Why are the irreplaceable minor keys in music considered “artificial”?

Everyone will have heard the word “tonality” at least once in their life, or, much more often, before listening to a piece, the declaration of the tonality itself (e.g., Toccata and Fugue in D Minor). However, not everyone is familiar with the “structural” differences, even though they know from experience, for example, that a song in a minor key is generally more melancholy and reflective than one in a primary key.

From a melodic point of view, tonalities are based on the related major and minor scales, which we will briefly analyze. Furthermore, in this short article, I will explain why, while major tones are absolutely “natural,” minor tones result from a deliberate choice that is not reflected in the physical phenomena associated with sound production.

A brief mention of stairs

It is helpful to open a brief parenthesis on scales to understand the critical concept of major and minor keys. For the sake of simplicity, from now on, I will always refer to C as the “base note.” The reasoning had no variation except for the need to adopt sharps and bemols. Moreover, I will not go into the details of the so-called “equable temperament,” to which I will devote a later article. However, I will not omit details to help you understand the concepts discussed here.

Suppose a C has a frequency of f = 261.6 Hz (i.e., the middle C of the piano). The following C will have a frequency of 2f = 523.2 Hz. This interval (f, 2f) is called an “octave” and is fundamental to our discussion. The reason why the frequencies f, 2f, 4f, …, 2nf are called the same goes back to the studies of the school of Pythagoras, who noticed a very strong consonance between the notes.

From a cognitive point of view, humans perceive two C’s an octave apart as different sounds in terms of their frequency (the former is lower than the latter), but by experimenting with different notes, all participants agree that the only two “C’s” have the same sound characteristics, that is, they are the same note. At the same time, the neurophysiological effect of listening to, say, C – B or C – C#, causes subjects to state without any doubt that the two sounds are different not only in frequency but also in “something else” (the nature of which is being studied by neuroscience).

The diatonic scale

The Pythagorean school was interested in philosophical and mathematical questions and sought numerical correspondences in natural phenomena. A significant number of these thinkers was precisely 7, representing the fulfillment of the union of man (3) and nature (4). The man was considered tripartite in body, soul, and spirit, while the four elements represented nature. Moreover, Pythagoras’ study of music was directed toward understanding the so-called “harmony of the spheres,” that is, the periodic motions of the known planets, which were precisely seven.

For this reason, the philosopher decided that the interval between two sounds having frequency f and 2f should contain exactly 7 notes (ed., the octave was precisely the repetition of the first at doubled frequency). Here, I will omit the process he used, but suffice it to say that he concluded that the octave should be divided into intervals (or, more accurately, ratios) “a” and “b” (with a > b), with the following succession:

a, a, b, a, a, b

If the first sound is C with frequency f, the second will have a frequency equal to “af,” the third equal “a²f,” “a²bf,” etc. The mathematical condition a × a × b × a × a × b = 2 must be valid because the overlap of all intervals must cover exactly one octave. Just as a purely informational matter, Pythagoras arrived at (no demonstration will be given now): a = 9/8 = 1.125 and b = 256/243 = 1.0535. You can prove for yourself that the equation is satisfied.

The “a” intervals were called “tones,” while the smaller “b” intervals were called “semitones.” It should be pointed out, however, that in the Pythagorean system, the semitone does not coincide with the square root of tone (i.e., since the superposition of two semitones must constitute a tone, it follows that b²=a, but we have b²=1.10986 ≠ a), and this caused a considerable series of tuning problems that led, precisely, to the almost universal adoption of equable temperament (where the semitone is equal to half of a tone, so that, being 5a = 10b + 2b = 12b, the octave was divided strictly into twelve semitones of equal amplitude).

Always starting with C, the diatonic scale is known by all:

C, D, E, F, G, A, B, (C)

It is good to clarify two basic concepts. The first is that the name of the notes is based on free choice, and the second is that only the interval between two consecutive notes matters. Hence, E and F are not a semitone apart for obscure reasons but simply because they are the names of two notes that, in the Pythagorean diatonic scale (and also in equable temperament), have a smaller ratio than, say, G and A.

Where do sharps and bemols come from?

If we wanted, for example, to construct the major scale of D, we would have to observe the same interval ratios defined earlier. So we would have:

D – x (where x is equal to “aD”) → D – E

At this point, however, a problem arises. The interval between E and F is a semitone (b), while we need a tone (a). The solution to the problem is to introduce sharps or bemolars. The former raises an interval by a semitone, and the latter lowers it. Thus, the ratio b# = a and, likewise, a♭ = b. So, the third note of the diatonic scale of D will be called F# (or G♭), and so on.

Where are the “minor” diatonic scales?

