Primary Music Theory

The art
Any harmony is a manifestation of the world unity. Therefore art can be defined as a specific attempt to express the world unity by certain means (poetry, paints, musical sounds, etc.). The failure of any attempt to formalize the aesthetics, caused by the necessity to perceive the unity of the world firstly.

White keys on the piano are marked with Latin letters A(la), B(si), C(do), D(re), E(mi), F(fa), G(sol), black keys are called altered notes and marked with accidentals. The sharp (#) denotes a note that has been raised in pitch by one chromatic half step. The flat (b) denotes a note that has been lowered in pitch by one chromatic half step. So, we have A, A#, B, C, C#, D, D#, E, F, F#, G, G#, A or A, Ab, G, Gb, F, E, Eb, D, Db, C, B, Bb, A.
A# or Bb (enharmonics)  is the same note. It is marked as A# in sharp keys and Bb in flat keys.

A diminished scale may be of two kinds - a symmetrical diminished scale (half-step, whole-step, half-step, whole-step, etc�) and an inverse symmetrical diminished (whole-step, half-step, whole-step, half-step, etc�).

A whole-step scale is a scale in which each note is separated from its neighbors by the interval of a whole step. It has only 6 notes.

Only main sounds of the musical system are presented in the diatonic range of tones. 
They form specific diatonic degrees inside every octave part. The degrees have definite interval correlation depending on the subkind of the diatonic scale (major, minor).

The scheme shows that natural musical intervals can be major (M) and minor (m). Perfect intervals are equal in Major and Minor. An octave has only seven diatonic degrees; the eighth note creates a chromatic alteration of the scale degree and complicates the diatonicism.
There is also a notion of diminished and augmented intervals.

The picture shows the correspondence between piano, guitar and 4&6-string bass ranges, and compliance of these ranges with the musical notation.
Harmonic interval - two notes sounded simultaneously.
Melodic interval - two notes sounded in succession as in a musical part.

The range of notes can be chromatic, diminished, whole-tone and diatonic. Melodic movement through the range of notes is called a scale.
A chromatic scale is a twelve-note scale including all the semitones of the octave.

1.1. Basic concepts
      2. Main structures
      2.1. Primary scales

2.2. Triads
We have examined four main scales: chromatical, whole-tone, diminished and diatonic. Triads are built on basis of scales. Taking I, III and V scale degrees, we will get a triad of this scale. A chromatic scale is not examined from the viewpoint of the triad building.  The augmented triad Aug is built on basis of the augmented scale.

  Diminished triad Dim is built on basis of the diminished scale.
The diatonic range of notes gives us two kinds of triads: a major triad Maj (major - large) is built on basis of  the major scale, and a minor triad min (minor - small) is built on basis of the minor scale.
One more kind of triads is examined in modern music. It is a suspended triad Sus (to suspend - to delay). The classical theory defines Sus as delaying the occurrence of the third scale tone; sounding tensely, demanding a resolution. I, IV, V ' I, III, V.  A sus triad does not have an own scale, but undoubtedly belongs to the natural diatonic range of tones.

Scale degrees are marked with Roman numerals and are called I - the first, II - the second, III - the third, IV - the fourth, V - the fifth, VI - the sixth, VII - the seventh, VIII - the eighth, etc� The same names are used to identify harmonic (two notes sound simultaneously) and melodic (two notes sound successively) musical intervals. For example, the interval between I (the first) and IV (the fourth) scale degrees is called the fourth, between I and III - the third, between I and VII - the seventh, etc�
The peculiarity of the diatonic range of tones is that playing diatonic degrees through a single note, we will steadily take major or minor thirds. Thus chords have a triadic structure. A seventh chord is a chord consisting of a triad (I, III, V) plus the seventh degree of a specific mode. A whole- tone scale has six degrees, so it is not examined as to seventh chords. The diminished scale consists of eight degrees; its seventh step is actually a major sixth. But still it is a seventh degree, so it is called the diminished seventh.

