The basic feeling of the major scale. Children's vocal and pop group Columbine

A minor scale (or simply minor) is a seven-step scale, the stable sounds of which form a small (minor) triad.

The word “minor” itself (Italian - minore) literally means “lesser”. This term is used in syllabic notation, while in alphabetic notation the word “minor” is replaced by the word moll (from the Latin molle, literally “soft”).

The main characteristic feature of the minor mode is the interval of the minor third (m. 3) between the I and III steps, which, in fact, determines the specificity, that is, the minor nature of the combined sound, both of the stable sounds themselves and of the mode as a whole, in any order of execution of its steps .

In principle, the properties and names of the scale degrees in minor will be the same as in major; only, in some cases, the intervallic relationships between them and, accordingly, the nature of their sound change.

The minor scale (like the major scale) has three main types: natural, harmonic and melodic minor.

The minor scale is built as follows: tone-semitone-tone-tone-semitone-tone-tone.

Key

The pitch level of the fret, determined by the sound of the tonic, is called tonality. Placing a fret on the same sounds, but in a different octave, does not have any effect on determining the tonality, since neither the structure of the fret itself, nor the names of its steps and their properties change from this.

The name of any tonality is determined by the name of the sound of the tonic itself (the first degree of the mode), but since the tonality is always inextricably linked with any particular mode (major or minor), an indication of the modal inclination is usually added to its name. Thus, the full name of a tonality, as a rule, contains two components: 1) the name of the tonic and 2) the name of the mode, regardless of which notation system - syllabic or alphabetic - is used: C major (C-dur), A minor (a-moll).

The names of major keys according to the letter system are written with uppercase (capital) letters, and minor keys with lowercase (small) letters. Sometimes, for brevity, the words dur or moll are omitted from the letter system, and then the modal mood is indicated by the spelling of the first letter (uppercase or lowercase).

Parallel and eponymous keys of major and minor

Although historically both main seven-step modes - both major and minor - developed completely independently, without losing their main specific features, there is still a certain kinship between them: the same number of steps, their similar functional meaning, the same directions of mode gravity and etc. The scales of some similar varieties of both modes (for example, harmonic major and harmonic minor, or melodic major and natural minor and, conversely, natural major and melodic minor), built from the same sound, will sound almost the same, differing only in the sound of the third degree - the main and only accurate sign of a particular mode.

Dedicated to L.G. and A.G., muses and fairies, who disenchanted my sense of beauty...

Low music began to play softly. Her unhurried minor chords smoothly flowed around, taking us somewhere into the deep distance. For some reason, there was a whiff of sadness... then the tempo began to increase, high notes were replaced by low ones, the tension gradually increased, and finally a bright, solemnly joyful, major denouement sounded. What happened to us? Mystery of nature...

To avoid ambiguity, I will give a few introductory phrases to clarify the terminology.

As is known, any audio signal of limited duration can be represented as an equivalent Fourier series (spectrum) as a sum of “pure” tones (sinusoidal oscillations) with different amplitudes, frequencies and initial phases. In this work we will consider mainly stationary sound signals that do not change over time.

According to the fundamental tone (first harmonic) of a sound, the lowest frequency of sound is called. All other frequencies above the fundamental tone are called overtones. That. The first overtone is the 2nd highest tone in the sound spectrum. An overtone with a frequency N times greater than the frequency of the fundamental tone (where N is an integer greater than 1) is called the Nth harmonic.

Musical (or harmonic) is a sound that consists only of a set of harmonics. In practice, this is a sound in which all overtones approximately fall within the harmonic frequencies, and some arbitrary harmonics may be absent, including the first. In this case, the fundamental tone is called “virtual” and its pitch will be determined by the psyche of the subject-listener from the frequency ratios between real overtones.

One musical sound may differ from another in fundamental frequency (pitch), spectrum (timbre) and volume. In this work, these differences will not be used, but all our attention will be focused on the mutual relationship of the pitches of sounds.

We will consider the effects of listening to one or more joint musical sounds taken outside of any other musical context.

As is known, the simultaneous sound of two musical sounds of different pitches (two-voice chord, dyad, consonance) can produce in the subject the impression of a pleasant (harmonious, harmonious) or unpleasant (irritating, rough) combination. In music, this impression of consonance is called consonance and dissonance, respectively.

It is also known that the simultaneous sound of three (or more) musical sounds of different pitches (three-voice chord, triad, triad) can produce in the subject an emotional impression of different colors. Different - according to the sign (positive or negative) and strength (depth, brightness, contrast) of the corresponding emotions.

The emotions evoked in people by listening to music, according to their type, among all known emotions, belong to aesthetic (intellectual) and utilitarian emotions. On the classification of emotions, incl. musical see more details.

For example, a triad from the notes “C, E, G” (major) and a triad from the notes “C, E-flat, G” (minor) have, respectively, a pronounced “positive” and “negative” emotional connotation, usually referred to as “joy” and “sadness” (or grief, sadness, suffering, regret, grief, melancholy, despondency - according to).

The emotional coloring of chords is practically independent of changes in the overall pitch, volume or timbre of their constituent sounds. In particular, we will hear an almost unchanged emotional coloring in chords of fairly quiet pure tones.

Looking ahead, we note that if some arbitrary chord can be defined as minor or major, then for the vast majority of subjects the emotions evoked by its sound will be utilitarian, i.e. refers to the category of “sadness or joy” (having a negative or positive sign of emotion). The emotional strength (brightness of emotion) of this chord will generally depend on the specifics of the situation (the state of the subject-listener and the structure of the chord). Essentially (in a statistical sense) it is possible to establish a one-to-one correspondence between major/minor and the emotions they evoke. And most likely it is the emotional coloring of these chords that allows “ordinary people” to recognize the major or minor key of individual chords.

That. Let us summarize that the aesthetic component of the sound “pleasant-unpleasant” (consonance and dissonance) arises in us when listening to two-voice chords, and the emotional component of the sound “joy-sadness” (major and minor) arises in us only when adding a third voice. Note that other types of chords (not major or minor) may not have the utilitarian component of the emotion “contained in them.”

CHORD PROPORTIONS

It is logical to make the assumption that when perceiving a different number of simultaneous musical sounds, the rule of transition of quantity (1, 2, 3 ...) into quality is triggered. Let's see what new qualities can emerge.

Even in ancient times, it was discovered that a chord of two (separately pleasant) sounds can be pleasant or unpleasant (consonant or dissonant) to the ear.

It has been established that such a chord sounds consonant if the ratio of the pitches of its sounds (with an error of, say, 1% or less) is a proportion of relatively small integer (natural) numbers, in particular the numbers from 1 to 6 and 8.

If this proportion consists of relatively large mutually prime numbers (15/16, etc.), then such a chord sounds dissonant.

I note that the accuracy with which whole proportions of musical sounds should be determined, as well as the choice of a specific proportion from a number of alternatives, may depend on the context of the situation. A brief historical excursion into musical intervals is given in.

A list of the ratios of the pitches of two musical sounds (musical intervals) in order of decreasing consonance accordingly looks like this: 1/1, 2/1, 3/2, 4/3, 5/4, 8/5, 6/5, 5/3, and further dissonances 9/5, 9/8, 7/5, 15/8, 16/15.

