HARMONIC WHEEL: UTILITIEShttp://www.harmonicwheel.com

In the next 2 large sections the main utilities of the Harmonic Wheel are explained in detail. They have been written in a simple and friendly way, so as to be useful even for the layman. In fact, a summary of main musical concepts is given, which may serve as an introduction to Music Theory and Harmony for anyone interested in this subject. Thus, all the concepts are explained in the simplest cases (for example, in the C Major key) and the rest of them are solved, automatically, with the aid of the Harmonic Wheel. In this development, a big effort has been made to logically connect all the concepts, to give the theory a solid and unified structure.All the explanations are given without using the musical notation, for it is unnecessary in this approach. Only one example of it is presented in one of the last chapters. Nevertheless, it is strongly recommended to learn it, for it is one of the few languages having the status as a universal language, along with the mathematical language, the symbols for the physical units and the symbols for the chemical elements. Moreover, once the contents of this Web page are acquired, it will be easier to understand this system of script.Even for those people having musical knowledge, this new way of connecting musical concepts will result very interesting. And, particularly, those musicians who play monophonic or percussion instruments will find, in Level 2, a very useful summary of Harmony foundations, which may be sufficient to satisfy their real professional needs, thus avoiding the necessity of reading complex and extensive texts.The contents have been organized into 2 Levels:Level 1 is devoted to fundamental aspects of Music Theory. This level deals with the musical notes, the intervals, the Major and minor scales, and the keys. The Harmonic Wheel allows us to easily obtain the interval between any two notes, as well as the notes belonging to any Major or minor scale, being it natural, harmonic or melodic, along with their corresponding key signature. As novelty, the Harmonic Wheel gives a complete and panoramic view of the existing relationships among all the keys. While the cycle of fifths shows the keys on a line (one dimension), the Harmonic Wheel shows them on a surface (two dimensions). So, besides the cycle of fifths, the changes of mode are also visible, which are not in the cycle of fifths. Thus, a real MAP OF THE KEYS is achieved.Level 2 is devoted to the foundations of Harmony. This level deals with the construction of Major, minor, Augmented and diminished chords, as well as all possible 4 note chords that can be obtained by superimposing Major and minor thirds. Then, the existing relationships among the different types of chords are explained by means of scales. On the Harmonic Wheel, all these chords have a simple graphical representation. This fact facilitates us to obtain the notes composing them, that is, their arpeggios. On the other hand, this instrument allows us to obtain, automatically, all the chords associated to any Major or minor scale. This function is named CHORD FINDER. As novelty, the particular representation of notes and consonant chords on the Harmonic Wheel explains, graphically, the construction of the Major scale, its relationship with its relative minor scale and the most important characteristics of their associated chords. All these things are difficult to see even in the conventional musical script. Moreover, thanks to this kind of representation, it is easy to determine which chords are more or less affine to any given scale. Finally, some special scales, containing 5, 6 and 8 notes, are explained, along with their associated chords.

LEVEL 1: FOUNDATIONS OF MUSIC THEORYhttp://www.harmonicwheel.com

1. MUSICAL NOTESThe Musical Notes are 12, 7 being natural and the other 5 are altered.The natural notes, sorted out by their pitch, from the lowest to the highest, are: C, D, E, F, G, A and B. These 7 notes form the so called C Major Scale and correspond to the white keys in the piano. After the B comes another C and so on. Nevertheless, the first and second C are not identical, as the latter is higher than the former. The thing is that, from the point of view of Physics, the vibration frequency of the second C is exactly double that of the first C, which makes those notes very affine when hearing them, to the extent of assigning them the same name. It is said that the distance between them is one octave, because there are 8 natural notes from the first to the second C, including both the initial and the last note in the count.The distance between two consecutive natural notes are not always the same, for in some cases the distance is one Whole step (W) and in others it is one Half step (H). Particularly, between E and F there is one half step, as well as between B and C. On the contrary, between any other two consecutive natural notes there is one whole step. This is the reason why the so called altered notes are placed between them, at a one half step distance, and they correspond to the black keys in the piano (Fig. 1). Therefore, a set of 12 different notes is obtained, there being one half step between any note and the next one. Thus, the distance between a given C and its octave (that is, the next C) is 12 H or 6 W. This set of 12 notes is known as the Chromatic Scale.