Thus, Sus is a triad of the indefinite function, composed of I, IV and V scale degrees. One of the five types of triads (aug, dim, maj, min and sus) is the basis of every chord we meet in music.
2.3.Seventh chords
Though primary scales (chromatic, whole-tone, diminished) are the basis of the equal temperament, they are very unfavorable for harmonization (building chords and chord harmonic sequences). The mode is the base of the harmony. In diatonic modes first four notes from the tonic to the fourth are called a lower tetrachord; notes from the fifth to the octave are called an upper tetrachord.

Seventh chords built on "natural" major and minor scale
The whole point is that all these modes are based on the natural major scale. The natural major formula (WWHWWWH) is the basis of the construction. It is clear on the piano in the key of C major. Only white keys are used. Then playing the diatonic range from d (re) to d (re) one octave higher, we will get a Dorian formula (WHWWWHW) and so on.

3.  Diatonicism
3.1.  Pentatonic scale
Music is rhythm (Or more precisely, rhythm is the breath of music). A pitch of sound is just a pleasant bonus, drummers and percussionists can go without it. At the same time, a chaotic sound being elicited by a monkey, (even from such beautifully organized instrument as a piano), is not music. Even if notes will be in one key. This example visually demonstrates the role of rhythm in our perception of music.

1.2. Rhythm and tempo
Rhythm is an execution of sounds in compliance with the rhythmic temperament. Rhythm has two main characteristics: time signature (2/4, 3/4, 6/8 etc�) and tempo (the speed at which a passage of music is meant to be played). These are usually indicated by a musical direction. Due to the rhythmic temperament we can play fast music pieces at slow tempo and slow music pieces at fast tempo. Usage of a metronome helps to understand the rhythmic temperament (real time) more properly.
3.2.  Principles of the chord construction
  4. Circle of fifth
1.     Introduction
1.1.  Basic ideas
1.2.  Rhythm and tempo

2.     Main structures
2.1.  Primary scales
2.2.  Triads
2.3.  Seventh chords
3.     Diatonicism
3.1.  Pentatonic scale
3.2.  Principles of the chord construction
4.     Circle of fifth
4.1.  Basic harmonic progressions

1. Introduction
  4. 1.  Basic harmonic progressions
4. 2. Notion of chord functions
4.2.  Notion of chord functions
                                                                             Hi everyone!
This textbook is written to help beginners, who would like to improvise by ear, to examine musical grammar and to choose exercises for efficient practice. The textbook consists of two parts: theoretical and practical. In the theoretical part one can find fundamentals of musical grammar and basic musical structures. The practical part contains detailed fingerings and examples of exercises for the guitar and 4,5,6-string bass with comments and endnotes.
As a matter of fact, all diatonic modes are called natural. But so as diatonic scale is mainly used in contemporary, especially "popular" music, only the Ionian major and the Aeolian minor are called natural nowadays. In the last paragraph we have called them "standard" ones. So, the Aeolian minor is built on the VI degree of the Ionian major, and the Ionian major is built on the III degree of the Aeolian minor. Such keys are called relative. So, the Ionian major and the Aeolian minor are relative keys.
From the above we can draw an obvious conclusion that a major, a minor and all other diatonic modes are the same scale played from different degrees. This is an extremely important fact for musical practice.

Notice that the Lydian major and the Dorian minor (outlined in green in the figure) and the Mixolydian major and the Phrygian minor (outlined in blue) are also relative like the natural major and minor.

                   Group of majors
                     Group minors
    Lydian major   (# 4)
    Dorian minor  (# 6)
    Ionian major
   Aeolian minor
   Mixolydian major   (b 7)
    Phrygian minor  (b 2)
As we can see, the Locrian, Half Diminished minor does not have its "relative" mode, but it is the center of the diatonic symmetry. It has a special melodic meaning in chord connections and is the most typical leading-tone chord resolving to the tonic.
The group of majors has two alterations (#4, b7). The IV degree is raised in the Lydian major; the Mixolydian major has its VII degree lowered. The group of minors also has two alterations (#6, b2). The Dorian minor has its VI degree raised; the II degree of the Phrygian minor is lowered. The Locrian mode is only relatively minor, thus it itself has two alterations (b2, b5). So, its main triad is diminished, but not minor. All this alterations altogether are called basic diatonic alterations.

