What is musical harmony definition. History of the musical concept

Music is an integral part of our lives and accompanies us almost everywhere - it sounds on TV and radio, in theater and cinema. People's taste preferences in this matter are different. Some people like classics, while others like hard rock or pop, or a varied combination of them.

Harmony as a science

In any matter there is such a thing as harmony. Many works have been written on this topic in different time periods. The main associations that arise with this word are calmness and tranquility. It can be traced literally in all spheres of human life, philosophical foundations of the universe.

Many cultural and religious principles of various peoples of the world praise and consider it the basis of human life. Harmony with oneself fills life with meaning and creates favorable conditions to develop it and maintain connections with other people.

Music and harmony

Harmony in music is no exception. The harmonious sound of instruments in an orchestra or group, pleasant to the ear motifs that you want to hum and listen to over and over again... The combination of many notes, tones and keys also includes this concept. There is even a whole science that answers the question of what musical harmony is.

It describes and studies certain rules and patterns of comparing notes in different styles and keys from a technical and compositional point of view. Their consonance determines the logical sequence. There are several areas in which the definition of “harmony in music” is applied:

• Musical mode.
• Style.
• Accordica.
• Individual specificity of author's works.

It reveals and denotes a peculiar relationship and interconnection of various rather specific musical and artistic elements and combinations inherent in music.

With all the simultaneous sincerity, eccentricity, and classical construction of the works, they are connected by the highest logic of the combination of sounds. It helps various compositions speak and convey the author’s message to listeners. Without such high organization and compliance with many rules, laws and concepts, numerous world musical masterpieces would not have been born.

History of the musical concept

Music has been around for a long time. People have been studying sounds and their combinations since ancient times. Although the concept itself was somewhat different from the modern one. It contained a deeper philosophical meaning. Therefore, harmony in music was considered as a combination of music and the Universe, cosmic harmony, which should coincide with human soul. There was even a comparison between music and architecture as its static manifestation, in which harmony and coherence of forms, materials and all elements also reigned.

Music in its various manifestations is a unique way of modeling the world, a way of life, which famous and not so famous musicians and composers saw in their own way and tried to embody. It is worth noting that the creation and study of music and its laws is directly related to human speech, its logic, consistency, purity, frequency and intonation.

Chronology of study this concept originates back in Ancient Rome, China, and is gradually gaining momentum in its expansion. This is also related to the study of the concepts of sound, intervals, keys, modes, and modalities. The greatest study and development of the concept of harmony appeared in the Middle Ages, in the mid-16th century, and develops until modern musical and scientific discoveries. As new ones open musical instruments The form of development from a single-voice sound combination to multi-voice music became more complex. And the term “harmony in music” also underwent changes.

Properties of harmony

Notes are transformed into sounds, then tones and chords arise, and a work is born. To a person who does not study carefully music device, it is quite difficult to determine the degree of combination of all elements. The ear is the perception of melody and motive. Tragic, romantic, comedy genre works... Through the close connection of sounds, the mood is conveyed, soul feelings heroes or an author who experiences everything.

For a long time, cinema was accompanied only by music played by musicians, conveying messages through it and revealing additional features actors who worked only with the help of facial expressions. In this vein, we can safely say that the harmony of the soul is music in any of its manifestations.

Manifestation of Harmony

If we talk about its manifestations, they are built on opposites and close connections of a number of elements. They seem to be repeating human nature, in which there is so much coherence and disagreement at the same time. This is what gives musical harmony such inconsistency, complete interconnection and complementation of sounds, tones, chords and modes.

Let's temporarily forget about temperament and remember that I promised you the connection between the overtone scale and harmony. The development of the overtone scale is associated with the development of musical culture generally.
In particular, with the development of the “warehouses” of musical texture (see Warehouse) - from “monody” to “homophony”

If we take professional music into account, then good example monody will be the music of the Middle Ages. At this stage we are dealing with the development of intervals, which occurs from the edges of the overtone scale. For clarity, you can listen to Gregorian chants. In them the melody is given top part overtone row - seconds (of course, major seconds predominate) and occasionally thirds (we are talking about horizontal presentation), and in the vertical section we find: octaves, fifths, fourths - and often in one composition one interval is chosen for parallel doubling of the main voice (usually an octave ), or two (fifth and octave) - and the duplication rarely changes in the process. When there are no duplications, we hear pure monody. Currently, this style of music can be heard in church singing.

In this example there are three duplication options, one in a row. At the end, the lower voice is partially released from the duplication function.

At the beginning of this example, there is still no “relationship of harmonies” in the sense we are looking for, there is no connection of chords or at least different intervals - the second voice duplicates the first at an interval higher - that’s all. Before us is still the same melody, with its own system, only more voluminous due to doubling. It turned out to be one layer. But at the end something more interesting happens, but for now the lower voice is tied to the upper one by rhythm. Surely this fragment was considered the pinnacle of mastery. :)

Then, after a couple of centuries, the second voice received freedom - rhythmic and melodic. And here, on each subsequent note of the melody, different intervals began to appear - from this moment we can talk about connecting intervals, although still within the framework of linear logic.

