Will science help improve music theory?

In my last article, I considered the question of how to build a musical theory on the basis of empirical observation of people — based on their tastes and musical preferences. Also, to build a musical theory, you can try to understand what is happening in our heads. This question was considered by Daniel Wilkerson in his article “Clarifying Harmony: Along the Path of Progress toward the Scientific Theory of Music.” This essay has a second charming old-fashioned title:

Major scale, standard terminology and differences in the perception of minor and major triads, which are explained by the basic principles of physics and mathematics. The failure of the Helmholtz theory, as well as the justification of the theory of Terhardt and others.
')
Wilkerson begins with a remark that books on music theory are read as medical texts of the Middle Ages and “they are full of unfounded judgments with funny characters, embellished with phrases in Latin.” We can write them more clearly.



Wilkerson suggests that the theory of harmony needs to be built on an understanding of how the brain works, and in particular how it processes audio signals. In the course of our evolution, we have acquired the ability to recognize sounds with natural harmonics, because they usually come from sources we know, the example is the throats of other animals. Musical harmony is a way we have invented to please our harmonic series resolvers.

How well can we perceive the harmonics of sound? So good that if we hear excerpts of a number of overtones, we can easily and absolutely unconsciously supplement it with missing overtones. For example, if we hear a harmonic series in which the basic tone is missing, then we automatically supplement them with a sequence. In particular, when we hear the consonance of several tones, then we can calculate the greatest common divider of their frequencies and we will consider it as a common tone. This “imaginary tone” phenomenon makes it possible to distinguish the bass line while listening to music in tiny headphones. Even though the size of the speaker is not large enough to reproduce the bass, we extrapolate it from overtones.



The idea is that our brain has special detectors of the harmonic series, which help to distinguish between the equivalence of the octaves .

While the tonalities of the various harmonic rows are different, the ratio of their frequencies to the fundamental frequency remains constant. Therefore, we believe that the brain normalizes tones by dividing them and obtains some of their ratios. Sound processing requires an analysis of frequencies differing from each other by several orders of magnitude. If these frequencies can be fully analyzed, then we have the opportunity to reuse this knowledge. Consider a conceptually simple process, when the brain reduces or doubles the frequency of the wave, until this value falls within a certain range.

Now the brain needs something to recognize the tones of the harmonic series in the frequency range that does not exceed the tone twice, and not on the entire sound spectrum. Thus, the problem is divided into two subtasks, where the first is normalization and the second is recognition itself, which greatly simplifies the design of the resulting recognizer. Therefore, we believe that, quite likely, our brain normalizes tones, halving or doubling them until they fall into a certain range of frequencies that differ by two orders of magnitude. It is also likely that the brain halves or doubles the frequencies into different degrees of two in parallel, and then conducts their simultaneous recognition. If any frequency coincides [with the base range], the desired harmonic is found.

So why do we like harmony? Wilkerson says it comes down to artificially enhancing a natural series of overtones. Hearing a chord is like listening to a magic voice with stronger and clearer harmonics than a single [natural] sound source. Wilkerson once said that harmony is “sweeter than sweet” for us.

Below is an illustration of what Wilkerson meant. The figure shows the spectrogram of two notes played on a violin: note C (“before”) on the left and G (“salt”) on the right.



Dotted lines mark notes that have almost the same spectrum. Each overtone from C can be found similar in note G. If you lose these notes at the same time, the harmonic recognizer in your brain will immediately activate and mark a striking correspondence.

This is a good explanation of consonance. But we also love and not consonant harmony. What does Wilkerson think about this? He blames it on our innate love of narration, comparing the chord sequence with the plot of the piece.

If to understand and predict the storyline of the story is quite easy, then the story seems boring, and if it is too difficult, the story turns out to be chaotic. If you hold the plot between these two extremes, it turns out an interesting story. Simplicity comes from the “theme” of the story, its main idea, and ambiguity is the absence of a simple explanation or “theme”, resulting in difficulties.