Skipping altered degrees of a major scale (#4, b7 in the group of majors) we will get a major pentatonic scale I, II, III, V, VI. The same is in the minor scale. Skipping altered degrees of the minor scale (b2, #6 in the minors' group) we will get a minor pentatonic scale I, III, IV, V, VII. We do not use degrees which are mode distinctions of minor and major scales. Thus major and minor pentatonics are unique musical structures. Pentatonics have in their composition not only whole and half steps but also minor thirds, so their sound is more interesting than the sound of a simple scale. Owing to these advantages pentatonics are widely popular with musicians of different styles and schools.

The main principle of the chord construction is their tertian structure. As we already know, a triad can be built on basis of any scale (except the chromatic one). The further continuation of a tertian line gives us a seventh chord (for example Cmaj7), then a ninth chord (Cmaj9), then an eleventh chord (Cmaj11), and finally a thirteenth chord (Cmaj13).Thus, the thirteenth chord is a full scale arranged in thirds. 
Here we examine just the principle of chords' construction. Similarly we can take any heptatonic scale and build the relevant chord on its basis. Chords built on basis of non-heptatonic scales (whole-tone, diminished) apply the same principle. All discrepancies are written by means of alterations. The more scale degrees are included in a chord, the more certain it becomes. The tonic (the main note of a mode) is a centre of the tonality. The third defines the quality of a mode, whether it is a Major or a Minor. The fifth is the basis of the mode. As we already know, one of the main features of a mode tonality is a perfect fifth. And finally the seventh degree of a mode defines a dominant function of a chord in a Major as well as in a Minor. Then extensions (ninth, eleventh and thirteenth degrees) follows, they influence the already formed (of 1, 3, 5 and 7 degrees) chord. The extensions brighten up the chord make its sound more definite.

We can take a tonic major triad and transfer it through all twelve keys. Comparing those triads with C major (for example) we can discover that the closest tonalities in sound composition are located five half steps higher (an interval of a perfect fourth, note fa = F in the C Major key) or five half steps lower, that is the same as seven half steps higher (an interval of a perfect fifth, note sol = G in the C Major key). The fourth and the fifth are inversions. The sound composition of those tonalities differs from the initial one in one note. There is a note Bb in the key of F Major and F# in the key of G Major. Those tonalities together with their relative keys and A Minor (relative to C Major) are called closely related keys.
The basic harmonic progressions are: a diatonic progression (chords are moving through the degrees of diatonic modes) Cmaj7/Dm7/Em7/Fmaj7/G7/Am7/Bm7b5/Cmaj7//, a circle progression (chords are moving through the circle of fourth) Cmaj7/Fmaj7/Bbmaj7/Ebmaj7/ Abmaj7/Dbmaj7/Gbmaj7/Bmaj7/Emaj7/Amaj7/Dmaj7/Gmaj7/Cmaj7//, and a chromatic progression (chords are moving through the chromatic scale) C7/C#7/D7/D#7//. Frequently, a harmonic progression consists of all the above mentioned kinds of movement. The diatonic progression always remains within a tonality. The circle movement can exceed the bounds of a tonality, but it can remain within the key with help of diatonic passages. The chromatic movement is almost always polytonal.
The notion of chord functions arises from the analysis of a circle chord progression. It is considered that an ascending movement by the interval of a fourth is more natural and harmonious then a descending by the interval of a fourth in the circle progression. Functions of all chords can be divided into stable and unstable in relation to the main tonal chord. Tonic chords refer to stable ones and are marked by letter symbol T - tonic. Unstable are divided in two parts: S - subdominant and D - dominant chords. T -> S-> D-> T this sequence of functional changes is considered to be natural. Tonic is the main degree of a mode. A chord built upon this degree is called the tonic chord. It represents the tonic function; it is a tonic center of all harmony movements inside the key. It generates a felling of repose and balance.
The so-called "chords' tree" also demonstrates a scheme of chords' construction. So there are four tertian triads C, Cm, Caug, Cdim and one quartal triad Csus. Altogether we have five triads: a major triad, a minor triad, an augmented triad, a diminished triad and a Sus triad. They are the basis of any chords' constructions.

