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, a 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 this case the risen eleventh or fourth (triton), a note G#, the risenfourth 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, theloweredVI - the sixth mode degree;
14 - a division into fourteen repeats the division into seven, we get a note C, theloweredseventh mode degree;
15 - one fifteenth part gives the natural seventh degree in the key of D Major, a note C# is thenaturalVII 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), let alone 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.
Overtones, harmonics, flageolets
The circle of fourth is the fundamental structure of our twelve-equally-halftones musical system. It runs through the Diatonicism like the Chromatic scale and like the Chromatic scale consists of twelve steps, those steps are not half tones, but fourths. 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 fourths, 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 is 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.
It is very useful to learn the circle of fourth in the first position. Next exercises are based on the circle progression of the one subkind of the seventh chords. We need seventh chords of two levels (from the E string and from the A string) to move them in a circle of fourth. Exercises help to learn the circle of fourth and simultaneously to practice chord's performing. Arrows on pictures show the direction of the movement. We start from the Cmaj7 chord on the E string and end on Cmaj7 on the A string. We use three subkinds of diatonic seventh chords. The half-diminished seventh chord never moves in the circle of fourth, because of its special status within the Diatonicism.
Circle of fourth within the tonality
We can 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 of fourth progression within the Major tonality
Circle of fourth progression within the Minor tonality