This list may not be completely complete (at least in terms of dissonances), because is based on possible musical intervals within the framework of an equal-tempered scale of 12 notes per octave (RTS12).

It is also known that the perception of consonance and dissonance occurs at an intermediate level of the human nervous system, at the stage of preliminary processing of individual signals from each ear. If you use headphones to separate two sounds into different ears, then the effects of their “interaction” (consonance peaks, virtual height) disappear.

Digging a little aside, I note that although today there are more than a dozen theories of consonance and dissonance, it is very difficult to give a clear explanation of why the interval 7/5 is dissonance, and 8/5 is consonance (moreover, more perfect than, for example, 5/3) .

However, by and large, we don’t need this here. Is it a good topic for a separate study?

So, let's note the following new fact. When moving from listening to one musical sound to two simultaneous sounds, the subject becomes able to extract information from the pitch ratio of these sounds. Moreover, the psyche of the subject especially distinguishes the ratios of heights in the form of proportions from relatively small natural numbers, which are ranked in one category - consonance/dissonance.

Now let's move on to considering chords of three sounds. In triads, compared to consonances, the number of (pairwise) intervals increases to three, and in addition, a new entity appears - the “monolithic” triad itself (as if a “triple” interval) - the general relationship between the pitches of all three sounds considered together.

This monolithic ratio can be written as the "direct" proportion A:B:C or in another form as the "inverse" proportion (1/D):(1/E):(1/F) of natural coprime triplets of numbers A, B,C or D,E,F. Purely mathematically, all such proportions can be divided into three main groups:

The direct proportion is “simpler” than the inverse, i.e. A*B*C< D*E*F

The inverse proportion is “simpler” than the direct one, i.e. A*B*C > D*E*F

Both proportions are the same (“symmetrical”), i.e. A*B*C = D*E*F (and so A=D, B=E, C=F).

That. a new quality of triad - information of a new type - can only be contained in these triple proportions, falling into one of the three categories described above.

Depending on the degree of consonance of all paired intervals, triads can be either consonant or dissonant. In some cases (when using various integer approximations), the choice of the specific composition of both proportions may be ambiguous. However, for consonantal chords such ambiguity does not appear.

According to musical practice, there are four main types of triads - major and minor (consonances), augmented and diminished (dissonances). Almost all consonant chords can be classified as major or minor.

The pitch ratios of the above-mentioned major triad with great accuracy are in direct proportion 4:5:6. The pitch ratios of the above minor triad with great accuracy are the inverse proportion /6:/5:/4. The direct and inverse proportions of the augmented and diminished triads are the same, because they consist of equal intervals (4-4 and 3-3 semitones RTS12), and these equal proportions look like /25:/20:/16 = 16:20:25 and, accordingly, /36:/30:/25 = 25: 30:36.

The pitch relationships of major triads are always more simply (using smaller integers) expressed in direct proportions, and of minor triads in inverse proportions, and this is a well-known fact. Already Gioseffo Zarlino (1517-1590) knew the opposite meaning of major and minor chords (“Istituzione harmoniche” 1558). However, even 450 years later, it is not so easy to find a serious work in which this fact is widely used for harmonic analysis or chord synthesis. The reason for this may have been the persistent but erroneous attempts of various authors to explain the phenomenon of major and minor (see below). Maybe the connection between chords and pitch proportions has become something of a forbidden topic of “perpetual motion”?

Based on simple mathematics and experimental data, we put forward a postulate: any major chord (it is simpler in direct proportion) can be turned into a minor chord (it is simpler in inverse proportion), if instead of a direct proportion we write the inverse of the same numbers. Those. if the proportion A:B:C is major, then the inverse (different!) proportion /C:/B:/A is minor. Of course, any direct proportion can (without changes!) be represented as an inverse, and vice versa. In particular, 4:5:6 = /15:/12:/10 and /4:/5:/6 = 15:12:10.

Summarizing all this, we can conclude that the three groups into which all proportions of triad pitches are divided really play an important role in musical practice, and correspond to the division of chords into major, minor and “symmetrical” (consisting of identical intervals).

One might ask: what is the “internal” representation of musical triads in the psyche of the subject? How does he use information about the above-mentioned “new quality” of the triad?

Taking into account the highly developed apparatus of the human auditory system, it can be assumed that although the human higher nervous system is quite capable of representing a minor triad in the form of a direct proportion (15:12:10), it is also (if not easier) capable of representing the same triad in in the form of an inverse proportion (/4:/5:/6), and “at the first comparison” of these proportions (to determine the category), “discard” the straight line because of its 15 times greater complexity (the product of three numbers of direct and inverse proportions is equal to 1800 versus 120).

We will further call the main proportion of a chord one of the two proportions of the pitches of its sounds (direct or inverse), which consists of smaller numbers (in the sense of their product), while the other proportion will be called secondary. That. The main proportion of a major chord will always be a direct proportion, and a minor chord will always be an inverse proportion.

And finally, we note that although the above-mentioned minor and major triads consist in pairs of the same intervals (4:5, 4:6, 5:6), they have the opposite emotional connotation, which is absent in any individual pair of their sounds. The only difference between monolithic triads (minor and major) is the fact of mutual reversal of their main proportions.

It is logical to conclude that the corresponding new “emotional” information of the chord is contained precisely in this last property (type of principal proportion), which can only appear when three or more sounds are combined, but cannot be detected when two are combined (since, say A:B is absolutely the same as /A:/B). There is simply no other source of (emotional) information contained in the triad (don’t forget that we are considering stationary sounds with a constant spectrum). Additional confirmation of this conclusion is that the sound of “symmetrical” chords does not have a utilitarian component of emotions.

Example 1. Sound of proportions

2:3:4 = /6:/4:/3 gives a soft major. 2:3:6 = /3:/2:/1 gives a soft minor.

3:4:5 = /20:/15:/12 gives a brighter (contrasting) major, and 20:15:12 = /3:/4:/5 gives a deeper (contrasting) minor.

4:5:6 = /15:/12:/10 gives the brightest major, and 10:12:15 = /6:/5:/4 gives the deepest minor.

To listen to chords, it is better to use pure tones with precise frequency relationships, using e.g. .

THEORIES OF MAJOR AND MINOR

Chords have been heard in music for many hundreds of years, and people have been thinking about the reasons for their euphony for almost as long.

For two-voice chords, the first explanation of this property was made a very long time ago (and it is captivatingly simple and clear, if you close your eyes to some dissonances - see above). For three-voice chords of major and minor, the above-described facts about direct and inverse proportions were also established quite a long time ago.

However, finding an answer to the question of why different chords have different emotional colors in sign (and strength) turned out to be much more difficult. And to the second question - why does a minor chord, with all its complexity (when presented in direct proportions - so to speak, in “major notation”) sound euphonious, and let’s say “almost the same” in terms of the complexity of the numerical proportion “dischord” (like 9:11 :14) sounds unpleasant - it was difficult to answer.

Generally speaking, it was not entirely clear how to justify “equally well” both major and minor?