Expand the topic at the very beginning - a way to get rid of the complexity still in the bud. Probably, the branches from the topic and the inexplicable residual complexity are “read” by our brain with the help of the “handler of expectations”. The art of managing harmony is, for the most part, just a game with a wait handler, when you give it such an amount of complexity that the initial information remains in an interesting area: between something obvious and something meaningless.

Next, let's see what Wilkerson thinks about how we get simple diatonic harmony from a natural series of overtones. We start by searching for the ideal harmonic series consisting of one note, say C4 - “to” the fourth octave (“to” the first octave on the piano) and relate it to the whole octave by dividing the frequency as necessary.

I will follow the Wilkerson conventions and will consider the first harmonic fundamental:

• The second harmonic has a frequency twice as large as C4, the note C5 is obtained - “before” the second octave.
• The third harmonic with a frequency three times greater than C4 is G5. When you divide its frequency by two, you get G4 - a clean fifth of “before” - “salt” of the first octave.
• The fourth harmonic with a frequency four times higher than C4 is C6. In fact, all even harmonics are just a note C (“before”), transposed two octaves higher.
• The fifth harmonic with a frequency five times as large as C4 is E6. Having normalized it, we get E4 - a large third of “do” - “mi” of the first octave.

So, the first few natural harmonics contain a large third and notes C, E, G (“do”, “mi”, “salt”), plus a note “do”, transposed a few octaves higher. By choosing another harmonic as the fundamental, you get the same frequency relationships.

Further, Wilkerson asks us to build another major triad based on the note in the series of overtones that are closest to the basic tone. The first note that you get from the harmonics of the note C (not counting C per octave higher) is G ("salt"). If you build a major triad on the basis of the note G, you get the notes G, B and D (“salt”, “si” and “re”). Then Wilkerson wants us to build a major triad from the nearest note, in which the third harmonic is note C. This note is F (“fa”) and its major triad is reproduced from a series of overtones F, A and C (“fa”, “la "And" before "). Having ranked all these notes in accordance with the frequency, you will get the sequence C, D, E, F, G, A, B - that is, the well-known major scale .

Once you get this set of notes, you can deduce from them all kinds of other interesting chords and scales. Using notes D, E or A (“re”, “mi” or “la”) as a basis, you will get minor triads . Our emotional reaction to minor chords is more complicated than just “Aha!”, As is the case with the recognition of major chords. The minor triad A (“do” - “mi” - “la”) has the same pair intervals of the harmonic series: a quint between A and E (“la” and “mi”) and a large third part between C and E. But we do not hear harmonic row itself. Wilkerson thinks that we find minor chords more interesting, because we only know them in part, and they tease our inner recognizer of harmonic series.

Subsequently, the theory was supported by the fact that there is only one major series, while the minor series are numerous. Let me remind you that in a major row built from a major triad, everything sounds clear to the human ear, while the sound of minor rows built from a minor triad often seems wrong and ambiguous.

A lot of our mental resources are spent on the elimination of ambiguity - this is, in essence, the search for the most probable and logical explanation of vague and contradictory information about the world. In this we differ from computers and transcend them in certain life situations. Music teases a part of our brain that is responsible for resolving ambiguous situations, causing us to easily recognizable images with non-obvious values.

Wilkerson likens complex harmony to cubist-style drawings:

Parts of the object can be depicted fairly reliably, so that they are easy to recognize, but the object as a whole is not clearly drawn. An interesting effect arises: we recognize the object by certain features that our perception catches, but we still have the feeling that we see the object not in its natural form, but in the form of disorderly, inappropriate or obscure.



What about more complex chords? Wilkerson says that logic applies here, which we used to explain the influence of minor chords.

The brain wants to hear a certain harmonic sequence. If we “miss” [in our sequence] more and more notes, and the brain tries to fill in more and more gaps, then soon enough there will be a moment when the reproduced notes will hardly suffice to recognize the harmonic series. What if there are not enough played notes to determine it? Does this mean that the gaps in the harmonic rows can be filled in more than one way?