Further development is difficult to describe in a nutshell, so I’ll go straight to the result: by the 16th century, rules of polyphony appeared, implying freedom and independence of voices within the framework of the rules for connecting intervals (that is, the chord was considered their sum) and melodic lines - by the 18th century this system reached its perfection and was replaced by another. Freedom of voices was achieved by rhythmic non-synchronization (see Counterpoint). (Rhythmic synchronicity was also used, but mainly in church music - in chorale. Hence the term “choral texture”). In the 18th century, the leading one became the homophonic-harmonic type of presentation of the material (the term itself hints that there is already harmony here in the very sense in which we need it) - dividing the texture into 2 layers: melody and accompaniment (complementary, harmonic). The accompaniment took on the function of harmonizing the melody. This system has absorbed some rules of polyphony - they gradually turn into rules for connecting chords in 4-voices. Here, for the first time, the functional connection of chords within one key is noticed.

By the 17th-18th century, chords of the tertian structure were fully mastered - the final capture of the overtone scale took place (that zone where we see successive thirds. Officially, triads and seventh chords appeared. They also appear in the melody (that is, the melody can now move along the sounds chords - in a row, or from one sound to another abruptly, in any direction).

This whole process is also associated with the expansion of the concepts of consonance and dissonance. I also advise you to read about this in elementary theory.

Also in the 18th century, the concept of tonality appeared. And then... Everything became even more complicated and at the same time there was even more freedom for creativity.

Registers

Above we looked at the importance of the overtone scale for harmony and music in general. Now let's look at it as a source of boringly perfect texture. What we see:

The lower register has wide consonantal intervals: octaves, fourths, fifths.

In the middle register is our main harmonic filling of the pieces - in this zone, “one-handed” triads and their inversions sound great (which cannot be said about them in the counter-octave, for example).

The high register is ideal for a melody that moves in a more detailed line, with seconds, embellishments, etc.

That is, if the entire texture is calculated in relation to the overtone series, we get an idealized version, acoustically monolithic.
The naturalness of a particular type of voice movement depends on the choice of register. In the lower register there is naturally a smoother, slower movement, in the high register there is a faster, more fluid movement, and in the middle there is something in between.

Nobody forces you to do this, it’s enough just to know that it is most convenient to perceive information in such placement and matching tempos and the listener’s brain is minimally strained. This placement is normal. They often make all sorts of relaxing things, ambient music for meditation, etc. In relaxation ambient music, everything sounds like this - there is a leisurely bass, there is a shimmering middle, there are some shimmers (often on oberons) in the high register. This option tells us about a natural state of peace, about complete and final ordinariness. The unusual begins where movement begins away from this position and rhythms - and this is where music begins, informing us of states different from rest. It turns out that all other uses of registers tell us about movement ( characters, objects or abstractions), anxiety, threat and other states.

For example, dense triads in a counter-octave, despite the feeling of harmony, will carry a threat or epic force ( minor chords will threaten, and the major ones will say, “Calm down, these are ours”), since each of the bass tones will give a number of overtones, and they, in turn, will give a feeling of dirt, dissonance, uncontrolled hidden movement (due to the resulting beats in the overtones). And the very intensification of sound in this zone has not promised us anything calm since the Paleolithic times, as well as the squeak of a violin or piccolo flute frozen on one note in the 4th octave.

The fast-moving bass also clearly carries an active character, an imperative. Typical for dance music. Also typical for showing the movement of large objects.

Let's summarize:
● There is a relationship between the pitch of a sound and the size of the intended source or carrier of the sound. In this sense, high-pitched sounds are more likely to be associated with small objects, and low-pitched sounds with large ones.
● Also, the pitch of a sound can actually be associated with the height of the object.
● Also, the pitch of the sound can be associated with the importance of an object or an ongoing voiced event. The lower, the more important, the more significant.
● There are types of object movement environment, which since ancient times inspire peace in a person - in the lower register nothing moves or moves slightly, in the middle - something slowly moves (a herd of sheep, for example), in the upper register - trees rustle, birds chirp, the wind howls.
● A rhythmic element placed in a certain register - bass, mid or high - will be evaluated by the brain according to its kinetic potential in the context of the register. Depending on the conditions, the sound will be accepted as the movement of an external object, or as a manifestation of its own activity (for example, in dance music The rhythm of the kick drum is associated with the movement of the legs - steps are perceived by the human senses as a blow, and from the music we transfer the movement to ourselves).

Check out this cute scene of the changing of the seasons in the first Harry Potter film.

At the beginning there is a scene of winter and serenity - the bass in the lower register stands still, the harmonies of T S slowly alternate for some time above, and a smooth melody sounds above all this.
Then (10th second) an active joyful fragment on the same harmonies (T S): lower case not busy, the middle one is occupied by moving chords (on A-flat of the small octave), and at the top it is frozen high note- E-flat 3 octaves. In total, this gave the promise of further movement.
At 18 seconds the promise is fulfilled - the music unfolds, occupying all registers. The bass moves, supporting magical shifts in harmonies that take us into new keys. The smooth melody of the horns moves in the middle register, more active figurations (shorter durations) are in the strings in the high register. The figures of the strings have something in common with the image of flight (what we see in the frame), this feeling is reinforced by the “flying up” passages of the harp and the strings that either gain height or decrease it. The sonorous bass in the lower register gives us a hint that the changes taking place are important, significant and “solid”. The image of change is mainly shown by movement through different tonalities.

Tuning by overtones, diagrams musical setting, harmony of intervals,
12 and 19 semitone scale - comparison.