Many authoritative researchers have tried to figure out this mystery of the nature of major and minor. And if the major was explained “quite simply” (as it seemed to many authors, for example, “purely acoustically”), then the problem of justification for the minor, which is similar in clarity, apparently is still on the agenda, although there are a great many different theoretical and phenomenological constructions trying to provide its solution.

The interested reader may refer to .

History- but it does not have to always be the case - for example in the case of a chord of pure tones.

Some authors, when “justifying” chords, also referred to the nonlinear properties of hearing, described for example. V . However, this undeniably true fact very rarely works in practice, because even a chord that is not too weak in volume will not generate discernible combination tones due to nonlinearity.

Other authors used very complex theoretical and musical constructions (or purely mathematical schemes, closed as “things in themselves”), the exact meaning of which was often impossible to understand without a detailed study of the specific terminology of these theories themselves (and sometimes this explanation was based on paraphrasing some abstract terms through others).

Some authors are still trying to approach this issue from the point of view of cognitive psychology, neurodynamics, linguistics, etc. And they almost succeed... Almost - because the chain of explanations can be too long and far from indisputable, and besides, there is no algorithmic formalization of theories, etc. basis for their quantitative experimental verification.

For example, in one of the most interesting, detailed and versatile studies of the phenomenon of major and minor, a hypothesis is given that the basis for the emotional content of sounds was laid down by nature in the instinct of higher animals, which was further developed in humans. It has been experimentally established that the dominance of a particular individual of a pack in the animal world is accompanied by the use of low or descending “speech” sounds, and subordination is accompanied by the use of high or rising sounds. It is further accepted that dominance is equal to “joy”, and submission is equal to “sadness”. Then a table is constructed of dissonant symmetrical triadic chords (with two equal intervals from 1 to 12 semitones RTS12) with a list of changes in these chords to minor when increasing or to major when the pitch of any sound of the original chord is decreased by one semitone.

Even aside from the fact that some of the changed chords cannot be unambiguously classified as major or minor, it is not clear why, when listening to a chord, a human subject must necessarily (and instantly) “think” that one of the sounds of this (consonantal) chord is shifted from the sound of another (unambiguously defined and also dissonant) chord for a certain fixed interval - a semitone? And how can this rather abstract thought turn into “innate” emotions? And why should the mind be limited only to the capabilities of the RTS12? RTS12 was also invented by Nature and put into instinct?

However, I agree that the emotional content of major and minor is based on the emotions available to many higher animals... although it is unclear - can they experience these emotions when listening to chords? I think it's unlikely. Because determining the relative proportions of the pitches of three or more sounds of a chord is a process of a higher order of complexity than determining the pitch of one sound (or the direction of change in this pitch).

The human hearing system has received particular development in connection with the advent of speech communication, which has given rise to the ability to perform detailed and rapid analysis of the spectrum of complex sounds, a by-product of which is most likely our ability to enjoy music.

Utilitarian emotions in higher animals (as well as in humans), however, may well be evoked through the perception of information from other senses - and above all - through the visual perception of events and their further interpretation.

A few words about the emotionality of human speech and monophonic music. Yes, they may “contain” utilitarian emotions. But the reason for this is the significant non-stationarity of the spectrum - changes in the height and/or timbre of these sounds.

And also about the individual differences of the subjects. Yes, with the help of special education (training) it is possible to accustom people (as well as some animals) to the fact that even one sound (or any chord) will evoke utilitarian emotions in them (grief from a reflexively expected stick or joy from a carrot ). But this will not correspond to the natural nature of things which we seek to establish.

Here is a phrase from a doctoral dissertation in musicology in 2008, apparently putting an end to the question of the well-known theories of major and minor: “despite the fact that many authors have described the perception of major/minor chords and scales, it still remains a mystery why major chords feel happy and minor chords feel sad.”

I think that the development of a correct theory of major and minor is possible only if two important conditions are satisfied:

Involving additional areas of knowledge (except music and acoustics), - using the mathematical apparatus of additional areas of knowledge.

We should remember history. The idea that the “meaning” of a chord should be sought outside the “old” space of music theory was first heard at least more than a hundred years ago.

Here are a couple of quotes.

Hugo Riemann (1849-1919) towards the end of his career abandoned the justification of major and consonance through the phenomenon of overtones and adopted the psychological point of view of Karl

Stumpf, considering overtones only as “an example and confirmation”, but not proof.

Karl Stumpf (1848-1936) transferred the scientific basis of music theory from the field of physiology to the field of psychology. Stumpf refused to explain consonance as an acoustic phenomenon, but proceeded from the psychological fact of “fusion of tones” (Stumpf C.Tonpsychologie. 1883-1890).

So, concluding this section, I note that most likely Stumpf and Riemann were absolutely right that it is impossible to substantiate a chord either acoustically, metaphysically, or purely musically, and what is necessary for this is the involvement of psychology.

Now let’s approach the question “from the other end” and ask the question: what is emotion?

THEORIES OF EMOTIONS

Let us briefly consider two theories of emotions, which, in my opinion, come closest to the level at which the possibility of applying their laws in such a complex issue as the psychological structure of the phenomena of music perception opens up.

For other theories and details, I refer the reader to a fairly extensive review in.

Frustration theory of emotions

In the 1960s L. Festinger's theory of cognitive dissonance arose and was thoroughly developed.

According to this theory, when there is a discrepancy between expected and actual performance results (cognitive dissonance), negative emotions arise, while the coincidence of expectations and results (cognitive consonance) leads to positive emotions. The emotions that arise during dissonance and consonance are considered in this theory as the main motives for the corresponding human behavior.

Despite many studies confirming the correctness of this theory, there is also other evidence showing that in some cases, cognitive dissonance can cause positive emotions.

According to J. Hunt, for the emergence of positive emotions, a certain degree of discrepancy between attitudes and signals, a certain “optimum of discrepancy” (novelty, unusualness, inconsistency, etc.) is necessary. If the signal does not differ from previous ones, then it is assessed as uninteresting; if it differs too much, it seems dangerous, unpleasant, annoying, etc.

Information theory of emotions

Somewhat later, an original hypothesis about the causes of the phenomenon of emotions was put forward by P.V. Simonov.

According to it, emotions appear as a result of a lack or excess of information necessary to satisfy the subject’s needs. The degree of emotional stress is determined by the strength of the need and the magnitude of the deficit of pragmatic information necessary to achieve the goal.

P.V. Simonov considered the advantage of his theory and the “formula of emotions” based on it to be that it contradicts the view of positive emotions as a satisfied need. From his point of view, a positive emotion will arise only if the received information exceeds the previously existing forecast regarding the probability of satisfying the need.

Simonov’s theory was further developed in the works of O.V. Leontiev, in particular, by 2008, a very interesting article was published with a number of generalized formulas of emotions, one of which I will describe in detail below. I quote further.

By emotions we mean a mental mechanism for controlling the behavior of a subject, assessing the situation according to a certain set of parameters... and launching the corresponding program of his behavior. In addition, each emotion has a specific subjective coloring.

The above definition assumes that the type of emotion is determined by the corresponding set of parameters. Two different emotions must differ in a different set of parameters or range of their values.

In addition, psychology describes various characteristics of emotions: sign and strength, time of occurrence relative to the situation - preceding (before the situation) or stating (after the situation), etc. Any theory of emotions must make it possible to objectively determine these characteristics.