The answer to this question is straightforward: they do not exist! Pythagorean diatonic scales are, in fact, neither major nor minor; however, they always include the interval between the first and third notes (fundamental – mediante), equal to 2 tones. In equable temperament, it can also be said that it equals four semitones. This is the crucial element that will be analyzed much later, leading to the definition of the major and minor scales (to be precise, the first step will be the definition of the so-called “ecclesiastical modes,” and later, it will come to the major and minor tonalities).

Harmony and tone

We could say that adopting a specific key means using only the notes of the corresponding diatonic scale. This is true, but at the same time, it is a statement that fails to provide a genuinely comprehensive explanation. For example, the G diatonic scale has only F#, and all other notes are unaltered; thus, it is familiar with the C scale. How could one tell the pitch if a song used only the unaltered notes?

Here again, of course, there is a tool available to composers: the key of a piece is declared by indicating the sharps or flats at the beginning of the score, that is, immediately after the clef (of violin, bass, tenor, etc.) and before the tempo declaration. Since there are no ambiguous situations, this little ploy resolves all doubts.

However, the natural way to strongly define a tone is provided by harmony. We can say with much more certainty that tonality has been established when the harmonic progressions “revolve around” the gravitational center given by the degree I chord (i.e., tonic or fundamental chord). The piece can evolve freely, but what can be seen is a periodic form of “closure” or recapitulation through what is called in the jargon “cadences,” which are harmonic passages (e.g., V – I) that create tension and immediately resolve it by moving toward the tonic.

Chords, thirds, and higher order harmonics: the ingredients for constructing tonalities

Suppose we consider the C (major) chord. Like all three-note chords (i.e., triads), it consists of a major third overlaid with a minor third. More precisely: Do – Mi – G. If you recall what was said earlier, the third interval plays a unique role; it is, in fact, the constituent “brick” of classical harmony. To give a more comprehensive overview, there are four combination possibilities:

• Major chords: major third followed by minor third (e.g., C – E- G)
• Minor chords: Minor third followed by major third (e.g., A – Do – E)
• Excess chords: Major third followed by major third (e.g., C – E- G#)
• Diminished chords: Minor third followed by minor third (e.g., B – D – F)

On a diatonic (major) scale, there will be major chords on the I, IV, and V degrees, minor on the II, III, and VI, and diminished on the VII. At this point, we can get to the heart of the matter by showing how, while major chords (and thus, major tones since the 1st degree is fundamental) are physically confirmed by the evidence, minor chords arise from a deliberate choice, namely, to use a minor third in the lower part of the triad (a condition that arises from the necessity of using only the 7 notes of the diatonic scale).

To clarify this concept, it is crucial to refer to the so-called higher-order harmonics, on which I have written a dedicated article. Here, I will not repeat what has already been explained in the article but only remind you that any instrument that produces pressure waves and is not an electronic wave generator possesses a timbre. This sound characteristic differentiates it from other media. Such timbre results from the action of notes at a higher frequency than the fundamental being emitted by the instrument. For example, in the case of a C, we have:

C(1) → C(2), G(2), C(3), E(3), G(3), B♭(3), …

With C(1), I indicated the fundamental sound, with C(2), at the upper octave, etc. As can be seen, the first harmonics consist of the octave (the quintessential perfect consonance), the major fifth (G, also a perfect consonance), C again, the major third (imperfect consonance), and so on. The sequence continues, and dissonances will also be found, but the intensity of the harmonics descends going forward, so the effect tends to fade. However, as is well noted, the first harmonics, i.e., the strongest ones, contain precisely the notes of the major triad (i.e., C – E – G).

This phenomenon occurs on any instrument (that does not just emit an electronically generated sinusoidal sound) and with any note. This is why, when we play a C major, we strengthen the action of the upper principal harmonics, creating a strongly consonant harmony. In contrast, C minor is based on a minor third (C – E♭ – G), so playing it will have the effect of dissonance between E(3) and E♭. This effect is minimal and helps to produce a more melancholy and calm atmosphere. Still, from a physical point of view, the problem remains that the minor triad (as well as the excess and diminished ones) has no natural basis but arises from human choice.

Conclusions

This article is not meant to stigmatize minor keys, which allow musical expressiveness to reach unreachable heights. However, it is essential to understand the fundamentals of harmony and the physical-mathematical elements that have enabled the extraordinary progress of Western music (up to the surpassing of tonality itself and the birth of the avant-garde).

As promised, the next musical article will be devoted to equable temperament, a fascinating and problematic concept that has quickly become a de facto standard adopted by composers and musical instrument makers.

For an in-depth study of harmony and tonality

Theory of Harmony

• A new critical foreword by Walter Frisch, H
• Harold Gumm/Harry and Albert von Tilzer Professor of Music at Columbia University, expands this centennial edition
• Frisch puts Schoenberg's masterpiece into historical and ideological context, delineating the connections between music, theory, art, science, and architecture in turn-of-the century Austro-German culture
• Leggi di più

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Last update on 2024-07-01 / Affiliate links / Images from Amazon Product Advertising API

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