"Pythagoras gave such a clear preference to strings
instruments that warned his students against
allowing the ears to listen to the sounds of the flute and
dulcimer. He further argued that the soul must be
purified from irrational influences by solemn singing,
which should be accompanied on the lyre."
link below

Dear readers. We will talk about harmony. Writing poetry, like singing or reciting it, is an exclusively harmonic activity. Therefore, in this article there is a little about harmony in music.
What lines of poetry, what sounds do we call harmonious? We know which ones - those that delight our ears. And if you look more closely, they have a certain symmetry. Rhythm and rhyme in versification, rhythm and harmony of overtones in music. We will talk about overtones.
Take a look at the picture. Imagine that the upper and lower long sections are strings of the same length, and stretched in such a way that they produce a sound up to the first octave in their normal sound. That is, when the middles of these strings vibrate up and down as a whole.
But they can sound differently - making the vibrations shown in the figure. The sound of these vibrations is higher, this is the sound of overtones, the third and fourth (we count by the number of antinodes). Stretched with the same force, but shorter strings with the sounds F and G can create the same sounding overtones. And therefore, we perceive the simultaneous sound of the notes Do and Sol, or Do and Fa, as harmonious.

If the harmony of the sound of strings was noticed a very long time ago, then its detailed development is associated with the name of Pythagoras. He is believed to be responsible for the creation of the diatonic series - a sequence of seven octave sounds arranged in harmony with the ratios of the numbers 1 2 3 and 4.
You can find a colorful (albeit not entirely correct from a physicist’s point of view) description of Pythagoras’ experiments here:
Pythagorean theory of music and color.

Read and enjoy. Because we will now begin a rather boring task - tuning the piano. Let me explain why I took up this matter in the first place.
I made a piano using computer keys. With the sound of keys, a mnemonic recording of the resulting sound, with the ability to edit, listen and store the notes of this recording in a regular text file. Since the recording uses note symbols familiar to musicians, such as C D E F G A B for notes of the first octave and the same letters, only lowercase, for notes in the second octave, reading and editing notes musical text It can also happen in a text file. If you open it, for example, using the Notepad program. It turned out, in my opinion, convenient and interesting. I will talk specifically about this program in the next article.
I was faced with the task of tuning a computer piano. Taking the frequencies of an evenly tempered series seemed too simple to me. And I began to understand tuning schemes.
You can view the setup details at this link -
www.tuning-piano-Kharkov.com

We will not create an evenly tempered tuning, but first of all we will turn to the classic tuning scheme of 2/3 or “down by fifths”. Usually this is the setting. I'll explain why.
A fifth is an interval of 5 tones, the upper degree of which, called the dominant, is known for its sensation of resolution into the tonic - the lower degree of the fifth. From A to Re.
This is where the setup begins. The frequency of the note A of the first octave is taken as the standard; it is equal to 440 Hz.
We multiply this frequency by 2 and divide by 3 (according to the 2/3 scheme), we find Re - a “fifth down” from the sound A.
And then we continue in the same spirit. If the frequency we obtain turns out to be lower than the frequency of A/2, then we multiply the frequency by 2, returning it closer to the original A. We do this mathematically, and the tuner does it all based on his hearing.
And now - a miracle. After 12 cycles we return to the sound A. Almost exactly, but not quite. We get not 440 Hz, but 434.078 Hz. The frequency difference (in our case, about 6 Hz) is called the Pythagorean comma.
Pythagoras did approximately the same thing, but somewhat differently, in his calculations. About the Pythagorean system, see here - http://www.px-pict.com/7/3/2/1/8/1.html

Trifle or not trifle - Pythagorean comma, can it be neglected?
6 Hz from a frequency of 440 Hz is 6/440 = 0.002348 that is, rounded, 0.23%. While the frequency rises from the note A to the next semitone (A sharp), increasing by 5.35% (I provide a table with frequencies below).
5.35/0.23=23.3 and, it would seem, the coma can be neglected.
But let's look at the intervals, looking ahead a little to the frequency table. Calculations show that the interval between A and A# is, as already mentioned, 5.53%. But this is not the case everywhere; the table also often shows intervals between semitones of 6.79%, for example, for the same A-A# interval, but for tuning in fourths. When moving along the chromatic scale, the frequency goes up and down from the midline.
This “out-of-tuning” is avoided by additionally slightly tightening each string when tuning, and obtaining an evenly tempered scale. In this series, the sound rises not by 5.35% and not by 6.79% per semitone, but by 5.94%.
Why did I put the word "upset" in quotation marks? Because it is precisely these changing intervals that enter integral part at other intervals of different keys, give originality to the sound of these keys. I’ll refer you to a wonderful description of the beauties of tonalities, here -
http://www.forumklassika.ru/showthread.php?t=90809
Figurative perception of “light” and “dark” tones in the 17th-20th centuries

I will quote some descriptions related to keys with a perfect dominant and subdominant (for example, C major) and to keys with a broken subdominant (E major), and with a broken dominant, and at the same time with many other intervals, with a perfect subdominant (A major).

C-dur “Completely pure. Innocence, purity, naivety, the voice of a child”
“Cheerful and militant”
“Joyful and pure”
“The state of nature, virgin purity and chastity, the charming innocence of youth”
“Easiness and nobility”
“Joyful and pure; innocence and simplicity"
“Simple, unadorned”
“Completely clean”
“Clean, confident, decisive; expressing innocence, firm resolve, manly seriousness and deep religious feeling.”
“Bold, energetic, commanding, suitable for war or enterprise”
“Pure, confident, decisive, full of innocence, seriousness, deep religious feelings”; “Simple, naive, frank” “White”
“Red” (Scriabin)

E-major “Bright exclamations of joy, joyful laughter, and at the same time unfinished, incomplete pleasure sound in E major”
“Uplifting”; "Bright"; “Radiating light, warm, joyful”
“Delight, splendor, brilliance; the brightest and most powerful tonality”
“Bright and clean, perfect for anything shiny”
“Expresses despair or mortal sadness with incomparable completeness; he is especially suitable in those extreme love affairs, where nothing can be helped and there is nothing to hope for, and under certain circumstances has in itself something so piercingly-farewell-sadly-penetrating that it can only be compared with the fatal separation of soul and body” (I. Matteson)
“Lush green, spring in full bloom. Swaying branches decorated with fresh foliage” (Riemann, KhTK) “Blue-whitish”
“Blue, sapphire, brilliant, night, dark azure”
“Night, very starry sky, very deep, promising”
=== pay attention to “unfinished, incomplete” - isn’t this an echo of the imperfection of the subdominant? D.M.