Some chords are ambiguous and therefore unstable. If the brain has more than one alternative to fill in the gaps, the sound will be "unstable" until the performer provides a sufficient number of notes capable of "breaking the disparity" and the ambiguity of the series.

If you hear notes C, F and G (“before”, “fa” and “salt”), you hear something like a natural series of overtones. But which of them? Both C (“before”) and F (“fa”) can act as the main tone. Musicians call such situations a “hanging chord,” which quite accurately describes the situation. You seem to be in a state of suspense between two acceptable situations in which either C or F can act as the main notes. If F is replaced by E ("mi"), then the situation will be resolved in favor of C, and if G is replaced by A ("la"), then in favor of F. In more modern music, this situation may not be resolved at all.

The ambiguity idea perfectly explains any of the exotic chords. When you hear an increased or diminished triad or a jazz chord with a lot of “extra” notes, then you hear pair intervals that are in tune with a number of overtones. But you do not hear the finished overtone row, but you may hear more than one such row. The result is less fun than a major chord, but it still sounds like something significant, you just need to make an effort to understand what is happening.

Probably, the brain has one module of the resolution of polysemy, which works in verbal communication and in other situations, as is the case with music. Thus, despite the fact that chords may sound strange separately, the musical theme that precedes these chords may bring a certain meaning to them. Recall the standard structure of the joke: the story (the creation of a theme), and then the zest. The highlight will not be funny without the context that history gives us, but we associate the joke itself with the highlight, and do not take into account the history that made this joke possible.

The Wilkerson theory and the standards of classical theory are consistent in that the harmonic series is the heart and soul of harmony. However, there are several points in which these theories diverge. For example, Wilkerson rejects the fifth circle as a method of achieving harmony, but does not reject everything that covers the principles of the nature of music.

The Quint Circle is a diversion that does not allow people to understand harmony, to realize how well it sounds.

I really like Wilkeson's theory, but in one place he is greatly mistaken - this is about his analysis of the triton .

The sound of the C and F # notes (“before” and “F-sharp”) on the piano is simply terrible. This interval is also called a triton, since the distance between C and F # is three whole tones (where the word “tone” means the distance from two semitones - in total, three tones are six semitones). We can analyze why this interval sounds so unstable: the ratio between [frequencies] F # and C does not correspond to any harmonic of the harmonic series. This interval deservedly got its name "devil interval".

Wilkerson did a great job in order to insulate his views from the influence of Eurocentric musical enhancement, but here he allowed himself to deviate a little. Those of us who love the blues and his musical heirs will not call the sound of a newt terrible. Its sound, of course, is not as pleasant as that of a quint or a large third, but a slightly less pleasant sound does not mean bad. Wilkerson could explain the newt better by resorting to his own ideas about ambiguity and complexity. A triton is a more “adult” sound, and it cannot be found within a series of overtones, but can be easily obtained from an interval that may be there. In C major there is a triton between F and B ("fa" and "si").

As one of many musical theorists, Wilkerson has a lot to say about harmony and says almost nothing about rhythm. But he can find this wonderful little paragraph here:

Recently, while listening to the sounds made by insects in the evening, I was amazed at how rhythm appeared in them as they “layered” a fading musical “theme” into a new, more complex sequence of sounds. At first, I heard a simple, predictable rhythm, and suddenly, contrary to expectations, the rhythm changed, beginning to sound “on top” of the main theme. Phenomena of narration, prediction and prediction of the topic can be attributed to both harmony and rhythm. The phenomenon of waiting as if generalizes the input data, and therefore waiting for harmony should work in a similar manner with rhythmic waiting.

Just like harmony is an idealized abstraction of a human voice, so a rhythm is an idealized abstraction of physical movements and dances. I would like to move Wilkerson’s theory one step further. A note is simply a very fast rhythm, and chords are very fast polyrhythms. Just as rhythm is the basis of music as a whole, so our theory of rhythm must be the basis of our theory of music.