Primary Music Theory
  The circle of fourth
Circle progression within the tonality

The position of the circle of fourth on the guitar
Thanks to the existence of overtones we have an orderly system of flageolets on every string. Actually, overtones elicited from the string are called flageolets (from the Old French "flageolet" - a flute, because of the similarity of a sounding). We will not delve deeply into the "physics" of processes, will mark only that a great Greek philosopher Pythagoras discovered all modern music structure studying positions of flageolets or overtones on a single string. Let's view it on the example of the D-string:
1 - the whole string, D-note, I - the first mode degree;
2 - half-and-half, a note D one octave higher, I - the first mode degree;
3 - one third, a note A, V - the fifth mode degree;
4 - a half in two, a note D two octaves higher then the initial one, I - the first mode degree;
5 - one fifth of the string gives a note F#, III - the third mode degree;
6 - one sixth repeats a division into three, a note A again one octave higher, V - the fifth degree of the mode;
7 - one seventh, a note C, a lowered VII seventh degree in the key of D Major;
8 - a division into eight gives us a note D again three octaves higher then the initial one, I - the first mode degree;
9 - one ninth gives E note, this is the ninth or  II - the second mode degree;
10 - repeats one fifth one octave higher, a note F#, III - the third mode degree;
11 - one eleventh, in the present case the risen eleventh or fourth (triton), a note G#, the risen fourth mode degree;
12 - is the note A again, one octave higher then one sixth, V - the fifth mode degree;
13 - a division into thirteen gives us an altered thirteenth chord degree, a note Bb, the lowered VI - the sixth mode degree;
14 - a division into fourteen repeats the division into seven, we get a note C, the lowered seventh mode degree;
15 - one fifteenth part gives the natural seventh degree in the key of D Major, a note C# is the natural VII seventh degree of the mode;
16 - and at last, one sixteenth part repeats the tonic D but it is four octaves higher then the initial one, I - the first mode degree.
Theoretically, we can continue the division of a string into all sequence of natural numbers; that is to infinity. But in practice, try to find all the viewed above flageolets (overtones, harmonics) on the guitar and you will see, the father we move, the higher in frequency and lower in sound flageolets become, and it gets harder to elicit them. So the elicitation of flageolets become problematic after the twelfth division (the fifth degree), it is not to mention a division into 17, 18, 19 and so on. If we produce the sound from the opened D string, its tone will be limited for our ear to the single note - D one-line octave. This is the main tone, the nucleus of this sound. But simultaneously with the D note a great number of other tones sound, which we call overtones or harmonics. Thus mostly overtones are indiscernible by ear, every adds individual inflexion to the common color or sound timbre. All instruments produce different overtones. Overtones explain the timbre color of certain tones and the uniqueness of an instrument's sounding.

The first picture shows the arrangement of notes of the circle of fourth in the first position. The second picture shows the arrangement of notes of the circle on two adjacent strings tuned in straight fourth.
Overtones, harmonics, flageolets
The closeness of a relation between any two keys is determined by the number and the meaning of common chords (chords which consist of sounds those are met in both keys). Different world schools have different approaches to the explanation of the closeness of a relation. We will not delve deeply into the labyrinth of a theory, but stop at a minimum we need to play music. Closely related keys are related as T, S, D (the main degrees of a mode - the tonic, the subdominant, the dominant), those are I - the first degree (T), IV - the fourth degree (S), V - the fifth degree (D), their relative keys. The closely relative keys in C Major are: F Major, G Major and their relative minors - A Minor, D Minor, E Minor. The same situation is in the Minor; there are I - the first degree, IV - the fourth degree, V - the fifth degree and their relative keys. Closely related keys in A minor are D Minor, E Minor and their relative keys - C Major, F Major, G Major. The easier definition of closely related keys is that they differ by one accidental.
In diatonic music keys interact in variable and complex way. Following relations are supported by terms:
Relative keys - major and minor keys, the sign of the relation is the same key signatures. Closely related keys have common set of pitches, but tonal functions of these pitches do not coincide. Scales of relative keys are the minor third away from each other: the Major is above, the Minor is below (for example: C Major - A Minor, F# Major - D# Minor, Bb Major - G Minor, so onů). Relative keys are the closest during modulation, so as they share seven common triads.  
Parallel keys - keys which share the same tonic but belong to different mode quality (G Major - G Minor). Major and minor parallel keys are three accidentals away from each other (major - in the sharp side, minor - in the flat side).
Keys with the same third degree - two keys which have the same third degree in the tonal triad (for example: C Major - C# Minor; E is the common third degree). Such keys belong to different modes and are a halftone apart (the Major is below).

The viewed here twelve-equally-halftones system contains twenty four keys. There are twelve Major and twelve Minor keys.
If we start moving from the tonic triad of the C Major by ascending fourth or descending fifth (the fourth and the fifth are inversions), then we will get back to the C Major in twelve "steps", and will pass all twelve major keys. If we move the Minor by fourth, we will pass all twelve minor keys. This is the circle of fourth. In the circle we can move in the opposite direction too, by ascending fifth and descending fourth. The circle progression is one of the basic harmonic progressions due to its naturalness.
If you master exercises introduced in this part and remember circle chord progressions, you won't have to learn the number and the position of key signatures in every key. If you know the circle progression, it is easy to count the necessary key starting from the beginning of a circle; that is from a tonality without key signatures. The position of key signatures is always the same and also based on the circle of fourth. Flats start from a note B flat and are added with every circle step on a fourth above or on a fifth below; B flat, E flat, A flat, D flat, G flat, so on. Sharps start from a note F sharp and are also added with every circle step but in the other way, on the fifth above or on a fourth below; F sharp, C sharp, G sharp, D sharp, A sharp, so on.

The offered exercises are circle sequences of seventh and ninth chords. The sequences are presented by arranged chords with basses on different strings. A work on these exercises will enable you to memorize the circle sequences, and, simultaneously, to learn basic fingerings of arranged seventh and ninth chords.
We can also arrange chords of a key according to the circle of fourth, having broken the circle progression in one place. On the example of G Major we can see, that the sequence breaks on the second circle "step". There is no F in the G Major, but there is F#, so to stay in the key we have to move by triton up- or downward instead of the fourth (two and a half steps). Then we move by fourths till we reach our first chord. This way we shorten the standard circle progression, squeezing it into the limits of one key. Pay attention that while playing the circle progression within the key, we use chords which are built on the degrees of this key.
Circle progression of the Major seventh chord
Circle progression of the Minor seventh chord
Circle progression of the Major minor seventh chord (dominant seventh chord)
Circle progression of the Major seventh chord

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