Figure 1. The musical notes and their location in a piano.In order to name the altered notes, the accidentals are introduced, which are the sharp () and the flat (). The sharp raises the natural note one half step and the flat lowers it one half step. Thus, for example, between C and D (where there is a whole step distance) an altered note is placed, which can be called Cor D. These two notes, corresponding to the same pitch but having different names, are called enharmonic. In the same way, Eand F natural, or B natural and C, are also enharmonic (the term natural means without any accidental). Occasionally, the double sharp () and the double flat () are used to raise or lower the natural note one whole step, respectively.In spite of the fact that the 12 notes are uniformly spaced, in Fig. 2 the different consideration given to the natural and the altered notes is evident, as well as the lack of uniformity in the distribution of whole and half steps among the natural notes. All these things, which seem to be strange and whimsical, are in fact the result of centuries of evolution of Music Theory, which in turn is the consequence of the affinity relationships that exist among sounds, along with the strong cohesion and unity that the set of 7 notes composing the Major scale have. All these questions will be properly explained in different chapters.Furthermore, as will be shown in Chapter 5, a Major scale can be built beginning with any of the 12 notes, thus obtaining, for example, the D Major scale, AMajor scale, etc., what makes the 12 notes equally important. The C Major scale is, simply, the one having exclusively natural notes. In the Guitar, for example, the 12 notes are given exactly the same treatment (Fig. 2).

Figure 2. Location of the notes on the 3rd string of the Guitar.

2. INTERVALSThe Interval is the distance in pitch between two notes, which is given by the number of whole and half steps between them. This number, however, does not indicate the degree of affinity between them nor their relative position in a Major scale. Therefore, it is most common to indicate the interval with a number and a quality. The interval number indicates the number of natural notes between the first and last note, both included. And, unless other thing indicated, it will be assumed that the interval is ascending, that is, that the second note is higher than the first one. Thus, for example, the interval between D and A is a 5th (D E F G A are 5 natural notes in ascending order).If we now consider the interval between Dand A, we find again a 5th (D E F G A are 5 natural notes in ascending order), although this distance is smaller than that between D and A. To take into account these differences, an interval quality is added to the interval number, which is related to the number of whole and half steps contained in the interval.The interval quality is based on the Major scale. So, every interval between the lowest C and any other note in the C Major scale in ascending order are called Major (M) or Perfect (P). Particularly, the interval between C and F, G or C is called Perfect, while the interval between C and any other note in this scale is called Major (in Chapter 3 an explanation of it will be given). Table 1 shows these intervals and also indicates the number of whole steps contained in each of them.Table 1. Intervals in the ascending C Major scale.Between C andCDEFGABC

IntervalP 1stM 2ndM 3rdP 4thP 5thM 6thM 7thP 8th

Number of Whole steps0122.53.54.55.56

The P 1st interval is called Unison and the P 8th, Octave. To name other different intervals, the following qualities are used:minor (m), if it has one half step less than the corresponding Major interval. For example, between C and Ethere is a m 3rd (1.5 W) and between C and Bthere is a m 7th (5 W).

Augmented (A), if it has one half step more than the corresponding Major or Perfect interval. For example, between C and Dthere is an A 2nd (1.5 W), between C and Fthere is an A 4th (3 W) and between C and Athere is a A 6th (5 W).

diminished (d), if it has one half step less than the corresponding minor or Perfect interval. For example, between C and Gthere is a d 5th (3W) and between C and Bthere is a d 7th (4.5 W).

When necessary, the term double Augmented (AA) is used for the interval having one half step more than the Augmented, and the term double diminished (dd) is used for the interval having one half step less than the diminished.

Table 2 shows the intervals between C and the altered notes and also indicates the number of whole steps contained in each of them.Table 2. Intervals between C and the altered notes.Between C andC D D E F G G A A B

IntervalA 1stm 2ndA 2ndm 3rdA 4thd 5thA 5thm 6thA 6thm 7th

Number of Whole steps0.50.51.51.5334455

The last two rows in Table 1 also serve to determine the interval between any two notes, irrespective of whether the first note is C or not. Let us see some examples:Between D and A there is a 5th (D E F G A are 5 natural notes in ascending order) having 3.5 W. It is, therefore, a P 5th.

