale of the octave.

Two tetrachords and a tonos

The small steps of a tetrachord should be chosen musically, and in fact many tetrachords were suggested and tried out. With the "diatonic tetrachord" (Η, Τ, Τ) the diatonic scales could also be recovered. Other tetrachord families were less balanced and sometimes had very small steps combined with a very large one.

The three tetrachord genders

Claudius Ptolemy considered in his work "Harmonics" from about 150 AD different moods of the genera, also a mood of the diatonic tetrachord, in which the two big steps together formed a pure major third ("diatonon syntonon"). Although this helped establish the major scale centuries later, it was not further emphasized by Ptolemy himself nor had it any influence at the time. Ultimately, the rambling tetrachord theory of the Greeks brought no real advancement in music.

The Pythagorean diatonic step scheme was popular with theorists because it was constructed in an elegant mathematical way and contained many fifths and fourths. There was no reason for the practitioners to reject these scales either. The exact frequency ratios of a scale were rather insignificant due to the lack of polyphony, after all it was only possible to judge whether an interval sounded clean or unclean if tones sounded at the same time.

Early Middle Ages - polyphony and thirds

With the spread of Christianity, a division of music into secular and sacred music became apparent in the West. While secular music was intended to entertain, church music was used for contemplation and worship. Rhythm, enjoyment, fun and spontaneity could only be part of secular music, banned from the strict and planned music of the monasteries.

Through frequent singing together, a certain polyphony developed by itself. Without the burden of any theoretical considerations, two-part music should simply sound good in secular music, for which at least two factors had to be fulfilled from today's perspective: First, the interval had to have a simple frequency ratio. But just as important was that the tones of the interval matched the scale used, and this was almost always diatonic. After all, diatonic scales formed, along with the simpler pentatonic scales, the de facto standard in Western music since antiquity. It turned out that two-step distances, what we now call thirds, sounded pleasant and interesting. Not as colorless and hollow as the fifth and fourth, but not as dissonant as the whole tone either. This was true for the major third T+T , but also for the minor third T+Η .

Mathematically, the euphony can be easily justified, because the Pythagorean major third 2Τ of the diatonic step scheme has almost the same frequency ratio as the pure major third (5:4)':

τt = (9:8) · (9:8) = 81:64 ≈ 80:64 = 5:4

It is unlikely that practical musicians were aware of this connection. The euphony of this interval was enough for them. The obvious two-part system developed, in which a main part was accompanied by an almost parallel second part, with an emphasis on the use of thirds and their complementary intervals. This early two-part system is linked to the term "Gymel" ^BRKlassik01 , and some authors see England or Scandinavia as their nucleus ^{Miller01, Stackexchange01, Geraldus01, Riemann01} . This means that England was probably already a pioneer of pop music in late antiquity.

Middle Ages - polyphony in monastic singing

In church music things were different. On the one hand, the music must not entertain, on the other hand, the music had to be mathematically pure. The development culminated in the monophonic "Gregorian chant " with different starting and resting points ("modes") in the Pythagorean diatonic step scheme. The church musicians did not completely shut themselves off from two-part writing, but it was quickly limited to fourths and fifths. As already indicated above, there were two reasons for this: For the strictly religious monks, who feared any form of distraction and sensuality, the major third simply sounded too interesting and colourful. On the other hand, the mathematically educated monks rejected the Pythagorean third 2Τwith their ratio of 81:64 as formally inharmonious. The major third was initially only accepted for transitional steps, for example to avoid the much worse “tritone” 3Τ , the “diabolus in musica” (ratio 729:512) in the parallel movement. This simple polyphony, which was initially essentially based on the parallel passage with fourths or fifths, became known as the first form of the "organum".

The practical use of such simple polyphony in the monastic environment is already documented for the year 900. In 2013, the earliest notation of a two-part song to date was discovered, made in a monastery near Düsseldorf, Germany. ^UniCam01

Over the centuries, however, the monks probably got used to the thirds. Eventually, based on their mathematical knowledge, they realized that 2Τ could be interpreted as a pure major third (5:4)'. This identification had become generally accepted around 1300, as noted by the English Benedictine monk Walter Odington around this time. With the justification that it was a matter of the pure major third (5:4)' and not of (81:64)', the monks could now use the major third in their two-part system without a guilty conscience. The identification of 2Τ with the pure major third also had an effect on the other third: almost inevitably Τ + Η was treated as a pure minor third (6:5)'.

At some point both secular and religious musicians used thirds in their polyphony, but the pedantic theorists struggled with it for a long time because of the apparent incompatibility of theory (Pythagorean major third 2Τ) and practice (pure major third). But exactly such a union was demanded more and more, because theory should no longer lag behind practice. In any case, the ideal of layering fifths as a principle for constructing the diatonic step scheme had cracked.

Middle Ages - triads

It was not far from two-part music with thirds to three-part music. The obvious triads were the two decompositions of the fifth into pure major third 5:4 and pure minor third 6:5, which we know today as major triad and minor triad. Their difference is that in major the major third sits on the root, and in minor the minor:

The triple frequency ratios of the pure triads in basic form are thus:

The major chord sounds more familiar, normal, and nicer. There are many theories as to why this is the case and whether this is just a habit. A strong argument for the importance of the major chord is that it resonates in the overtone series of every natural tone at a lower octave in basic form (namely the double octave). This means that we always inevitably hear major chords, even if they are not played intentionally.

Overtone series of a single trumpet tone on an interval scale (logarithmic frequency scale)

The major triad, with its frequency ratio of 4:5:6, is also mathematically simpler than the minor triad, with its ratio of 10:12:15. In fact, the major triad is actually the simplest conceivable triad in terms of number ratios: 1:2:3 and 2:3:4 are essentially combinations of fifth and octave, and 3:4:5 is practically the major triad , only not in the basic form but as an inversion (lowest tone of 3:4:5 one octave up gives 4:5:6, i.e. 3:4:5 ≡ 4:5:6).