Time signatures may be simple and complex. Simple time meters - 1/4, 2/4, 3/4. All others can be received by combination of simple meters. For example, 5/4 = 3/4 + 2/4. 2/8, 3/8 or 2/16 and 3/16 etc� are simple times too. The upper figure represents the number of beats per bar and the lower one - the time value of each beat. 
The continuation of this movement discovers tonalities which differ from the initial one in two notes. Those are Bb Major and D Major (two flats and two sharps correspondingly). Moving to a "flat side" we reach the Gb key in six steps. Moving to a "sharp side" we reach F# key in six steps too. Gb and F# are enharmonics (the same note with different names).  Thereby, the circle closes up. As we see, the circle of fourth is a natural formation created with the twelve-half steps chromatic scale and the diatonic scale.
The scheme shows a functional dependence of the diatonic degrees. The left scheme displays the Major, the right one - the Minor. The first, third and the sixth degrees are tonic degrees (outlined in red colour), the second and the fourth are subdominant degrees (outlined in green), the fifth and the seventh are dominant degrees (outlined in blue). Notice that the third and the sixth degrees are of a variable functionality. The third degree (besides of a tonic function) can present the dominant function, and the sixth degree (besides of a tonic function) can present the subdominant function.
Since you start learning to play any music instrument, you should organize your practice 'in time'. The more you will play scales and other exercises 'in time' the clearer will be your ability to hear time values marked with notes. Always keep 'in time' while practice. Listening to the music, try to feel its rhythm, tap your hand or foot. Remember that a musician desist to be a beginner as far as he understood that there is no music without rhythm.
Analyzing specific compositions of different genres and styles, three main components of music can be singled out: the rhythm, the melody and the harmony. Speaking in these terms we examine "standard" rhythmical patterns meaning peculiarities of their styles, "standard" ways of phrasing (melodic figuration) and also chords and "standard" chord (harmony) sequences.

5.     Conclusion

Piano School

The diatonicism consists of chords marked in the "chords' tree". For example in the key of C Major: C/Dm/Em/F/G/Am/Bm-5// or Cmaj7/Dm7/Em7/Fmaj7/G7/Am7/Bm7-5//. It is obvious on the guitar that Dm and Em have the same position of notes in the chord and accordingly the same fingerings (position of left hand fingers on the neck). As a matter of fact an alphanumeric chord notation appeared thanks to the guitar or to be more exact thanks to its wonderful structure which includes standard chord's positions.  Thus to play a diatonic chord sequence in the Major or in the Minor we need to know three different fingerings (triads), and four fingerings for seventh chords. It is interesting that there are only three fingerings for pentatonics, though it consists of five notes. This circumstance emphasizes its universality.

Elementary school guitar
School playing bass guitar
School playing keyboards
School playing percussion instruments
The class of musical esthetics
The picture presents a comparative analysis of the major (upper string) and minor  (lower string) structures. The Ionian major has all major intervals (except perfect 4th and 5th). The Phrygian minor has all minor intervals (except perfect 4th and 5th).
School Druming
Bass guitar School
Six-string guitar School
Augmented triad
Diminished triads
    Major and minor triads
Diminished seventh chords
    © 2010-2012 GennadiyKlyuevskiy

The main task of "theory" is to make "practice" more efficient.
The main components of music are: tempo (time) and a pitch of tone (notes). The scale is divided into octave parts. The names of sounds equivalent to sound meanings in another octave are repeated in every octave (octave correspondence). The octave consists of twelve exactly equal halftones. This pitch is called equal tempered or chromatical. A half step (halftone) is an interval between two adjacent sounds of the chromatic scale (C & C#). Two half steps make a whole step 0,5 + 0,5 = 1; three half steps make a step and a half 0,5+0,5+0,5=1,5 etc. Thus 12 half steps make 6 whole steps.
The bases of the rhythmic temperament are duplet and triplet. Let's take a quarter note as a basic rhythmic unit. A kick to the floor is "one", a rise of the toe is "and", down again is "two", rise - "and" so on... This way we divide every kick in two equal time pieces. This is an eighth duplet. We can divide a quarter note into three equal time pieces and this is an eighth triplet. The picture shows that it works on every level of time division. 