Between Dand Aalso there is a 5th, but now having 2.5 W, so it is a dd 5th.

Between Band Gthere is a 6th having 5 W. It is, therefore, an A 6th.

Between Aand Cthere is a 3rd having 3 W, so it is an AA 3rd.

Those intervals not bigger than an octave, as those studied up to this point, are called simple, while those bigger than an octave are called compound. Every compound interval can be reduced to a simple one by eliminating the proper number of octaves. The interval quality of a compound interval is that of its corresponding simple interval.

3. INVERSION OF INTERVALSThe inversion of an interval consists, simply, in interchanging the order of its notes. Thus, if we invert the interval D A, which is a P 5th, we obtain the interval A D, which is a P 4th. And, if we invert the interval E G, which is a M 3rd, we obtain the interval G E, which is a m 6th. In the inversion of intervals, the following two rules apply:The sum of the interval numbers of a given interval and its inversion is always 9.

When inverting intervals, the Major is transformed into minor, the minor into Major, the Augmented into diminished, the diminished into Augmented, the double Augmented into double diminished and the double diminished into double Augmented. The Perfect interval, however, remains Perfect.

We can check these two rules in the two previous examples. So, in the first one, the P 5th becomes a P 4th (5 + 4 = 9 and both of them are P); and, in the second one, the M 3rd becomes a m 6th (3 + 6 = 9 and M becomes m).In practice, the inversion of an interval is achieved by raising the lower note one octave or by lowering the higher note one octave. But the same result is reached by changing the ascending character of the interval to descending. Thus, between D and A there are 5 natural notes in ascending order (D E F G A) and a distance of 3.5 W, so it is a P 5th. But, in descending order, there are 4 natural notes (D C B A) and a distance of 2.5 W, that is, a P 4th. Therefore, the inversion of the interval can be understood in these two different ways.It is very illustrative to verify that, in the descending C Major scale, every interval between the highest C and any other note in the scale is minor or Perfect. Table 3 shows these intervals and also indicates the number of whole steps contained in each of them.Table 3. Intervals in the descending C Major scale.Between C andCBAGFEDC

IntervalP 1stm 2ndm 3rdP 4thP 5thm 6thm 7thP 8th

Number of Whole steps00.51.52.53.5456

If we compare Table 3 with Table 1 in Chapter 2, we will see that, apart from the 1st and 8th intervals, only the 4th and 5th have the same number of whole and half steps in both the ascending and the descending scale. On the other hand, notes F and G are very affine to note C, due to the particular relationships among their frequencies, from the physical point of view. As a consequence of all these things, these intervals are called Perfect. Remember that the P 4th and the P 5th are the inversion of each other.Although to a smaller extent, those notes forming M 3rd or m 3rd intervals with C are also very affine to it, as well as their corresponding inversions, the m 6th and M 6th, respectively. This concept of affinity several times mentioned is technically known as Consonance and is related to the physical phenomenon of vibration, which originates the sound. In summary, we can say that, apart from the unison and the octave, the Consonant intervals are the P 5th, the M 3rd and the m 3rd, along with their inversions, the P 4th, the m 6th and the M 6th, respectively. The rest of the intervals are Dissonant. In practice, this means that if two notes forming a consonant interval are simultaneously heard, a sensation of harmony, rest and stability is perceived. On the contrary, hearing at the same time two notes forming a dissonant interval produces a sensation of tension and instability.Finally, in Tables 1 and 3 we can also observe that all possible interval distances appear, except the 3 W or Tritone, a highly dissonant interval having special properties. Actually, with the notes of the C Major scale it is possible to form only one tritone, the F B (A 4th) or its inversion B F (d 5th).The concept of Octave and its division in 12 parts, the concept of Consonance and the concept of Major scale are 3 fundamental principles which the Western Music is based on, irrespective of the musical style considered.