The dependence of an emotion on its objective parameters is called the formula of emotions.

One-parameter formula of emotions

If a person has a certain need of value P, and if he manages to obtain a certain resource Ud (for Ud > 0) that satisfies the need, then the emotion E will be positive (and in the case of loss of Ud< 0 и эмоция будет отрицательной):

E = F(P, UD) (1)

The resource Ud is defined in the work as the “Level of Achievement”, and the emotion E is defined as a stating one.

To be specific, you can imagine a person playing a new game and having no idea what to expect from it.

Joy.

If a player wins a certain amount Ld > 0, then a positive emotion of joy arises with force

E = F(P, Ud).

Grief.

If the player “won” the amount< 0 (т.е. проиграл), то возникает отрицательная эмоция горя

force E = F(P, Ud).

Another method of formalizing emotions is proposed in the work.

According to him, emotions are considered as a means of optimal control of behavior, directing the subject to achieve the maximum of his “target function” L.

An increase in the target function L is accompanied by positive emotions, a decrease - by negative emotions.

Since L depends in the simplest case on some variable x, emotions E are caused by a change in this variable over time:

E = dL/dt = (dL/dх)*(dх/dt) (2)

It is also noted that, along with the above-described (utilitarian) emotions, there are also so-called. “intellectual” emotions (surprise, guess, doubt, confidence, etc.), which arise not in connection with a need or goal, but in connection with the intellectual process of information processing itself. For example, they can accompany the process of observing abstract mathematical objects. A feature of intellectual emotions is their lack of a specific sign.

At this stage, we will stop quoting and move mainly to the presentation of the author’s original ideas.

MODIFICATION OF FORMULAS OF EMOTIONS

First of all, we note that formulas (1, 2) are very similar, if we take into account that the resource parameter Ud is actually the difference between the current and previous value of a certain integral resource R. For example, in the case of our gambler, it is logical to choose his total capital as R , Then:

BP = R1 - R0 = dR = dL

However, both formulas (1, 2) are “not entirely” physical - they equate quantities that have different dimensions. You can’t measure, say, time in kilometers or joy in liters.

Therefore, firstly, the formulas of emotions should be modified by writing them in relative quantities.

It is also desirable to clarify the dependence of the strength of emotions on their parameters, i.e. to increase the plausibility of the results over a wide range of changes in these parameters.

To do this, we will use an analogy with the well-known Weber-Fechner law, which says that the differential threshold of perception for a variety of human sensory systems is proportional to the intensity of the corresponding stimulus, and the magnitude of the sensation is proportional to its logarithm.

In fact, the joy of that same player should be proportional to the relative size of the win, and not to the absolute size. After all, a billionaire who lost one million will not grieve as much as the owner of a million or so. And the heights of the “most similar” musical sounds are related by octave ratio, i.e. also logarithmic (increasing the frequency of the fundamental tone of the sound by 2 times).

I propose to write the modified emotion formula (1) as follows:

E = F(P) * k * log(R1/R0), (3)

where F(P) is a separate dependence of emotions on the need parameter P;

k is some constant (or almost constant) positive value, depending on the subject area of ​​the resource R, on the base of the logarithm, on the time interval between measurements R1 and R0, and also possibly on the details of the character of a particular subject;

R1 is the value of the target function (total useful resource) at the current point in time, R0 is the value of the target function at the previous point in time.

You can also express the new formula of emotions (3) through the dimensionless quantity L = R1/R0, which is logically called the relative differential objective function (the current value of the integral objective function relative to some previous moment in time, always located at a fixed distance from the current moment).

E = F(P) * Pwe, where Pwe = k * log(L), (4)

where in turn L = R1/R0, and the parameters k, R0 and R1 are described in formula (3).

Here the value of the power of emotions Pwe is introduced, proportional to the “flow of emotional energy” per unit of time (i.e., the everyday meaning of the expression “intensity of emotions”, “strength of emotions”). The expression of the strength of emotions in units of power allocated by the subject’s body to emotional behavior is known from the works of other authors, so we should not be surprised at the appearance of such a (somewhat unusual) term as “power of emotion.”

As is easy to see, formulas (3 and 4) automatically give the correct sign of emotions, positive when R rises (when R1 > R0 and thus L > 1) and negative when R falls (when R1< R0 и т.о. L < 1).

Now let's try to apply new formulas of emotions to the perception of musical chords.

INFORMATION THEORY OF CHORDS

But first, a little “lyrics”. How can the above information theory of emotions be expressed in simple human language? I’ll try to give a few fairly simple examples to clarify the situation.

Let’s say that today life has given us a “double portion” of some “life benefits” (against the average daily amount of “happiness”). For example - twice the best lunch. Or we had two hours of free time in the evening versus one. Or we walked twice as far as usual on a mountain hike. Or we were given twice as many compliments as yesterday. Or we received double bonuses. And we are happy because the function L today has become equal to 2 (L=2/1, E>0). And tomorrow we will receive all this fivefold. And we rejoice even more (we experience more powerful positive emotions, because L=5/1, E>>0). And then it all went as usual (L=1/1, E=0), and we no longer experience any utilitarian emotions - we have nothing to be happy about, and nothing to be sad about (if we have not yet gotten used to happy days). And then suddenly a crisis broke out and our benefits were cut in half (L = 1/2, E<0) - и нам стало грустно.

And although for each subject the goal function L depends on a large set of individual sub-goals (sometimes diametrically opposed - for sports opponents or fans, for example), what is common to all is the personal opinion of each - whether this event brings him closer to some of his goals, or moves away from them.

Now let's get back to our music.

Based on verified facts of science, it is logical to assume that when listening to several sounds simultaneously, the subject’s psyche tries to extract all kinds of information that these sounds may contain, including those located at the highest level of the hierarchy, i.e. from the pitch ratios of all sounds.

At the stage of analyzing the parameters of triads (as opposed to consonances, see above), individual streams of information from different ears are already used together (which is easy to check by sending any two sounds to one ear, and the third to the other - the emotions are the same).

In the process of interpreting this combined information, the subject’s psyche tries to use, among other things, its “utilitarian” emotional subsystem.

And in some cases she succeeds in this - for example, when listening to isolated minor and major chords (but other types of chords can apparently generate other types of emotions - aesthetic / intellectual).

Perhaps some fairly simple analogies (at the level of more/less) with the meaning of “similar” information from other sensory channels of perception (visual, etc.) allow the subject’s psyche to classify major chords as carrying information “about benefit”, accompanied by positive emotions, and minor chords - “about loss”, accompanied by negative ones.

Those. in the language of the emotion formula (4), the major chord should contain information about the value of the objective function L > 1, and the minor chord should contain information about the value of L< 1.

My main hypothesis is the following. When perceiving a separate musical chord, the value of the target function L is generated in the subject’s psyche, which is directly related to the main proportion of the pitches of its sounds. In this case, major chords correspond to the idea of ​​an increase in the target function (L>1), accompanied by positive utilitarian emotions, and minor chords correspond to the idea of ​​a decline in the target function (L<1), сопровождаемое отрицательными утилитарными эмоциями.