A-dur “This key is an innocent declaration of love, satisfaction with the state of affairs; hope to see your beloved again after separation; youthful vivacity and faith in God”
“Joyful and pastoral”; “Golden, warm and sunny”;
“Must have a very strong impact; he is brilliant, but more inclined to express sad and mournful feelings than to have fun.”
“The most beautiful, modest, kind, soft, delicate, gentle, devoid of sharp insolence La. Each author noticed the charm of this tonality, reserving it for the most sincere feelings”
“Filled with faith and hope, radiates love and simple, genuine joy, surpasses all other tones in expressing sincere feelings”
“Fresh and healthy, vital”; “Sincere, sonorous” “Green”
“Clear, spring, pink; this is the color of eternal youth, eternal youth"
“More of a joyful, intoxicating mood than a luminous sensation, but as such approaches D major”
=== end of quotation D.M.

So, these “errors” of harmony and violations of intervals ultimately turn out to be unusually expressive in music. But isn’t it the same in poetry, where deviations from the norm create unique semantic and emotional accents.
Equally tempered tuning is called bland and soulless. But isn’t this how things are in poetry, in which a verse that is completely smooth in rhythm sometimes turns out to be just as disgusting as a clumsy verse?

Now I want to demonstrate the computer calculation of the 2/3 setting as I received it from the computer:
1 A lja 1 440
2 D re .6666666865348816 293.3333435058594
3 G sol .8888888955116272 391.1111145019531
4 C do .5925925970077515 260.7407531738281
5 F fa .790123462677002 347.6543273925781
6 - lja# .5267489552497864 231.7695465087891
7 - re# .7023319602012634 309.0260620117188
8 - sol# .9364426136016846 412.0347595214844
9 - do# .6242950558662415 274.6898498535156
10 - fa# .8323934078216553 366.2531433105469
11 B si .5549289584159851 244.1687622070312
12 E mi .7399052977561951 325.558349609375
13 A lja .9865403771400452 434.0777893066406
\ in this column are relative frequency heights, in the next - frequencies

Exactly the same calculation can be carried out according to the 3/4 scheme - multiply by 3, divide by 4
The difference is in the order of tones - instead of ADGCF, etc. - AEB, etc. only not 434.077 for A at the output, but 446.003 - that is, the same comma, but already positive. This setting is called “down by fourths” or “up by fifths”, which is the same thing:

1 A lja 1 440
2 E mi .75 330
3 B si .5625 247.5
4 - fa# .84375 371.25
5 - do# .6328125 278.4375
6 - sol# .94921875 417.65625
7 - re# .7119140625 313.2421875
8 - lja# .533935546875 234.931640625
9 F fa .8009033203125 352.3974609375
10 C do .600677490234375 264.298095703125
11 G sol .9010162353515625 396.4471435546875
12 D re .6757621765136719 297.3353576660156
13 A lja 1.013643264770508 446.0030517578125

It would be possible not to show this picture, but it curves very beautifully on decimal numbers. This is because we do not divide by 3 (which is not very consistent with the 10-digit counting system) but by 4.

Let's go further, 2/3, 3/4 - it is clear that these numbers mean the ratio of the number of overtones on the arching strings, and why not go further and try other overtones in the form of simple fractions and other tuning schemes? What if it comes together? This is me in relation to the comma))
Let's supplement the Pythagorean series with the number 5.

Calculations show that for a ratio of 4/5 there is no chance of convergence in a reasonable number of cycles. The same as for the number 7, if you try to include it in the Pythagorean series. But for 3/5 and 5/6 it turns out to be quite a decent tuning of 19 semitones. And I want to discuss it, along with other settings.
And also my impressions of implementing all five settings on the computer keys.
These 5 settings are the following - p-t (equally tempered scale) and patterns 2/3, 3/4, 3/5, 5/6.

TABLE 1
All frequencies of different settings, Hz (first octave)