4. INTERVALS AND THE HARMONIC WHEELDetermining the interval between any two notes is one of the first obstacles arising when one begins the study of Music. Nevertheless, this task is highly simplified by using the Harmonic Wheel, as we will see in this chapter.In the Harmonic Wheel, the Musical Notes are represented in BLACK and the couples of enharmonic notes are represented by placing one of them just above the other. On the other hand, the notes are placed in such a way that each note is connected, by means of red lines, with the 6 notes with which it forms Consonant Intervals.As previously explained, the Consonant Intervals are combinations of 2 notes that, when played together, produce a sensation of harmony, rest and stability. And, apart from the unison and the octave, they are the P 5th, the M 3rd and the m 3rd, as well as their inversions.In the Harmonic Wheel, each of these consonant intervals is represented with a different type of line. Thus, Circumferences are used for P 5th intervals (and their inversions, P 4ths); Radii, for m 3rd intervals (and their inversions, M 6ths); and Spirals, for M 3rd intervals (and their inversions, m 6ths). Fig. 3 shows these three types of lines.

Circumferences: P 4th, P 5th

Radii: m 3rd, M 6th

Spirals: M 3rd, m 6th

Figure 3. Lines used to represent the consonant intervals.As an example, Fig. 4 shows the note E and the 6 notes with which it forms consonant intervals, which are:A and B, at intervals of P 4th and P 5th, respectively, and located on the circumference going by note E. If we move to the right along this circumference, we find P 5th intervals, while if we move to the left, we find P 4th intervals. (The enharmonic Cis discarded, for it does not form a 5th interval with E, but a 6th).

G and C, at intervals of m 3rd and M 6th, respectively, and located on the radius going by note E. If we move upwards along this radius, we find M 6th intervals, while if we move downwards, we find m 3rd intervals. (The enharmonic Dis discarded, for it does not form a 6th interval with E, but a 7th).

C and G, at intervals of m 6th and M 3rd, respectively, and located on the spiral going by note E. If we move to the right along this spiral, we find M 3rd intervals, while if we move to the left, we find m 6th intervals. (The enharmonic Ais discarded, for it does not form a 3rd interval with E, but a 4th).

Figure 4. The note E and the 6 notes with which it forms consonant intervals.NOTE: All the indications in LIGHT BLUE are included for illustrative purposes and do not appear on the Harmonic Wheel.Furthermore, these 3 types of lines allow us to easily know the interval between any two notes. To do this, we only have to pay attention to the Major (3rd and 6th) and Perfect (4th and 5th) intervals. The orientations of these intervals with respect to a given note are always the same and are easy to memorize. Fig. 5 shows these intervals from note E.

Figure 5. M 3rd, P 4th, P 5th and M 6th intervals from note E.By comparison with these 4 intervals, it is possible to determine which is the interval between E and any other note, except in the case of a 2nd or its inversion, a 7th. But, for these two cases, it is sufficient to remember that the M 2nd contains 1 W.The final procedure consists in firstly determining the interval number of the given interval and then its quality. If keeping to this order, it will not be necessary to deal with enharmonic notes. Let us see some examples:Which is the interval between E and G? It is a 3rd. So, we follow the M 3rd line and compare with the G at its end. As it is a G, this means that between E and Gthere is a M 3rd, so between E and G there is a m 3rd.

Which is the interval between E and C? It is a 6th. So, we follow the M 6th line and compare with the C at its end. As it is a C, this means that between E and Cthere is a M 6th, so between E and C there is a m 6th.

Which is the interval between E and B? It is a 5th. So, we follow the P 5th line and compare with the B at its end. As it is a B natural, this means that between E and B there is a P 5th, so between E and Bthere is a d 5th.

Which is the interval between E and A? It is a 4th. So, we follow the P 4th line and compare with the A at its end. As it is an A natural, this means that between E and A there is a P 4th, so between E and Athere is an A 4th.

Which is the interval between E and D? It is a 7th. Its inversion, from Dto E, contains 0.5 W, so it is a m 2nd. Therefore, between E and Dthere is a M 7th.

In case the first note in the interval is altered, we will begin by considering this note natural and then we will include the effect of the accidental. Let us see a couple of examples of this:Which is the interval between Eand G? We begin by considering the interval between E and G, which is a 3rd. And, following the previous procedure, we see that it is a M 3rd. Therefore, between Eand Gthere is an A 3rd.