As a first approximation, we can assume that the value of L is equal to some simple function of the numbers included in the main proportion of the chord. In the simplest case, this function can be some kind of “average” of all the numbers of the main proportion of the chord, for example, the geometric mean.

For any major chords, all these numbers will be greater than 1, and for any minor chords they will be less than 1.

For example:

L = N = "average" of the numbers (4, 5, 6) from the major proportion 4: 5: 6,

L = 1/N = “average” of the numbers (1/4, 1/5, 1/6) from the minor proportion /4:/5:/6.

With this representation of L, the amplitude of the strength of emotions (i.e., the absolute value of Pwe) generated by the major and (its inverse) minor triad will be exactly the same, and these emotions will have the opposite sign (major - positive, minor - negative). A very encouraging result!

Let us now try to clarify and generalize formula (4) for an arbitrary number of voices of the chord M. To do this, we define L as the geometric mean of the numbers from the main proportion of the chord, ultimately obtaining the final form of the “formula of musical emotions”:

Pwe = k * log(L) = k * (1/M) * log(n1 * n2 * n3 * ... * nM), (5)

where k is still some positive constant - see (3),

Let's call the value Pwe (from formula 5) the “emotional power” of the chord (or simply power), positive for a major and negative for a minor (analogy: the flow of vital forces, for a major there is an influx, for a minor there is an outflow).

For consistency with the logarithmic frequency scale (remember the octave), we will use the base 2 logarithm in formula (5). In this case, we can set k = 1, because in this case, the numerical value of Pwe will be in a completely acceptable range near the region of “unit” amplitude of emotions.

For further analysis, along with the “main” one, we may also need the “side” power of the chord, corresponding to the substitution of its side proportion into formula (5) (see above). If not specified, the “main” Pwe is used throughout below.

The appendix to the article shows the meanings of the main and secondary powers of some chords.

THE DISCUSSION OF THE RESULTS

So, having put forward a number of fairly simple and logical assumptions, we have obtained new formulas (3, 4, 5), which connect the general parameters of the situation (or specific parameters of the chords for formula 5) with the sign and strength of the utilitarian emotions they evoke (in the context of the situation).

How can we evaluate this result?

I quote the work:

“There have probably been no attempts to objectively determine the strength of an emotion. However, it can be assumed that such a definition should be based on energy concepts. If an emotion causes some behavior, then this behavior requires a certain expenditure of energy. The stronger the emotion, the more intense the behavior, the more energy is required per unit of time.

Those. We can try to identify the strength of an emotion with the amount of power that the body allocates to the corresponding behavior.”

Let's try to approach the new result as critically as possible, since there is nothing to compare it with yet.

Firstly, the power of emotions Pwe from formulas (4, 5) is proportional to the “subjective strength” of emotions, but their relationship may not be linear. And this connection is only a certain average dependence along the entire continuum of subjects, i.e. may be subject to significant (?) individual variations. For example, the “constant” k can still change, although not too much. It is also possible that instead of the geometric mean in formula (5), some other function should be used.

Secondly, if we keep in mind the specific form of the formula for musical emotions (5), then it should be noted that although formally in it M can be equal to 1 or 2, we can talk about the emergence of utilitarian emotions only when M >= 3. However, already with M = 2, the presence of aesthetic/intellectual emotions is possible, and with M > 3, there is a possibility of additional factors (?) somehow influencing the result.

Thirdly, apparently the range of valid values ​​of the Pwe amplitude for the categories of major and minor has an upper limit of 2.7 ... 3.0, but somewhere already from the value of 2.4 the area of ​​saturation of the utilitarian-emotional perception of chords begins, and the lower limit of the range passes approximately there possible “invasion” of dissonances.

But this latter is rather a general problem of the “non-monotonicity” of a number of dissonant intervals, not directly related to the emotional perception of chords. And the limited dynamic range of the power of emotions is a general property of any human sensory system, easily explained by the lack of analogies with events in “real life”, which correspond to too rapid changes in the target function (7-8 times or more).

Fourthly, “symmetrical” (or almost symmetrical) chords, in which direct and inverse proportions consist of the same numbers (even in the absence of obvious dissonances in them) apparently fall out of our classification - their utilitarian-emotional coloring is practically absent, corresponding to the case Pwe = 0.

However, it is possible to supplement the formal result of applying formula (5) with a simple semi-empirical rule: if the main and secondary powers of some chord (almost) coincide in amplitude, then the result of formula (5) will not be the main power, but half the sum of the powers, i.e. (approximately) 0.

And this rule begins to work already when the difference between the amplitudes of the main and secondary Pwe is less than 0.50.

Most likely, a very simple phenomenon is taking place here: since it is impossible to distinguish the direct and inverse proportions of a chord by complexity, then the classification of this chord in the categories of utilitarian emotions (“sadness and joy”) is simply not carried out. However, these chords (like intervals) can generate aesthetic/intellectual emotions, e.g. “surprise”, “question”, “irritation” (in the presence of dissonances), etc.

With all its imaginary or real shortcomings, formula (5) (and, apparently, formulas 3 and 4) still gives us very good theoretical material for numerical estimates of the strength of emotions.

At least in one specific area - the area of ​​\u200b\u200bthe emotional perception of major and minor chords.

Let's try to test this formula (5) in practice, by comparing a pair of different major and minor chords. A very good example is the chords 3:4:5 and 4:5:6 and their minor variations.

For the purity of the experiment, pairs of chords composed of pure tones should be compared at approximately the same average volume level, and for both chords it is better to use such pitches that the “weighted average” frequency of these chords (in Hertz) is the same.

A pair of major triads can consist of tones of frequency e.g. 300, 400, 500 Hz and 320, 400, 480 Hz.

From the ear, it seems quite noticeable to me that the emotional “brightness” of the major 3:4:5 (with Pwe = 1.97) is indeed somewhat less than that of the major 4:5:6 (with Pwe = 2.30). In my opinion, approximately the same thing happens with the minor /3:/4:/5 and /4:/5:/6.

This impression of correct transmission of the power of emotions by formula (5) is also preserved when listening to the same chords composed of sounds with a rich harmonic spectrum.

TOTAL

In total, in accordance with the information theory of emotions, the work proposes modified formulas that express the sign and amplitude of utilitarian emotions through the parameters of the situation.

A hypothesis has been put forward that when a musical chord is perceived, the value of a certain target function L is generated in the subject’s psyche, which is directly related to the proportion of the pitches of the chord sounds. In this case, major chords correspond to direct proportions, giving rise to the idea of ​​an increase in the target function (L>1), causing positive utilitarian emotions, and minor chords correspond to inverse proportions, giving rise to the idea of ​​a decline in the target function (L<1), вызывающее отрицательные утилитарные эмоции.

A formula for musical emotions has been put forward: Pwe = log(L) = (1/M)*log(n1*n2*n3* ... *nM), where M is the number of voices of the chord, ni is an integer (or reciprocal fraction) from the general proportion of pitches corresponding to the i-th voice of the chord.

A limited experimental test was carried out, the limits of applicability of the formula of musical emotions were explored, in which it correctly conveys the sign and (in my opinion) their amplitude.

CODA

The fanfare sounds joyfully!

Then everyone stands up - and holding hands - a cappella they sing the Hymn to Reason!