The main condition of the modal system is the definite symmetry of lower and upper tetrachords. All heptatonic modes have this symmetry. Modes with other number of degrees do not maintain the tonality, they are atonal. Consequently every musical composition can be written either in major, or in the minor key. Thus we should devote much attention to the examination of the diatonicism.

A natural major (WWHWWWH) and a natural minor (WHWWHWW) formulas are "standard". Comparing all derived formulas with standard ones, we can speak about alteration. For example, the configuration of a Phrygian scale is almost the same as a natural minor's structure, but its second degree is lowered. It is obvious on guitar and bass.
Another significant point is that the diatonic "structure" (which we are examining now) is actually a "structure" of a key. We can play in the key of G (sol major) or Dm (re minor) or any other F, C#m, Eb� the "structure" will not change. So, the diatonicism displays the fact that one modal system may have two tonal centers: a major (Ionian) and a minor (Aeolian). Anyway, this is the main point.

The next scheme is an effort to display visually a circumstance that playing a thirteenth's degree arpeggio in one octave we skip sounds called extensions in the next octave (ninth, eleventh and thirteenth degrees). So, the ninth degree is actually the same as the second degree but in the next octave, the eleventh is the same as the fourth, the thirteenth is the same as the sixth. Thus we have a basic seventh chord 1, 3, 5, 7 degrees and extensions 9 - 2, 11 - 4, 13 - 6.
It should be noted that all constructions are viewed on the example of the major chord in its basic form (from its first degree alternately 1, 3, 5, 7, 9, 11, 13). But in practice chord inversions are used more frequently. We will deal with chord inversions in the practical part. The function of a chord will not change because of the rearranging, transposition into other octaves, duplication (doubling, tripling, etc.) of its degrees. In short, the function of a chord and its name remain the same in spite of a transposition. Rejecting doublings and arranging notes in thirds we can bring almost every complex harmonic combination to its basic form.

Of course, none of the viewed above notations is perfect so as it does not indicate the position of notes in a chord. But in contrast to the plain staff notation symbol (tabular) notation leaves much more space for creativity.
There is also a term "add" (to append) in a chord symbol notation. It is used to emphasize a specific position of notes in a chord. For example for the guitarist D9 is a basic position of a dominant seventh chord without a fifth degree. And the chord Dadd9 is a major triad with the added ninth degree.