Equal t. 2/3 3/4 3/5 5/6 3/5-Hg. 2/3-Hg Pythagorean harmonious
difference pitch intervals for 3/5
frequencies, Hz overtones, deviation
-1- -2- -3- -4- -5- -6- -7- -8- -9-
261.63 260.74 264.29 264.00 264.42 to 2.37 -0.89 260.74 (2/3)
277.17 274.68 278.43 273.71 274.16 ==# 32/31 0.438%
277.17 274.68 278.43 283.78 284.24 ==b 274.69 (2/3)fr.
293.66 293.33 297.33 294.23 294.71 re 0.57 -0.33 293.33 (2/3) 10/9 0.31%
311.13 309.02 313.24 305.05 305.55 ==#
311.13 309.02 313.24 316.80 317.31 ==b 309.03 (2/3)fr. 6/5 0%
329.63 325.55 330.00 328.45 328.99 mi -1.18 -4.08 330 (3/4) 5/4 -0.47%
329.63 325.55 330.00 340.54 341.09 fa b
349.23 347.65 352.39 353.07 353.65 fa 3.84 -1.58 347.65 (2/3) 4/3 0.30%
369.98 366.25 371.25 366.07 366.66 ==#
369.98 366.25 371.25 380.16 380.77 ==b
392.00 391.11 396.44 394.14 394.79 salt 2.14 -0.89 391.11 (2/3) 3/2 -0.47%
415.29 412.03 417.65 408.65 409.31 ==#
415.29 412.03 417.65 423.69 424.38 ==b 412.03 (2/3)fr. 8/5 0.31%
440.00 440.00 440.00 440.00 440.00 la 0 0 440 5/3 0%
466.16 463.53 469.86 456.19 456.93 ==#
466.16 463.53 469.86 472.98 473.74 ==b 463.54 (2/3)fr. 9/5 -0.47%
493.88 488.33 495.00 490.38 491.18 si -3.5 -5.55 495 (3/4)
493.88 488.33 495.00 508.43 509.25 to b 31/16 -0.60%
fr. - Phrygian scale
=======================================

Pythagorean scale - diatonic scale do - re - mi - fa - sol - la - si - do
Phrygian scale do - reb - mib - fa - sol - lab - сib - do
calculations of the frequencies of these scales were carried out based on the materials of the article
http://www.px-pict.com/7/3/2/1/8/1.html

First of all, I want to congratulate you and myself, the tuning according to the 3/5 and 5/6 schemes converged and demonstrated at the 19th step a comma of not even 6 Hz, but only 0.7 Hz, completely competing with equal temperament. And although with such uniformity of tuning there is no expected difference in the sound of keys, the subtleties of melody are added, due to the new sounds that have appeared - 7 flats from each step of the diatonic series, in addition to the already existing 5 sharps. The enharmonism of sounds on sharps and flats has diverged. Now the sound F sharp is different from the new sound G flat.

I would like to draw your attention to the fact that the Pythagorean scale, together with the Phrygian scale, fits perfectly into the classical 2/3 tuning, with the exception of the two sounds Mi and Si, which can be classified as 3/4 tuning.
Why did the Pythagoreans do this with these particular sounds?
To answer this question, let's turn to the 7th column of the table. It shows the difference in frequencies between the 2/3 tuning and the equal tempered tuning. We see that it is for Mi and Si that the difference in Hertz is noticeable. And the tuning of 3/4 of these notes is closer to equal temperament. That’s why the Greeks took the frequencies of the 3/4 circuit for Mi and Si. They strived not only for harmony, but also for uniformity of sound.

But what about the uniformity of the diatonic scale in 19-semitone tuning? We will not distinguish between settings 3/5 and 5/6, since the difference in frequencies for them is negligible, not the same as between 2/3 and 3/4.
*** By the way! If you tune the piano not classically - “down by fifths” but by fourths, then all that magnificent description of the imagery of musical tonalities that you could read above will shift 2.5 tones lower - the attributed C-dur will have to be attributed to G-dur. The tonality of C-dur will be colored in the colors of F-dur.

The diagram demonstrates the alternation of semitone intervals in an octave, indicated by _ 5.35% - 6.79% Hz/semitone
La
| ? ? - this is Mi
|_-_ _-_-_ -_ _-| |_-_ _-_-_-_ _-_-_ _-_-_-_ _-| 2/3 tuning from E in fifths
|_-_ _-_-_ _- _-| |_-_ _-_-_ _-_-| 3/4 is equivalent to tuning from A to fourths
| |

It is clear from the diagram that if you trace the alternation of intervals in the 2/3 tuning in 2 octaves, then you will find a place that coincides in the alternation of intervals with the octave of the 3/4 tuning

Let's return to the diatonic scale in 19 semitone tuning and look at column 6 of the table. Do, Sol, Si and, especially, Fa stick out.
The sound of the scale is unusual, although the resolution to the tonic is clearly felt.
You can read about scales on Wikipedia, in particular about the whole tone scale, which can be implemented as a scale in 12 semitone tuning. Tuning at 19, having an odd number of semitones, does not allow this. But you can try other scales.

Diatonic scale in normal tuning, number of semitone steps shown (sum=12)
| 2 2 1 2 2 2 1 | if from Do, then this is the usual “do-re-mi-fa-sol-la-si-do”

Diatonic scales in 3/5 tuning (19). I found two acceptable scales.
| 3 3 1 4 3 3 2 | - “C major on white keys” with F replaced by Fab to improve harmony, I tried it, it sounds almost familiar.
and from D flat - not all on flats, but with penetration into sharps, I tried it, it also sounds good.

| 3 3 2 3 3 3 2 | - from D#, sounds more uniform in the middle of the scale, but with worse resolution in the tonic.
the same scale from C - in execution "regular C major on white keys"