The centuries-old mystery of major and minor has finally been solved! We won...

LITERATURE AND LINKS

1. Audiere sound system, Download archive Use wxPlayer.exe from the bin folder.
2. Trusov V.N. Materials from the site mushar.ru 2004 http://web.archive.org/http://mushar.ru/
3. Mazel L. Functional school. 1934 (Ryzhkin I., Mazel L., Essays on the history of theoretical musicology)
4. Riman G. Musical dictionary (computer version). 2004
5. Leontyev V.O. Ten unresolved problems in the theory of consciousness and emotions. 2008
6. Ilyin E.P. Emotions and feelings. 2001
7. Simonov P.V. Emotional brain. 1981
8. Leontyev V.O. Formulas of emotions. 2008
9. Aldoshina I., Pritts R. Musical acoustics. 2006
10. Aldoshina I. Fundamentals of psychoacoustics. A selection of articles from the site http://www.625-net.ru
11. Morozov V.P. The art and science of communication. 1998
12. Altman Ya.A. (ed.) The auditory system. 1990
13. Lefebvre V.A. Formula of man. 1991
14. Shiffman H.R. Sensation and perception. 2003
15. Teplov B.M. Psychology of musical abilities. 2003
16. Kholopov Yu.N. Harmony. Theoretical course. 2003
17. Golitsyn G.A., Petrov V.M. Information - behavior - creativity. 1991
18. Garbuzov N.A. (ed.) Musical Acoustics. 1954
19. Rimsky-Korsakov N. Practical textbook of harmony. 1937
20. Leontyev V.O. What is an emotion? 2004
21. Klaus R. Scherer, 2005. What are emotions? And how can they be measured? Social Science Information, Vol 44, no 4, pp. 695-729
22. BEHAVIORAL AND BRAIN SCIENCES (2008) 31, 559-621 Emotional responses to music: The need to consider underlying mechanisms
23. Music Cognition at the Ohio State University http://csml.som.ohio-state.edu/home.html Music and Emotion http://dactyl.som.ohio-state.edu/Music839E/index.html
24. Norman D. Cook, Kansai University, 2002. Tone of Voice and Mind: The connections between intonation, emotion, cognition and consciousness.
25. Bjorn Vickhoff. A Perspective Theory of Music Perception and Emotion. Doctoral dissertation in musicology at the Department of Culture, Aesthetics and Media, University of Gothenburg, Sweden, 2008
26. Terhardt E. Pitch, consonance, and harmony. Journal of the Acoustical Society of America, 1974, Vol. 55, pp. 1061-1069.
27. VOLODIN A.A. Abstract of doctoral dissertation. PSYCHOLOGICAL ASPECTS OF PERCEPTION OF MUSICAL SOUND
28. Levelt W., Plomp R. The appreciation of musical intervals. 1964

ACKNOWLEDGMENTS

I express my gratitude to Ernst Terhardt and Yury Savitski for the literature kindly provided to me for writing this work. Thank you very much!

AUTHOR'S INFORMATION

Feedback.

Any constructive criticism, comments and additions to this work will be accepted with

thanks to the email address: author(at)vmgames.com

License.

Free copying and distribution of the text of this work in unchanged form is permitted in the absence of any material income from these actions. Otherwise, prior written permission from the author is required. Any quotation of this work or retelling in your own words must be accompanied by a WWW link: http://www.vmgames.com/ru/texts/

Version.

APPLICATION

Emotional power Pwe of the main proportions of some chords, calculated using formula (5).

The main part of the proportions are direct proportions corresponding to major chords.

Minor chords can be made from proportions that are the reverse of major chords by simply changing the Pwe sign of the major proportion (as in a couple of examples).

The side power of some chords is given in parentheses if it approaches the main power in amplitude.

For symmetrical chords, both of these powers differ only in sign.

Home Side Pwe main (side) Note proportion proportion proportion

Some symmetrical [pseudo]chords

1:1:1 1:1:1 0 (0)

1:2:4 /4:/2:1 1 (-1)

4:6:9 /9:/6:/4 2.58 (-2.58) “fifth” triad

16:20:25 /25:/20:/16 4.32 (-4.32) increased triad

1:2:3 /6:/3:/2 0.86 (-1.72)

2:3:4 /6:/4:/3 1.53 (-2.06)

2:3:5 /15:/10:/6 1.64

2:3:8 /12:/8:/3 1.86

2:4:5 /10:/5:/4 1.77

2:5:6 /15:/6:/5 1.97

2:5:8 /20:/8:/5 2.11

3:4:5 /20:/15:/12 1.97 /3:/4:/5 20:15:12 -1.97

3:4:6 /4:/3:/2 -1.53 (2.06)

3:4:8 /8:/6:/3 2.19 (-2.39) almost symmetrical

3:5:6 /10:/6:/5 2.16 (-2.74)

3:5:8 /40:/24:/15 2.30

3:6:8 /8:/4:/3 2.39 (-2.19) almost symmetrical

4:5:6 /15:/12:/10 2.30 major triad

/4:/5:/6 15:12:10 -2.30 minor triad

4:5:8 /10:/8:/5 2.44 (-2.88)

5:6:8 /24:/20:/15 2.64

Some dissonant triads

4:5:7 /35:/28:/20 2.38

5:6:7 /42:/35:/30 2.57

1:2:3:4 /12:/6:/4:/3 1.15

2:3:4:5 /30:/20:/15:/12 1.73

3:4:5:6 /20:/15:/12:/10 2.12

In music, there are two main musical modes: major and minor. Let's talk in more detail about what a major scale is, what a major scale is, and how a major scale differs from a minor scale, as well as how this word is used in modern slang.

What is a major: definition

Major is a set of sounds of a musical instrument, implying a cheerful and cheerful tonality. This formulation is due to the fact that all music is inherently created to convey human feelings and emotions. The composer who composes a certain musical composition or symphony is based on his feelings and puts them into the music.

The process of developing a musical composition is quite complex and is often intertwined with the experimental method of selecting the necessary sounds. Each note recorded is followed by a heavy selection of subsequent notes and an attempt to harmonize them with each other so that the music sounds pleasant to the human ear.

Major mode

Due to the fact that the set of sounds is limitless, and the human ear is capable of perceiving sounds in the range from 20 to 20,000 Hz, it became necessary to classify certain simple sets of tones. The main ones, which can be played on any musical instrument, are the major and minor scales. It turns out that the major mode is a sequence of any composition that, on a psychological level, evokes in a person associations with a cheerful and cheerful mood. Also, any composition can be recorded in a minor mode to get the opposite effect of sound and perception on a psychological level. A minor scale is a scale whose sounds form a small and minor triad.

Differences between major and minor modes

Any set of notes can be written as a major or minor scale. That is, any musical composition consisting of a certain set of notes can be played in both a minor and a major sound.

The difference between a major mode and a minor one is that between the second tone and the first in minor there is a minor third, and in major there is a major third. In simple terms, major-minor is a harmonious expression of the states of the soul - joy and sadness.

Musical term in everyday life

In youth slang, the word major is used in a context indicating rich people living in prosperity and abundance. It is believed that people who have great material wealth get almost everything they want. And the person who gets what he wants is in joy and euphoria. And vice versa: people who do not have a large amount of material wealth do not always get what they want and therefore are not always in a state of joy and pleasure.