Major or minor triads appeared at almost every step of the diatonic step scheme, when once again identifying 2Τ with the major third, and Τ+Η = Β−2Τ with the minor. Strictly speaking, the triads were always slightly out of tune.

Middle Ages - tonality

With the triads, the new concept of harmony found its way into music. A partnership between the melody voice and a rhythmically freer accompaniment with chords began. Even more: the accompaniment developed a life of its own. The harmonies not only referred to the melody part, but also related to each other.

Already with melodies and two-tone sequences there were arcs of suspense and resting points. Certain tones tended toward still points. With the two-tone sequence [cg], [dg], for example, a tension arises that wants to be resolved into [cg]. Or also with [cg], [hf], whose tension wants to be resolved to [ce]. With triads, these tension tendencies are even more pronounced. Thus the triad sequence [ceg], [hdf] demands the resolution back to [ceg].

The functions of a basic chord (tonic) and a partner chord on the fifth striving strongly towards the tonic (dominant or dominant seventh chord) developed - initially in major, as well as a more neutral second partner chord on the fourth (subdominant). These functions and correlations of chords, today called tonality or tonal harmony, were also never discovered theoretically, but only through practical music-making with accompanying triads.

It was noticed that the associated diatonic scale contained the major triads on root, fourth and fifth (i.e. tonic, subdominant and dominant) only at a single starting point. In the pre-tonal time, all starting points were still equal. But tonality and the importance of the major triad now led to a preferred starting point ('root') in the cyclic pattern of the diatonic step scheme.

It is the diatonic scale

0 , T , 2T , B−T , B , B+T , B+2T , A

1 , τ , ττ , β/τ , β , βτ , βττ , 2

with the tone steps

T , T , H , T , T , T , H .

Due to its special tonal properties, this diatonic scale became more and more popular. Today we call it the major scale.

Note, however, that as a Pythagorean scale, it also contained no pure thirds. After all, it was only distinguished from the other diatonic scales by the choice of a special root.

It should also be clear that the emergence of the major scale has so far essentially been a stroke of luck and by no means derived theoretically. After all, it was pure coincidence that in the layering of fifths the double whole tone step 2Τ almost resulted in a pure third. And yet this was the occasion for the development of tonal harmony and thus for the major scale.

Renaissance – Pure major scale

The problem that still had to be solved was to accommodate thirds officially and well-founded in a scale. After some preliminary work by others, Gioseffo Zarlino presented a major scale in 1558 that was no longer based on the layering of 6 fifths (i.e. was not Pythagorean), but on the layering of 3 pure major triads: the pure major triads on the root, on the fifth and on the fourth (i.e. on the tonic, dominant and subdominant) are united. Obviously, this construction was strongly influenced by the newly discovered tonality.

The pitch (i.e. frequency ratio of a tone to the scale fundamental) is the product of the frequency ratio of the triad fundamental to the scale fundamental and the frequency ratio of the tone to the triad fundamental:

By combining all the tones, eliminating the double ones and sorting them by size, you get a scale that is very similar to the Pythagorean major scale in terms of the size of the frequency ratios, i.e. it is also a major scale, only tuned a little differently:

It also contains 5 whole steps and 2 half steps, but there are two sizes of whole steps (τ and τ⁻), and the half steps are slightly larger (η⁺ ):

In summary, the new major scale has the following pitches and steps:

As might be expected, this construction yields a major scale tuning that contains many perfect thirds, in addition to many perfect fifths and fourths:

That is why today the tuning is called the just intonation or pure tuning of the major scale. Another pure intonation was also found, but that is better suited for minor triads.

Apart from the fact that the combination of the major triads on tonic, dominant and subdominant provides many pure intervals, it also forms an independent justification of the major scale without any reference to the layering of fifths and their lucky hit:

This construction can be viewed as the sought-after elegant definition of the major scale, assuming only the importance of the major triad and the tonic, dominant, and subdominant degrees. One has to ask oneself, however, whether and when the major scale would have come into being at all without the bumpy detour via the layering of fifths.

Or even earlier?

There is evidence that early English polyphony, in addition to the use of thirds, also intuitively anticipated tonality and preference for the major scale, centuries before the very development took place in pan-European music. ^Sanders01

A historical peculiarity should also be mentioned: The first mainland European to anticipate the special position of the major scale within the diatonic scales - i.e. the special choice of the fundamental tone - and perhaps also to recognize its special feature, was the Benedictine monk and music theorist Guido von Arezzo around the year 1020. He used the first six tones of the Pythagorean major scale, i.e. the hexachord 0 , Τ , 2Τ , Β−Τ , Β , Β+Τ , as the basis of his later widespread music theory. To what extent he already equated 2Τ with the pure major third remains unclear, as does the motivation for choosing precisely this part from the diatonic step scheme. His choice was probably just a coincidence. However, this choice certainly shaped our listening habits.

outlook

Anyone who now thinks that the pure major scale was the solution to all scale problems is greatly mistaken. There was now a satisfactory derivation of the major scale, but the lack of perfect tuning became obvious. It became even clearer that a scale or tuning that delivered pure intervals on all degrees and all intervals was what actually was wanted. However, it also became obvious that this is not possible. For example, just intonation on the second degree delivered an unclean fifth with the ratio 40/27 ≈ 1.481. Many tunings were tried out in search of the optimal compromise. Only gradually did the musical world begin to come to terms with the dilemma it had discovered and to work with several coexisting moods, knowing full well that there was no optimal solution.

summary

In retrospect, one can say that three components contributed significantly to the development of the major scale:

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