One important remark must be made here. Diatonic scale in classic setting turns out to be harmoniously unified. That is, the sounds included in it follow the tuning sequence. This corresponds to the fact that the major diatonic scale occupies a continuous sector on the fourth-fifth circle.
For a 19-half-tone setting, you can also build a similar circle, reflecting the mutual harmony of its neighbors. However, this will be harmony not in fourths and fifths, but in minor thirds and sixths. Seven close neighbors on this circle can also be identified, and among them there are those that form the dominant and tonic (with a frequency ratio of 3/2), but the subdominant is far from this continuous sector. The notes of this continuous sector in the form of a scale sound very interesting, with an oriental motif, and are safely resolved into the tonic.
The scheme of this fret is as follows - | 1 4 1 4 1 4 4 |
In the article to which I refer just below, the six steps of this mode are located in consonant minima (Fig. 14a) in pairs, while the last step falls on the dissonance and gravitates strongly toward the first step.
The degrees are unevenly distributed across the octave - three close pairs and one note distant from these pairs. But those uniform diatonic scales, examples of which were given above, turn out to be broken in a circle of tuning 3/5, although they contain intervals of classical tuning.
All this constitutes a very interesting object for the study of harmony.
It must be said that people are engaged in such research. In the practical sphere, (you can look at the search engine) - who tunes a guitar by thirds, who discusses temperament by thirds (not as deeply as we do, but in the form of a potential attempt), who tunes a piano by thirds, tightening the strings and listening to the number of beats ( to obtain an evenly tempered series, but we do the opposite - so that there are no beats on the thirds). A discussion of the possibilities of constructing a scale other than a scale with 12 semitones is also ongoing on the Internet - http://forum.buza.su/viewtopic.php?f=54&t=2165

After writing this article, I found something that I advise those seriously interested in this problem to read, there is also something about the 19-tone scale - http://unism.narod.ru/arc/2006gs/gs.pdf with many drawings, formulas and explanations.
But the proposed keyboard seems unsuccessful and inconvenient to me. But the computer keyboard turned out to be very suitable for this purpose. In addition, the proposed alteration signs may be more correct from the point of view music theory, in my opinion, are completely inconvenient in practical terms - two sharp marks (increased by 1 semitone and raised by 2 semitones) and two flat marks. Who will look at them?))

Let's continue. After the described experiments, it is interesting to turn to the next topic - how are things going with the harmony of intervals in 12 semitone and 19 semitone settings? For comparison, let's take the p-temperature settings. and 3/5 - both are uniform, and therefore we will compare them. Due to the uniformity of the settings, it is enough to compare intervals in the same key - C major. Moreover, all intervals for comparison can be taken in relation to a single note - C. Due to the uniformity of settings, any interval shifted in semitones by any number of semitones will not change its harmonic quality.

We will search for harmonious intervals and evaluate their quality as follows.
We take two prime numbers, starting with small ones, and form them into a simple fraction, greater than one, but less than two. The first similar fraction is 3/2 = 1.5, multiply this number by the frequency of the lower sound C.
For a 3/5 setting it's 264Hz. 264 x 1.5 = 396 Hz We look in the table for the frequency closest to this value. We find 394.14 Hz note Sol. This is an approximate dominant. Let's evaluate its quality.
Divide 394.14/264 = 1.49295, but it should be 1.5. What is the percentage difference?
Divide 1.49295/1.5=0.995303, subtract one and multiply by 100. We get -0.4697, round up => -0.47%
So, in the 3/5 setting, the imperfection of the dominant, expressed as the fraction 3/2, is -0.47%

We select the following suitable prime numbers. See table.
We see that in the 5/3 tuning, the 4/3 subdominant on Fa is not completely perfect.
Let's continue the process. Note that fractions with the number 7 do not give harmonious intervals, while fractions 5/3 and 6/5 give intervals that are ideal in harmony. As expected in a setting with the number 5.

With 19 semitone tuning, the dominant and subdominant look worse, but the harmony of other intervals improves. Let's see this in the table:

TABLE 2 Deviations of harmonic intervals from the ideal value
for equal temperament tuning and 3/5 tuning by 19 semitones
p-temp. 3/5
C-D small whole tone 10/9 1.02% small whole tone 10/9 0.31%
do-mib minor third 6/5 0.90% minor third 6/5 0%
do-mi big third 5/4 0.79% major third 5/4 -0.47%
do-fa quart 4/3 0.11% quart 4/3 0.30% subdominant
C fifth 3/2 -0.11% fifth 3/2 -0.47% dominant
do-lab small sixth 8/5 -0.79% small sixth 8/5 0.31%
do-la sexta 5/3 0.91% sexta 5/3 0%
do-sib minor septima 9/5 -1.01% minor septima 9/5 -0.47%

Average absolute value 0.705% 0.291%, that is, 2.4 times less

Yes, we saw that despite the relatively less perfection of the dominant and subdominant, in general, the intervals in the 19-semitone tuning turn out to be almost 2.5 times more perfect than the intervals of the equal-tempered scale in terms of harmonic consonance. But will the existing imperfections be perceived keen ear. Let's give some clarification on this matter.

The perception of frequency by ear depends on the duration of the presented sound.
According to the formula
cos(x)+cos(y) = 2·cos((x+y)/2)·cos((x-y)/2) - the sound of two close frequencies is perceived as the sound of a medium frequency, accompanied by amplitude beats. The frequency of the beats is the half-difference of frequencies, and the duration of one beat is the period during which the sound increases from zero and then decreases. This period is equal to 1/(x-y) in seconds if x and y are frequencies in Hz. Thus, the difference from the ideal frequency at a detuning of 2 Hz is perceived only when playing quarter notes (sound duration 0.5 seconds) or when playing at a slower tempo.
A detuning of 0.5 Hz or less can be considered insignificant, since it can only be noticed when notes of a whole duration are sounded.
The Pythagorean comma for classical piano tuning is 6 Hz on the A note of the first octave, this is 13.6% of the reference frequency (and it is quite audible). And it will be heard even more in the second octave, where on the note A it will be 12 Hz. Therefore, false tuning, or its imperfection, is more audible in the upper octaves.
2 Hz in relation to 440 Hz is 0.45%. Tuning imperfections of this magnitude will be noticeable when playing notes in the first octave at a tempo of 2 notes per second (1/4th note) or slower.
Yes, the 0.47% imperfection of the dominant and subdominant in 3/5 tuning will be noticeable in the first octave when playing quarter notes, but let's take a look at the key of A major in classical tuning. What did they say about her? “Every author noticed the charm of this tonality, saving it for the most sincere feelings.” And the dominant in it is broken - 1.35%

It's not about harmony. But the point is how composers will be able to extract divine sounds and give them expressiveness, using the features of the harmony of musical instruments that these instruments have.
The same applies to poetry.