The fret represents system of relationships between stable and unstable sounds. The mode is the organizing principle for sounds, thanks to which they are united into a hierarchical and functionally interconnected system. A scale is also a system of sounds, but unlike a mode, the scale does not determine how sounds should interact with each other.

The two most widely used modes in music are major and minor. These frets consist of seven diatonic steps located in such a way that an interval of a major or minor second is formed between adjacent steps. Diatonic can be interpreted as a transition from one step to another.

Example 1: Transition before->re in diatonicism is possible, since in this case a transition occurs from one adjacent level to another. Transition B-sharp->C-sharp is also possible, since this involves a transition between various adjacent altered (increased or decreased) steps. Transition before->C sharp in diatonics is impossible, since this involves a transition from one level to the same elevated level. Instead of before->C sharp in diatonic the transition should be indicated as before->D-flat. Transition B-sharp->D-flat in diatonic is also unacceptable, since this skips a step before, located between si And re. Therefore, instead of B-sharp->D-flat the correct transition will be B-sharp->C-sharp.

Understanding the sequence of steps in the diatonic system allows you to correctly build tonalities, which will now be discussed. That is, in one key the first secondary step will be called C sharp, and in a different key - D-flat.

The order of the seconds determines the inclination of the scale - major or minor.

In the major mode, the seconds are arranged in the following sequence:

b.2, b.2, m.2, b.2, b.2, b.2, m.2

It is very easy to remember this sequence. You can use the phrase "b-b-m-b-b-b-m" or look at the piano keyboard. White keys from one stage before to the next step before just form this sequence: do-re-mi-fa-sol-la-si-do. The first four degrees of the mode are I-II-III-IV - the lower tetrachord, and the four upper degrees - V-VI-VII-VIII - the upper tetrachord. The intervallic structure of these tetrachords in natural major is identical and equal to b-b-m. The interpretation of the 7-step fret as a sequence of two tetrachords is not used in modern theory (it is considered outdated), but thanks to them, interesting points can be seen. For example, the structure of the lower tetrachord determines the inclination of the mode (major or minor), and the structure of the upper one determines its type (natural, harmonic or melodic).

The height of the fret is set tonality. The tonality is specified as follows: first the tonic of the mode (the first degree of the mode) is called, and then its type, for example: C major .

Rice. 1. Natural C major

The natural minor scale differs from the major mode in a different sequence of seconds:

b.2, m.2, b.2, b.2, m.2, b.2, b.2

White keys from the stage la form a minor scale: la-si-do-re-mi-fa-sol-la.

Rice. 2. Natural A minor

Varieties of the natural mode are harmonic and melodic modes (Fig. 3 and 4).

Harmonic major - lowered VI degree. The sound of the upper tetrachord takes on an oriental hue.

Melodic major - the VI and VII degrees are lowered. Lowering these steps leads to the fact that the upper tetrachord of a major becomes minor (has the structure of the upper tetrachord of a natural minor) and acquires a melodic sound.

Rice. 3. Types of major modes

Harmonic minor - raised VII degree. An increase in the VII degree leads to the appearance of an oriental shade in the upper tetrachord and an increase in the gravity of this level towards the tonic. The upper tetrachord in harmonic minor sounds more intense compared to its sound in natural minor.

Melodic minor - the VI and VII degrees are raised. Increasing the VI degree makes it possible to smooth out the transition from V to VII of the increased degree (increased second) and gives the upper tetrachord a melodious quality.

Rice. 4. Types of minor modes

The steps of the fret have their own names, and some have a letter designation. The serial numbers of the steps are indicated by Roman numerals:

Stage I - tonic, T (main note)
Stage II - descending introductory sound (resolves into tonic by lowering)
III degree - mediant or upper mediant (from the Latin media - middle, since it is in the middle between the I and V degrees above the tonic)
IV degree - subdominant, S (the Latin prefix sub means under)
Stage V - dominant, D (from the Latin “dominant”). This sound is the highest.
VI degree - submediant or lower mediant (located in the middle between IV and I degrees and located below the tonic)
VII stage - ascending introductory sound (resolves into tonic by rising)

According to functionality, fret stages are divided into main and secondary. I, IV and V are the main degrees of the mode, and II, III, VI and VII are secondary degrees. From the main steps, the main chords of the mode are built, which will be discussed further. Now let's just remember them.

Rice. 5. Main and secondary steps of the mode

Each fret level is characterized by a degree of stability. We can say that an unstable sound is a comma in a sentence, and a stable sound is a period. The instability of sound to the ear is expressed in the fact that when it appears, tension arises, which one wants to eliminate by playing a stable sound. Unstable sounds tend to gravitate towards stable sounds, just as objects on the surface of the earth are attracted to the globe. The transition from unstable sound to stable sound is called resolution.

Rice. 6. Stable and unstable fret degrees

The stability and instability of the steps are different. The most stable stage is the 1st stage of the mode (tonic). Stages III and V are significantly less stable. Stages II, IV, VI and VII are unstable. The degree of gravity of unstable steps towards stable ones is different. It depends on:

• degree of stability of sustained sound
• the interval formed between an unstable and a stable sound. Halftone gravity is stronger than tone gravity.

In the case of a natural major, the situation looks like this:

• VII -> I - the strongest gravity, since the I degree (tonic) is the most stable sound in the mode, and the gravity is semitone
• II -> I is stronger than II -> III, since stage I is more stable than III
• semitone gravity IV->III is stronger than tone IV->V
• VI is allowed only in V, since VII is an unstable stage

Stages II and IV have two resolutions, stages VI and VII have one each (Fig. 6). Knowing the relationship between stable and unstable steps allows you to play the simplest motives (the smallest possible meaningful fragment of a melody): first the unstable sound is played, and then it resolves into a stable sound.

In the fret system, sound acquires new characteristics - a degree of stability and functionality. The fret itself is part of the scale system, and the scale is part of the scale system (Fig. 7).

Rice. 7. The relationship between sound and its surrounding systems

Intervals and modes

The interval consists of two stages, so the stability of the interval will depend on the stability of the stages included in it. Tritones have the greatest instability and tension in their sound. In the natural major and minor modes there are two of them: a diminished fifth of the VII-IV degree and an augmented fourth of the IV-VII degree. The interval VII-IV resolves into the major third of I-III, and the interval IV-VII into the minor sixth of III-I.

Intervals can be constructed either from a given note, or in a given key. On the previous page all simple intervals constructed from the step were considered before. In kind C major there are no degrees that correspond to black keys, so in this key impossible(without using chromaticisms - higher or lower steps) build intervals from the step before, which includes a black key. But this does not mean that, for example, there is no small second in natural C major. Such an interval exists and is built from stages III and VII: mi-fa, si-do. Constructing intervals without relative to tonality is an exercise. The ability to construct intervals in tonality provides practical benefits. If chromaticisms cannot be used, then constructing an interval in a key can only be done using diatonic levels.