Which is the interval between Eand A? We begin by considering the interval between E and A, which is a 4th. And, following the previous procedure, we see that it is a d 4th. Therefore, between Eand Athere is a dd 4th.

5. MAJOR SCALESIn Chapter 1, we saw that the C Major scale is composed by the following succession of notes and intervals: C Major scaleThe first note in the scale is called Tonic (in this example, C) and it is normally repeated at the end of the scale. The numbers representing the order of each note in the scale are called scale degrees and are written in Roman numerals:

C Major scale, along with its degrees The names of the scale degrees are the following:DegreeDegree Name

I Tonic

II Supertonic

III Mediant

IV Subdominant

V Dominant

VI Superdominant

VII Leading tone

VIII Octave or Tonic

If we want to build a Major scale beginning, for example, with note A, that is, the A Major scale, we only have to keep to the same succession of whole and half steps seen for the C Major scale, that is, W W H W W W H:

A Major scaleTherefore, we can define a Major Scale as a set of 7 notes characterized by the whole and half step succession W W H W W W H. Another equivalent way to build the Major scale consists in finding the notes forming Major or Perfect 2nd to 8th intervals with the tonic.In the A Major scale, we can observe that there are 3 sharp notes (C, F and G), which is necessary to keep to the whole and half step succession. The number of sharps or flats in a Major scale is known as the Key Signature. So, it is said that the A Major key signature has 3 sharps.The order in which the sharps appear in Major scales is always the same: F, C, G, D, A, E, B (that is, by P 5th intervals). This means that, in a Major scale having 3 sharps, they will correspond to the notes F, C and G, as seen in the A Major scale. In the same way, if a scale only has 2 sharps, they will correspond to the notes F and C, as occurs in the D Major scale:

D Major scaleOn the other hand, the flats appear in the reverse order with respect to the sharps, that is, in the order: B, E, A, D, G, C, F (that is, by P 4th intervals). So, in a scale having 2 flats, they will correspond to the notes B and E, which occurs in the BMajor scale:

BMajor scaleBuilding the Major scales, determining the key signature of each of them or knowing the order in which the tonics appear when augmenting or diminishing the number of sharps or flats, are complex questions that require much time of study. For this reason, the Harmonic Wheel is provided with a very easy system to solve all these questions directly and in a very educational way. To do this, we only have to rotate the two discs forming this instrument, as we will see in the next chapter.

6. MAJOR SCALES AND THE HARMONIC WHEELThe Major scales are obtained with the Harmonic Wheel by simply rotating its two discs until the sign matches the desired tonic. The remaining notes in the scale are then automatically marked with the sign . As an example, Fig. 6 shows the D Major scale.

Figure 6. D Major scale.On the other hand, the lines in DARK BLUE indicate the region in the Harmonic Wheel embraced by this scale (the use of which will be seen in Chapter 13). For every Major scale, this region is a curved rectangle.The notes in the scale thus marked are represented in black (because this colour always represents notes) and they are placed by following the consonance relationships. Therefore, they are not sorted out by their pitch. If we want to have these notes sorted out by their pitch, we must use the red notation (although what this colour actually represents are chords, as will be explained in Level 2). This way, the notes are found to the right of the Roman numerals, which indicate the scale degrees, as in Fig. 6. When using this notation, we must neglect the letters and symbols appearing to the right of the notes.In Fig. 6 we can also see, next to the I degree (Tonic), the key signature of this scale, that is, 2 sharps. If we continue rotating the two discs forming the Harmonic Wheel, we will see how the different scales appear, along with their tonics and key signatures. So, note that, when augmenting the number of sharps or reducing the number of flats, the tonics appear, precisely, by Perfect fifths. And, when reducing the number of sharps or augmenting the number of flats, the tonics appear in the reverse order, that is, by Perfect fourths.An interesting property of the Major scale is that, if we represent its notes on a circumference, they will be located one next to the other and they complete a half circumference. As an example, Fig. 7 shows this kind of representation for the D Major and A Major scales. This fact means that the 7 notes of a Major scale can be sorted out by Perfect fifths, although in this case the first note is the IV degree, that is, the Subdominant (G in the first scale and D in the second one). The fact that the notes in a Major scale are connected by Perfect fifths indicates the high degree of consonance among them. In Fig. 7, notice that those enharmonic notes not following the series of P 5th intervals are discarded.