Thank you for your attention.
======================== 17.01.2015

I express my gratitude to Nikita Skiba, a 3rd year student of the Moscow Conservatory in the viola class, for the interest he showed in the sound coming from the computer, which inspired me for all this activity, including the creation of a program, the conveniences and features of which we discussed together.

The article can be downloaded from here - https://yadi.sk/i/umsaM6J-e6Sbp Word format
========

P.S. Dear reader, if you don’t understand why I was so enthusiastic about this work and laid out a lot of numbers in the article, maybe. and boring ones, I'll explain. Since the time of Pythagoras, we have been using a 12-semitone scale to tune our musical instruments, either in a “pure” form or in a form adjusted by equal temperament. However, as it turned out, there is another scale with 19 sounds.
It is quite harmonious, and an order of magnitude more uniform than the traditional one. In addition, as it turned out, other similar scales (except 12 and 19) with a reasonable number of tones do not exist in nature.

You can object that since the time of Pythagoras, many have practiced harmonies, and if there was anything worthwhile in 19 sounds, then this harmonic series would certainly have been discovered.
But I will also object. Since the time of Pythagoras, musicians have tuned their instruments not just by ear, but, thanks to Pythagoras, according to his system. And bearing in mind the sufficient scholasticism of the Middle Ages, it is difficult to imagine that anything could change, and musical material a significant amount has already been developed in 12 tones. Why did you have to leave him? The 19-tone sound series is currently, apparently, known only theoretically - http://unism.narod.ru/arc/2006gs/gs.pdf and this information appeared relatively recently.

A musical composition consists of several components - rhythm, melody, harmony.

Moreover, if rhythm and melody are like a single whole, then harmony is what decorates any musical composition, the stuff that makes up the accompaniment you dream of playing on the piano or guitar.

Musical harmony is a set of chords, without which not a single song or piece will be complete, full-sounding.

Properly chosen harmony caresses the ear, ennobles the sound, allowing us to fully enjoy the wonderful sounds of a piano, guitar or instrumental ensemble. The melody can be sung, the harmony can only be played. (By the way, you can also sing harmony, but not for one person, but for at least three, provided that they can sing - this is what choir and vocal ensemble artists are trained to do).

A play or song without harmony is like an uncolored picture in books for children - it is drawn, but there is no color, no tints, no brightness. That is why violinists, cellists, domrists, and balalaika players play accompanied by an accompanist - unlike these instruments, you can play a chord on the piano. Well, or play domra or flute in an ensemble or orchestra, where chords are created due to the number of instruments.

IN music schools, colleges and conservatories there is a special discipline - harmony, where students study all the chords existing in music theory, learn to apply them in practice, and even solve harmony problems.

I will not delve into the jungle of theory, but will tell you about the most popular chords used in modern compositions. Often they are the same. There is a certain block of chords that wander from one song to another. Accordingly, a lot of musical works can be performed on one such block.

First, we determine the tonic (the main note in musical composition) and remember - together with the tonic, the subdominant and the dominant. We take a scale step and build a triad from it (one note at a time). Very often they are enough to play a simple piece. But not always. So, in addition to the triads of the main steps, triads of the 3rd, 2nd and 6th steps are used. Less often – 7th. Let me explain with an example in the key of C major.

Examples of chord progressions

I put the chords in descending order of their popularity:

C major

• C major, F major, G major (these are the main triads of the mode);
• Li minor (this is nothing more than a triad of the 6th degree);
• E major, less often - E minor (triads of the 3rd degree);
• D minor (2nd degree);
• si – diminished triad of the 7th degree.

And this is another option for using the 6th degree triad in musical compositions:

But the fact is that these musical harmonies are characteristic only if the note DO is taken as the tonic. If suddenly the key of C major is inconvenient for you, or the piece sounds, say, in D major, we simply shift the entire block and get the following chords.

D major

• D major, G major, A major (1st, 4th, 5th steps - main triads)
• B minor (6th degree triad)
• F# major (3rd degree triad)
• E minor (2nd degree)
• to # reduced 7th stage.

For your convenience, I will show the block in minor key, slightly different degrees are popular there and it can no longer be said that chords of the 3rd and 2nd degrees are rarely used. Not that rare.

La Minor

A standard set of chords in A minor looks like this

Well, in addition to the standard ones - 1, 4 and 5 steps - the base of any key, the following harmonies are used:

• A minor, D minor, E major (main);
• E seventh chord (related to E major, often used)
• F major (6th degree triad);
• C major (3rd degree triad);
• G major (2nd degree triad);
• A major or A seventh chord (the major of the same name is often used as a kind of transitional chord).

How to find tonic

A question that torments many. How to determine the tonic, that is, the main tonality from which you need to start when searching for chords. Let me explain - you need to sing or play a melody. The note it ends on is the tonic. And we determine the mode (major or minor) only by ear. But it must be said that in music it often happens that a song begins in one key and ends in another, and it can be extremely difficult to decide on the tonic.