In music, as already mentioned, everything is relative. The new environment can make additions and adjustments. By now you know that intervals can exist in a scale system (they can be built from any degree of the scale) or in a mode system in a particular key (intervals are built on diatonic degrees). In a scale system, the main characteristic of a harmonic interval is the nature of its sound (consonant and dissonant intervals). In the mode system, intervals acquire a new important property - functionality.

Figure 8 shows two intervals - a perfect fourth and a perfect fifth. The stable steps of the fret are highlighted in green.

Rice. 8. Intervals from the 1st degree of the natural major scale (C major)

Both of these intervals belong to the same group - intervals of perfect consonance and for this reason in the scale system it is impossible to determine which of these intervals sounds more melodic. Than intervals built from step before, differ from the same intervals constructed from the foot re? Nothing, if you do not take into account the range of frequencies included in them.

If we consider these intervals in the mode system, then it is correct to say that they are not built from the degree before, and from the 1st degree of the fret. A perfect fifth is formed by stable scale degrees - I and V, and a pure fourth - by stable and unstable modes - I and IV. In the mode system, intervals formed by stable steps sound, accordingly, less tense. A perfect fifth from the 1st degree sounds more stable (less tense) than a perfect fourth. That is, the mode system made it possible to see the difference between the intervals under consideration.

In Fig. 9. The same intervals are shown, but built from the 2nd degree of the fret.

Rice. 9. Intervals from the second degree of the natural major scale (C major)

Now the fifth is formed by unstable degrees II and IV, and the fourth includes a stable degree (V). Now the fourth sounds more stable than the fifth.

The functionality of intervals is most clearly demonstrated by the example of the minor and major thirds. In the scale system, these intervals relate to imperfect consonance, that is, in terms of the degree of euphony they are inferior to the pure fourth and fifth. However, in the mode system, these intervals determine the type of triad: if the first interval in the triad is a major third, then a major sound is obtained, but if the first interval is a minor third, then a minor sound is obtained. A perfect fourth is not used in a triad, and a perfect fifth does not affect the appearance of the chord. The question arises, which characteristic has greater priority in the mode system - the degree of euphony or functionality? In the mode system, the functionality of the interval is of great importance.

Relationship between keys

From 12 chromatic steps of the octave, you can build a large number of major and minor keys. In practice, they are limited to 15 major and 15 minor keys, guided by the principle of reasonableness. Why complicate something when the same thing can be done simply? No one forbids, for example, using the key of B-sharp major (Fig. 10).

Rice. 10. Key of B-sharp major

Try playing this scale (crosses in front of the steps - double-sharp - increase by 1 tone). Not easy? But in fact this is the key of C major . In C major there are no signs at the key, but in the key of B-sharp major there will be twelve of them.

How to learn to build any key without memorizing them? First, you need to remember the order of the signs on the key. The sharps follow in this order: fa-do-sol-re-la-mi-si. The flats follow in this order: si-mi-la-re-sol-do-fa. Secondly, you need to remember well the tonal structure of the major and minor modes. Thirdly, you need to remember that the major and minor modes are modes consisting of seven diatonic degrees.

Example 2. It is required to construct the key of E major and determine the signs for it. Let us remember that the major mode has the tone structure “b-b-m-b-b-b-m”. After this, we begin building the E major key from the note E:

The key of E major has four sharps: F, C, G, D.

Example 3. It is required to construct the key of E minor and determine the signs for it. Let us remember that the minor scale has the tonal structure “b-m-b-b-m-b-b”. After this, we begin building the E major key from the note E:

In the key of E minor there is one sign - F sharp. Why F sharp and not G flat? Because in the latter case the following will happen:

That is, instead of seven steps, six steps are used, and the minor mode should consist of 7 steps. In addition, in diatonic a transition with skipping a step is unacceptable, and in this example a step is skipped F.

Example 4. It is required to construct the key of F major and determine the signs for it.

In the key of F major there is one sign - B-flat

15 major keys is also a lot. Why do we need so many major keys? From the point of view of construction, all major modes are the same and have the same intervallic composition (b.2, b.2, m.2, b.2, b.2, b.2, m.2). But from the point of view of the nature of the sound, one major mode is different from another major mode. This is easy to check using the formula for determining the frequencies of steps in an evenly tempered tuning F=440*2 i/12. Let's determine what frequency range is contained in one tone between the steps of the first octave la And si,before and re. In the first case, 440*2 0/12 - 440*2 1/12 = 53,8833 Hz, and in the second 440*2 -9/12 - 440*2 -7/12 = 32,0392 Hz The difference is visible to the naked eye. In practice, this means that a song in one major key will sound more fun and brighter than in another major key. The same can be said about any tonality. The transition to another key during transposition does not change the harmony, but leads to a change in the color and character of the sound of the composition. You can check this by comparing the sound of a midi file in different keys.

Musical mode– another concept from music theory that we will become familiar with. Mode in music is a system of relations between stable and unstable sounds and consonances, which works for a certain sound effect.

There are quite a lot of modes in music, now we will consider only the two most common (in European music) - major and minor. You have already heard these names, and you have also heard their banal decodings such as major - a cheerful, life-affirming and joyful mode, and minor - sad, elegiac, soft.

These are only approximate characteristics, but in no case are labels - music in each of the musical modes can express any feelings: for example, tragedy in a major key or some bright feelings in a minor key (you see, it’s the other way around).

Major and minor - the main modes in music

So let's analyze the major and minor modes. The concept of mode is closely related to scales. The major and minor scales consist of seven musical steps (that is, notes) plus the last, eighth step repeats the first.

The difference between major and minor lies precisely in the relationship between the degrees of their scales. These steps are spaced from one another by a distance of either a whole tone or a semitone. In major, these relationships will be as follows: tone-tone semitone tone-tone-tone semitone(easy to remember - 2 tones semitone 3 tones semitone), in minor – tone semitone tone-tone semitone tone-tone(tone semitone 2 tones semitone 2 tones). Let’s look at the picture again and remember:

Now let's look at both musical modes using a specific example. For clarity, let’s build a major and minor scale from the note before.

You can see that there is a significant difference in the notation of major and minor. Play these examples on instruments - you will find a difference in the sound itself. Let me make one small digression: if you do not know how tones and halftones are calculated, then refer to the materials of these articles: and.

Properties of musical modes

Mode in music exists for a reason, it performs certain functions, and one of these functions is regulating the relationship between stable and unstable steps. For major and minor, stable degrees are the first, third and fifth (I, III and V), unstable - the second, fourth, sixth and seventh (II, IV, VI and VII). The melody begins and ends with steady steps if it is written in a major or minor mode. Unstable sounds always tend towards stable sounds.

The first step is of particular importance - it has a name tonic. Stable steps together form tonic triad, this triad is an identifier of a musical mode.

Other musical modes

The major and minor scales in music are not the only options for scales. In addition to them, there are many other modes that are characteristic of certain musical cultures or artificially created by composers. For example, pentatonic scale- a five-step mode in which the role of tonic can be played by any of its steps. The pentatonic scale is extremely widespread in China and Japan.

Let's summarize. We defined the concept, learned the structure of the scales of major and minor modes, and divided the steps of the scales into stable and unstable.

Did you remember that tonic is basic level of musical mode, basic sustained sound? Great! You've done a good job, now you can have a little fun. Look at this cartoon joke.