Figure 7. The D Major and A Major scales represented on a half circumference.Furthermore, in this representation it can be clearly seen how the altered notes appear. Thus, in the D Major scale, which has a key signature with 2 sharps, the notes Fand Cappear at the end and following the order of sharps. And, when going from the D Major to the A Major scale, we can see that the note G is replaced by G, as must occur, since the A Major key signature has 3 sharps. This way, it can be observed that the sharps appear by Perfect fifths, beginning with F. And, when augmenting the number of sharps in the key signature, the tonics also appear by Perfect fifths. On the other hand, if we represented the scales containing flats, we would see how the flats appear by Perfect fourths, beginning with B; and that, when augmenting the number of flats, the tonics also appear by Perfect fourths.Learning the scales takes its time. Firstly, the order of sharps and flats must be known. And then, it is necessary to know how the key signatures are related to the tonics. To do this, the following two rules apply and can be easily proved by representing the scales as in Fig. 7:In key signatures containing sharps, the last sharp note is the Leading tone. For example, in the key signature having 3 sharps, they correspond to F, Cand G. Thus, Gis the Leading tone and A is the Tonic. Therefore, this key signature corresponds to the A Major scale.

In key signatures containing flats, the next to last flat note is the Tonic. For example, in the key signature having 5 flats, they correspond to B, E, A, Dand G. Thus, Dis the Tonic and this key signature corresponds to the DMajor scale.

We only have to add to these rules that the key signature having no accidentals corresponds to the C Major scale and that having one flat corresponds to the F Major scale. A musician must be able to play every scale both ascending and descending, by only knowing the tonic and the key signature, finding the notes mentally. Although the Harmonic Wheel allows us to understand the scale construction and to solve any doubt related with them, it cannot replace the necessary practice to fluently play all the scales.It is difficult to exaggerate the importance of Major scales in Music. In fact, the whole musical script system is conceived to easily write this type of scales. Thus, when learning this system, one can realize that, for a given staff with any clef and any key signature, if there are no accidentals, all the written notes necessarily belong to a certain Major scale. On the contrary, the notes out of this scale will have accidentals. This way, the accidentals serve to easily recognize the notes not belonging to the considered Major scale. On the other hand, all the diatonic instruments (some types of flutes and harmonicas, almost all the toy instruments, etc.) only contain the notes of a Major scale.7. MINOR SCALESIf, in any Major scale, we change the tonic but keep all its notes, what we obtain is a new Mode. For example, if in the C Major scale, we choose the note E as the tonic, the resulting scale is: E, F, G, A, B, C, D, E. This is not a Major scale, since it does not keep to the interval succession W W H W W W H, but it is a mode associated to the C Major scale. Therefore, there are 7 possible modes for any Major scale, one for each note in the scale that is considered the tonic. In fact, the name Major is, precisely, the name given to the mode with the interval succession W W H W W W H.Western music has evolved with time to the so called Major-minor System, which means that, from the 7 possible modes, only two of them remained: the Major and the minor modes. The minor mode is that obtained by considering the VI degree in the Major scale as the tonic. So, from the C Major scale, the so called A natural minor scale is obtained:

A natural minor scale, along with its degreesThese two scales, C Major and A natural minor, are said to be relative to one another. In the same way, from the BMajor scale, the G natural minor scale is obtained. And, from the A Major scale, the Fnatural minor scale is obtained.In general, we can define a natural minor scale as a set of 7 notes characterized by the following interval succession: W H W W H W W. Logically, a Major scale and its relative minor have the same key signature. In Chapter 13, the special characteristics of these two modes will be studied, as well as the reason why they prevail over the rest of the modes.Nevertheless, the natural minor scale has an inconvenience: the distance between its VII and VIII degrees is one whole step, while in a Major scale this distance is one half step. So, it turns out that, when playing the natural minor scale and passing from the VII to the VIII degree, it does not produce the sensation of having reached the end of the scale. In Music, it is said that the VII degree of the natural minor scale does not have the Leading tone character and so it is called Subtonic instead. In order to avoid this inconvenience, it is common to raise one half step the VII degree of this scale by means of an accidental, which results in the harmonic minor scale.