Only hearing, musical intuition and knowledge of theory will help here. Often the completion of a poetic text coincides with the completion of a musical text. Tonic is always something stable, affirming, unshakable. Once the tonic has been determined, it is already possible to select musical harmonies based on the given formulas.

Well, the last thing I would like to say. The flight of creative inspiration of a composer can be unpredictable - seemingly completely unpredictable chords sound harmonious and beautiful. This is already aerobatics. If only the main steps of a scale are used in a musical composition, then this is called a “simple accompaniment.” It is really simple - even a beginner can pick them up with basic knowledge. But more complex musical harmonies are closer to professionalism. That’s why it’s called “picking” chords for a song. So, to summarize:

1. We determine the tonic, and for this we play or hum a melody and look for the main note.
2. We build triads from all degrees of the scale and try to remember them
3. We play chords in the blocks indicated above - that is, standard chords
4. We sing (or play) a melody and “pick” a chord by ear so that they create a harmonious and beautiful sound. We start from the main steps; if they are not suitable, we “feel” for other triads.
5. We rehearse the song and enjoy our own performance.

As a tip, it’s convenient to select musical harmonies along with the sound of the original on music center, computer or tape recorder. Listen to it several times, and then take a fragment, say 1 verse, and pause it, play it on the piano. Go for it. Selecting musical harmonies is a matter of practice.

Harmony in music (Greek, from harmozo - to put in order) - musical agreement, euphony. Harmony or chord is the combination of three or more different sounds in thirds (see Chord). Harmony or harmony is the part of musical grammar devoted to the construction of chords, their connection, presentation of tonalities, modulations, as well as the study of intervals, consonances, and dissonances. The word G. among the Greeks meant consonance, i.e. interval. Although composers of the 15th century, for example, Josquin des Pres, appeared combinations similar to our simple harmonic addition, they were considered uniform counterpoint (counterpoint of the first category - note against note). With the development of homophony in XVII century G.'s development in the sense of chords begins. Agostino Agazzari already has a bass with numbers indicating chords. In this century the general bass arose. In the 18th century, the creator of a new harmonic system was Rameau, who established the law of tertian addition of chords, and mainly their inversion. F. Marpurg and Kirnberger, theorists of the same century, can be considered followers of Rameau's theoretical views. Thanks to them, the path followed further development G., was scheduled.
Soloviev.

encyclopedic Dictionary F. Brockhaus and I.A. Efron. - S.-Pb.: Brockhaus-Efron. 1890-1907 .

See what “Harmony in Music” is in other dictionaries:

HARMONY, expressive means of music based on the combination of tones into consonances (see CONSONANCE) and on the connection of consonances in their sequential movement. The main type of consonance is a chord (see CHORD). Harmony is built according to certain laws of harmony (see... ... encyclopedic Dictionary

This term has other meanings, see Harmony (meanings). Harmony (ancient Greek ἁρμονία connection, order; structure, harmony; coherence, proportionality, harmony) is a complex of concepts in music theory. It is called harmonious (including... Wikipedia

This term has other meanings, see Harmony (meanings). Harmony is a global principle of harmonizing disparate and even opposing, conflicting elements, bringing them into a single whole. In philosophy, harmony is a category... Wikipedia

Music of the spheres, antique. the doctrine of music. the sound of the planets (including the Sun and Moon and planetary “spheres”) within the framework of the geocentric ideas of Eudoxus, Ptolemy and others (astronomy before Eudoxus did not know spheres, Plato speaks of “circles”, Aristotle... ... Philosophical Encyclopedia

Harmony of the world, 1806 Harmony of the spheres, harmony of the world (Greek ἁρμονία ἐν κόσμῳ, ἡ τοῦ παντὸς ἁρμονία; lat. harmonia mundi, harmonia universitatis, etc.), world music(lat... Wikipedia

Harmony, expressive means of music based on unification musical sounds in consonance and succession of consonances in terms of mode and tonality. The most important in G. are chords of consonance, the sounds of which are located or can be ... ... Big Soviet encyclopedia

- (Greek harmonia, from harmoso to put in order). 1) musical consonance, coordination, the study of the relationships between intervals, scales, chords, modulations, etc. 2) proportionality of parts with the whole and among themselves in works of art … Dictionary foreign words Russian language

- (Greek harmonia - connection, proportionality) consonance, agreement, corresponding aesthetic laws coherence of parts in a dissected whole. The idea of ​​harmony was still at the basis of the Pythagorean idea of ​​harmony of the spheres, it continues to exist in... ... Philosophical Encyclopedia

1. HARMONY, and; and. [from Greek harmonia connection, consonance, proportionality] 1. Expressive means of music based on combining tones into consonances and on their relationship and sequence; section of music theory and academic subject, studying these... ... encyclopedic Dictionary

HARMONY, the area of ​​expressive means of music based on the combination of tones into harmonies and the connection of harmonies in their sequential movement. Harmony is built according to certain laws of mode in polyphonic music of any kind of homophony,... ... Modern encyclopedia

Books

• Harmony in pop and jazz music CD, Peterson A., Ershov M.. This manual is addressed to students of higher and secondary specialized music educational institutions, as well as to a wide range of musicians. First of all, performers on keyboard instruments...
• Harmony in pop and jazz music. Study guide (+CD), Peterson Alexander Valterovich, Ershov Maxim Viktorovich. This manual is addressed to students of higher and secondary specialized music educational institutions, as well as a wide range of musicians. First of all, performers on keyboard instruments...