There is one more important principle in the chords' construction and notation. Chord tones depending on their positional relationship are called chord voices. The main form for every chord is the four-part structure corresponding to a vocal quartet, a brass band or a mixed chorus. The four-part structure divides in two parts: the group of upper voices is a harmonic extension relatively to the bass which is the basis of a harmony (upper voices/bass). This approach to a chord notation is widely used now, so as it is very convenient and sometimes almost irreplaceable. For example, instead of "hazy" Gsusb5(9)(13) we use definite A/G. Here the first letter signifies the "whole" triad or even the seventh or ninth chord. The second letter defines only a bass tone (Dm9/C, E7/G# or F/A).
And finally, there are two "contingent" chords, so as they do not have their own scale. In classical music theory those triads and seventh chords are called "suspended" and are not viewed independently, their role is only situational. The third degree of suspended chords is (suspended) risen or lowered by a step or a half step depending on the expected resolution (a major or minor chord). That is the third degree can be substituted for the second or the fourth degree. So there are two kinds of a Sus chords: sus9 and sus11 (the same as sus2 and sus4). Seventh chords may be built on basis of suspended chords then every sus's subkind will include a major or minor seventh. Thus there are four different sus seventh chords: sus#7(9), sus7(9) and sus#7(11), sus7(11). These chords are widely used in contemporary music due to their wide and tense sound. Actually if we remove the third degree from any major or minor chord, it can be viewed as Sus. And it will not loose its main function (thought getting altered sound) in a specific harmonic sequence.
As we already know the third degree defines the quality of a mode and identifies whether the chord is major or minor. The fifth degree of a chord detects its ability (major or minor) or inability (diminished, augmented) to be the main tonal chord. The seventh degree determines whether the sound of a chord is stable (the tonic function; major seventh, Cmaj7) or unstable (dominant function; minor seventh, C7). The unstable sound demands the further harmonic movement. It is interesting that sevenths change their functions in minor; the minor third has stable sound and the major seventh has dominant sound. So, there are seven tertian seventh chords. There are two chords in atonal modes - Caug7 (whole-tone scale) and Cdim7 (diminished scale); two chords in the Major - Cmaj7 (major seventh chord) and C7 (dominant seventh chord); three chords in the Minor - a minor seventh chord Cm7, a half-diminished seventh chord Cm7b5 and a minor seventh chord with  the risen seventh degree Cm#7 (this chord is built on the first degree of the harmonic minor). Those were seven tertian seventh chords.
The main representative of a subdominant function is a chord built upon the fourth degree of a mode (a fourth above the tonic). Proceeding from the name (prefix "sub"), a subdominant is subordinate to a dominant, or rather the subdominant moves to the dominant. The main chord of a dominant function is built upon the fifth degree of a mode (a fifth above or a fourth below the tonic); it is called the dominant seventh chord. It markedly requires a resolution (it has a minor seventh) to a tonic (a fourth above). This way the circle closes up. A Tonic moves (in a circle progression) to the Subdominant, the Subdominant moves to the Dominant, the Dominant moves to the Tonic, the Tonic moves to the Subdominant again. A kind of perpetuum mobile! The Tonic establishes the tonality, subdominant tonalities are ancillary for the Tonic. They establish a harmony movement within the mode and via the Dominant resolves to the Tonic.

As it was mentioned above only modes with a natural fifth degree (an interval of a fifth relative to the tonic) can be the basis of a tonality, that is to act as a main tonal mode. All diatonic modes except the Locrian are of this kind. We all know from childhood that there are only minor and major keys.  All the above conclude that the Diatonicism is the tonal basis of music. In the language of cookery the diatonicism is the main dish. All other kinds of modes (augmented, diminished so on) are seasoning.
Of course, it takes time to master such unusual instruments, but meanwhile you will become a real musician.
Not in the sense that you can learn some (10-20) compositions of respectable people of the past and present, but in the sense that thanks to music you will be able to express (share) your feelings and emotions. This way you will be able to create a definite mood. "To master" means to learn to play by ear. To play by ear means to play (in time, preferably with a metronome) an exercise as many "million" times as necessary to stop thinking about it as about an exercise and interpret it as a musical phrase. A technical part and a method are examined in greater length in the practical part of this textbook. And now let's sum it up.

The research was carried out first of all to define basic tools we use while playing music. That is musical structures which help us to remain within a specific tonality. Scales, of course, are the basis of any mode constructions, so they are the first instrument of our tools. The next instrument is a triad built upon a scale. One can say, a triad is the shortest scale consisting of three notes. Then there is a seventh chord, a scale of four notes. And at last there is a pentatonic scale (a scale of five notes). Ones more: 1) a scale, 2) a triad, 3) a seventh chord, 4) a pentatonic scale.

Proceeding from the above you should make a list of exercises consisting of all the main structures (scales, triads, seventh chords, pentatonic) and begin a regular practice. Some examples of exercises are also offered in the practical part. It is advisable to work at all main musical structures in the process of education.
In conclusion it should be pointed that this rather superficial research was carried out not for the deep analysis of details but rather to "glance the action field over" and to single out main structures which are the basis of musical thinking. The experience and practice show that the diatonicism is a backbone of music. So the main list of exercises should contain: two scales (major and minor), three triads (major, minor and diminished), four seventh chords (big major seventh chord "Xmaj7", dominant seventh chord "X7", minor seventh chord "Xm7", half-diminished seventh chord "Xm7-5") and two pentatonic scales (major and minor).