A harmonic minor scale, along with its degreesHowever, in the so built harmonic minor scale, an A 2nd interval appears between its VI and VII degrees, that is, 1.5 W, which has a strange and unnatural effect, for it is a too big interval for being between two consecutive degrees. So, sometimes the VI degree is also raised one half step by means of another accidental, thus solving this problem. This results in the melodic minor scale, where the interval between two consecutive degrees is always one whole or one half step.

A melodic minor scale, along with its degreesFinally, since the Leading tone character associated to the VII degree is only needed in the ascending scale, but not in the descending, it is also common to use the melodic minor when ascending the scale and the natural minor when descending the scale. This combination is occasionally known as the classical melodic minor scale, although sometimes it is called, simply, the melodic minor scale, what can lead to confusion. In this context, it will be assumed that the melodic minor scale is the one having the VI and VII degrees raised one half step, both ascending and descending.By means of a similar procedure to that seen for the Major scales, the Harmonic Wheel also allows us to obtain the notes of any minor scale, being it natural, harmonic or melodic. Moreover, if we pay attention to the red notation, we will see that each key signature is next to two notes, one of them followed by the letter m for minor (actually, what the red notation represents are chords, as will be seen in Level 2). These two notes are, precisely, the tonics of the Major and its relative minor scales having this key signature (all of which will be explained in detail in Chapter 12).As examples of this, Fig. 8 shows the following cases: For 2, the tonics of the BMajor and G minor scales; for 2, those of the D Major and B minor; for 3, those of the A Major and Fminor; and, for no accidentals, those of the C Major and A minor scales. This information must be sufficient to play all the Major and minor scales, with all the minor scale variants (natural, harmonic, melodic and classical melodic), by mentally obtaining the notes.

Figure 8. Tonics of the Major and their relative minor scales, along with their corresponding key signatures.

8. MAP OF THE KEYSThe Major or minor character of a scale is named Mode. The fact that Western music has evolved to the so called Major-minor System means that most musical works or passages are composed, basically, by the notes of a certain Major or minor scale.On the other hand, the tonic of the Major or minor scale becomes the tonal center and determines the Key. So, a key represents a tonic and a Major or minor mode. Thus, it is said that certain work or passage is written in the C Major key or in the Fminor key.This way, each Major and minor scale has an associated key with the same name. The difference between the scale and the key is that, in the scale, the notes are sorted out by their pitch, while in the key the notes appear in any order and even several of them can be played at the same time. Traditionally, the keys have been sorted out by means of the so called Cycle of Fifths. Basically, this means that, when passing from one key to another having one more sharp or one less flat, the new tonic is a P 5th above the previous one. In the Harmonic Wheel, the Cycle of Fifths in found along any circular band (Fig. 9).

Figure 9. Cycle of Fifths, along a circular band.The cycle of fifths is also used to determine how close two given keys are. Thus, for example, two relative keys, as D Major and B minor, are very close to each other. In the same way, those keys which key signatures only differ in one accidental are also close; for example, F Major and C Major, or F Major and BMajor.Nevertheless, there is a particular case of close keys that is not visible in the cycle of fifths. It is the one corresponding to the Change of Mode. It is well known that keys such as C Major and C minor, or E minor and E Major, etc., are very close to each other. It is also said that they are Parallel Keys. However, this fact is not seen in the cycle of fifths, since their key signatures differ in 3 accidentals. On the contrary, in the Harmonic Wheel the proximity between parallel keys is clearly shown, since they are one next to the other in the radial direction (Fig. 10). Moreover, because each radius begins and ends with the same note, when an end is reached, it is possible to continue from the other one, thus closing another cycle in this direction.

Figure 10. Cyclic Changes of Mode, in the radial direction.So, while the cycle of fifths shows the keys on a line (that is, in one dimension), the Harmonic Wheel shows them on a surface (that is, in two dimensions). Therefore, the Harmonic Wheel provides a complete and panoramic view of all the existing relationships among the Keys. This is, as a matter of fact, one the most important characteristics of the Harmonic Wheel: the so called MAP OF THE KEYS.